Design of a digital holographic interferometer for the M. P. Ross, U. Shumlak, R. P. Golingo, B. A. Nelson, S. D. Knecht, M. C. Hughes, R. J. Oberto University of Washington, Seattle, USA
Abstract The experiment at the University of Washington investigates the use of sheared axial flow to stabilize a Z-pinch plasma. A variety of diagnostics characterize the plasma. In the past, holographic interferometry (HI) has measured radial electron density profiles at a fixed axial location. As the ZaP experiment prepares to re-focus itself as a high energy density plasma experiment, a redesign of the HI system is underway. The existing HI system relies on holographic film to record interferograms to measure density profiles. The use of film mandates time-consuming development and reconstruction procedures. The new digital HI system will employ a complementary metal-oxide-semiconductor (CMOS) camera and numerical reconstruction techniques to eliminate physical postprocessing, providing operators with electron density data between each plasma pulse. Additionally, a new traverser system will allow operators to easily change the axial measurement location.
Electron density induces phase shift A plasma s refractive index is proportional to electron density. Two identical laser beams travelling the same distance where one passes through a plasma and one passes through air will exit their respective media phase shifted relative to each other by ΔΦ. When the n c >>n e, the phase shift is given by n c = plasma critical density n e = electron density λ = laser beam wavelength dl = laser beam path Measurements of the phase shift are essentially measurements of line-integrated electron density, which can be used to approximate the radial profile of actual electron density.
Holograms record phase, intensity Holography requires a coherent light source because interference patterns are recorded. We will employ a Korad ruby laser. Recording: At the hologram plane, we record the interference pattern of a reference beam and a scene beam effected by a plasma positioned between the hologram and object planes. Reconstruction: Using the recorded interference pattern as an aperture at the hologram plane and shining the reference beam through the pattern, real and virtual images of the reconstructed hologram will form in the image and object planes respectively. Holograms store intensity and phase: Because holograms store phase information, reconstructing the holograms allows us to extract the phase shift induced by the plasma.
Correct object plane location allows phase measurement Positioning the Z-pinch between the object and hologram planes allows us to measure the full phase shift caused by the plasma.
Off-axis holography measures phase shift Past phase shift measurements were taken with an off-axis holographic interferometer as shown below. This interferometer combined the scene and reference beams on holographic film at a 15 degree angle. S. Jackson, Ph.D. dissertation, University of Washington, 2006.
Off-axis technique relies on holographic film Baseline shot and plasma shot were recorded on a single holographic film. Including baseline shot allows the removal of any effects of windows or optics consistent between the shots. Reconstructed holograms were photographed; the phase shift distributions were measured by determining the fringe shifts. One such photograph is shown to the right; this reconstruction shows a peaked phase shift distribution with gradients of phase in the vertical direction. S. Jackson, Ph.D. dissertation, University of Washington, 2006.
Abel-inversion provides approximation of radial profile Chord-integrated density can be determined directly from the phase shift. Abel-inversion can compute radial profile of number density from the chord-integrated density. The results below are determined from the hologram above. S. Jackson, Ph.D. dissertation, University of Washington, 2006.
Replacing film with digital camera is advantageous Recording interferograms with a digital camera allows for faster data collection and post processing. Only one baseline shot is needed for multiple plasma shots. No film/plate development requiring wet chemistry is necessary. No physical reconstruction set-up required. Numerical reconstruction techniques acquire phase distributions without having to measure fringe shifts.
Using digital camera requires on-axis holography Using a digital camera compels the use of on-axis holography as opposed to off-axis holography. The resolution limitations of digital CCD/CMOS sensors prevent them from measuring the tiny fringe patterns created in off-axis holography interferograms. The fringe spacing, h, of the interference pattern of two incident wave fronts is given by the following in which θ is the angle between the wave fronts and λ is the wavelength. On-axis holography seeks to increase the fringe size, d, by aligning the reference and scene beams before they strike the sensor.
On-axis holography will measure phase shift The on-axis interferometer requires one more beamsplitter than the off-axis diagnostic because it must realign the two laser beams. A single plano-convex lens will resize both the scene and reference beam to form the interferogram on the camera s CMOS sensor.
Preliminary set-up will observe candle flame Preliminary on-axis holographic interferometer set up will allow the instrument to be tested with a candle flame instead of plasma before the diagnostic is moved into the experiment.
