Shock-wave distortion cancellation using numerical recalculated intensity propagation phase holography

Highlights

• Shock-waves and index gradients distort quantitative holographic measurements.

• RIPPH is a practical numerical technique for phase distortion cancellation.

• Simulations and experiments demonstrate successful distortion cancellation.

• RIPPH cancel distortions in supersonic and explosive shock-wave environments.

• RIPPH shows benefits over DIH, PCDIH, electric field, and iterative phase methods.

Abstract

Digital holography is a three-dimensional (3D) measurement technique that can be used to quantitatively determine the size and 3D location of the objects inside a field-of-view. However, in systems where refractive index gradients are present, variations in optical phase due to high-speed shock-waves or low-speed thermal gradients can cause distortions that obscure objects. While techniques like phase-conjugate digital in-line holography and iterative phase measurement techniques have been developed in the past for phase removal or phase measurement, they require either nonlinear four-wave-mixing or iterative algorithms to operate. In this paper, we demonstrate a novel recalculated intensity propagation phase holography (RIPPH) method that captures distorted holograms using low-power continuous lasers, refocuses the hologram to the plane of the distortion, and cancels the phase distortion numerically in a single post-processing step. The resulting hologram can be numerically refocused to provide distortion-free 3D information describing objects that absorb or scatter light. In RIPPH, only the approximate $z$-locations of the phase distortions are needed, making this method significantly faster to compute than phase retrieval methods. Theoretical simulations are first used to describe and assess the distortion removal process. Experiments are then conducted to demonstrate at least $3\times$ lower edge distortion for RIPPH compared to traditional digital in-line holography. We demonstrate, for the first time, how phase distortions from supersonic air jets, particle-laden spherically expanding shock-waves, and convectively-driven thermal gradients are numerically cancelled and show how objects of interest are accurately recovered.

Introduction

Digital holography is a non-intrusive optical technique that enables three-dimensional (3D) analysis of a wide range of systems. This is accomplished by numerically refocusing diffraction patterns generated by objects that absorb, refract, or delay light propagation [1], [2], [3], [4]. Digital in-line holography (DIH) has been successfully applied in the study of several multi-phase flow systems [3] including liquid breakup systems [5], [6], [7], aluminum particle combustion in solid propellants [8], [9], [10], and velocimetry in complex flows [11], [12], [13].

As experimental conditions become more extreme, shock-waves [14], [15], flames [16], [17], high-speed flows [18], and other phenomena that generate density gradients produce optical distortions in digital in-line holograms, as demonstrated in Fig. 1(a). These distortions make locating the true edge of an absorbing feature more difficult, greatly limiting the ability to characterize quantities such as particle sizes and positions. One new experimental technique, phase conjugate digital in-line holography (PCDIH) [19], [20], [21], [22], was recently developed to remove a large portion of the phase-distortions associated with shock-waves. PCDIH passes a laser beam over a region of interest containing both objects and distortions. Then, the light is passed into a phase conjugate mirror, which inverts the phase [23]. When the reflected beam passes through the distortion a second time, the forward and inverted phases cancel to remove distortions. While PCDIH is an effective in-situ method that has been demonstrated at up to 5 million-frames-per-second [22], it does have several drawbacks. First, the technique requires nonlinear four-wave mixing to generate the phase conjugate signals. This requires high laser pulse energies, scales poorly at ultra-high-speeds, and double-passes light through the object and distortion field [22]. By double-passing the light through the phase distortion, additional imaging artifacts due to object or shock-wave motion during the laser light time-of-flight are generated. Outside of PCDIH, there are currently no alternative techniques or examples in the literature that demonstrate shock-wave distortion removal for particle tracking applications.

Phase measurement techniques, on the other hand, have been explored in other microscopy-related application areas. This includes electric field propagation (EFP) techniques such as two-step phase measurement [24], [25], [26] and parallel phase shifting [27], [28], [29], [30], [31], [32], [33]. In these techniques, the amplitude and phase information is separated using iterative numerical post-processing. This includes phase estimation using trained neural networks [34] and iterative phase retrieval (IPR) [35] methods. Because phase measurement techniques focus on accurately determining phase, the algorithms can be more computationally intensive, requiring iterative processing and the application of phase unwrapping algorithms [36].

Rather than cancelling the phase distortions in-situ or iteratively determining phase, we propose removal of the phase distortion contributions numerically in a single post-processing step. Numerical phase distortion cancellation has not been previously explored in the literature. Additionally, electric field hologram, phase measurement, and numerical distortion cancellation methods have not been applied previously in the literature for tracking particles in high speed flows. To remove shock-wave distortions, we propose an innovative recalculated intensity propagation phase holography (RIP phase holography or RIPPH) method for numerical phase distortion cancellation. The first step of the technique is to capture the electric field of the hologram using a single camera and exposure. Then, the electric field is propagated to the center of the phase distortion source. At this point, the intensity is calculated to generate a RIPPH image, effectively removing phase contributions in a simple, elegant, and