Camera / Laser parameters laser head power supply cooler Q-switch shutter wavelength specified pulse width Korad K-1 Ruby Laser Korad K-1 Ruby Laser Head Korad K-1 Power Supply (5 kv) Korad KWC Laser Cooler Korad K-QS2 Pockels cell Q-switch assembly Korad K-QS2 Pockels cell shutter Electronics 694.3 nm < 15 ns Canon Rebel T2i (D550) Image sensor size: 22.3 x 14.9 mm Sensor type: CMOS Effective Pixels: 18 MP Pixel size (linear dimension): ~4.3 microns operating pulse width specified pulse energy operating pulse energy polarization 50 ns 1.1 J 600 mj horizontal Pixel spacing: negligible thanks to gapless micro lenses between pixels Image type: JPEG (8-bit), RAW (14-bit Canon original)
Alignment necessary to record interferograms The minimum resolvable spatial frequency of the CMOS is half its sampling frequency. Thus, if pixel size is Δx, The reference and scene beams must be aligned within angle θ max to ensure all generated fringes are detectable. Bayer array of filters used on CMOS to detect red, green, and blue light intensities. In a Bayer array, each 2 x 2 square of pixels contains one red, one blue, and two green filters. Doubling Δx during this analysis accounts for this filter arrangement. θ max 2.3 degrees. Alignment within 2.3 degrees ensures the CMOS will detect all fringes, but that does not ensure it can effectively resolve the densities we are trying to measure.
Better alignment improves resolution The closer we can align our scene and reference beams, the better the resolution we can achieve because better alignment increases the size of the fringes. Alignment within 0.1 degrees should be attainable due to over 1 meter of space available to extend alignment beams. Considering the CMOS spatial resolution as the limit on our resolving power, the minimum resolvable line-integrated density 7.6x10 19 m -2. Assuming a uniform pinch with radius 0.01m, this equates to a number density of 3.8x10 21 m -3. Should allow for detailed resolution of ZaP experiment s present electron density profile. Past line-integrated densities 10 21-10 22. Maximum measureable line-integrated density 1.2x10 23.
Numerical reconstruction can be treated as diffraction Physical reconstruction is not possible with digitally recorded interferograms. Instead, by modeling each interferogram as an aperture for a diffraction process, we can numerically reconstruct holograms. Huygen s Principle: Each point in a light wavefront is the source of a secondary spherical wave. Superposition Principle: Because the wave equation for electromagnetic waves is linear, all the secondary waves superpose to form the observed intensity distribution. Illustrated by Young interferometer (right).
Diffraction model obtains phase distribution The diffraction process is a convolution equivalent to summing all the secondary waves emitted at all the points in the aperture. T. Kreis, Handbook of Holographic Interferometry (Wiley, Germany 2005). This equation is called the Fresnel-Kirchoff integral, and the resulting complex amplitude Γ can be used to find the hologram s intensity and phase distributions in the object plane.
Notes on the Fresnel-Kirchoff integral Point spread function, U(x,y): the quatitative representation of the light passing through the aperture; the product of the hologram function, h(x,y), and the conjugate reference wave, E R* (x,y). Hologram function, h(x,y): describes the transmittance of the recorded interferogram. For our purposes of determining phase shifts, the hologram function is simply the intensity of the recorded interferograms obtained from the DSLR camera. Conjugate reference wave, E * R (x,y): complex conjugate of the mathematical representation of the reference wave. The exponential term: the source function of a spherical Huygen s wavelet.
Convolution method eases numeric reconstruction The Fresnel-Kirchoff integral is a convolution of the point spread function and a source function of a spherical wave. Including the specifics of the point spread function, let s use the convolution theorem to rewrite the integral in terms of Fourier transforms. In this form, g is the spherical wave source function. This form allows for easy numerical implementation because readily available and efficient FFT algorithms can take the Fourier transforms and inverse transforms.
Phase shift determination used in code The full process of determining the phase shift by numerical reconstruction is outlined below. Record interferograms Compute complex amplitudes Baseline shot: h 1 (x,y) Plasma shot: h 2 (x,y) Compute Phase Distributions Subtract phase distributions to find phase shift
Code compares favorably to other algorithms Tested validity of code against the performance of algorithm by S. Grilli et al. Generated synthetic interferograms to simulate the real-life case shown below, supplied reconstruction results. Can see how changing the reconstruction (object) plane location changes the reconstruction. / S. Grilli et al., Optics Express, Vol. 9 Issue 6, 299 (2001)
Synthetic inteferograms used for code validation Concentric circular interference fringes mimic the expected pattern caused by light from the lens. These synthetic interferograms closely match those actually recorded by Grilli et al with a CCD. S. Grilli et al., Optics Express, Vol. 9 Issue 6, 300 (2001)
Intensity varies with reconstruction location d = 0.18 m Moving the reconstruction (object) plane to the focal plane of the lens shrinks the intensity down to a point. Illustrates holography s ability to record and recreate depth of field information. d = 0.25 m S. Grilli et al., Optics Express, Vol. 9 Issue 6, 300 (2001)
Reconstructed phase distributions match Phase distributions wrapped in [-π, π ] interval appear identical. These distributions represent the phase difference between the scene and reference beam at points in the chosen object plane. d = 0.18 m S. Grilli et al., Optics Express, Vol. 9 Issue 6, 301(2001)
Summary Digital holographic interferometry provides a simple alternative to using holographic film / plates. Resolution should be sufficient to provide detailed measurements of the ZaP flow Z-pinch even before its upgrade to a high energy density experiment. Preliminary set up of the digital holographic interferometer will allow for testing with a candle flame. Development of a numerical reconstruction code is currently underway and initial validation tests are promising.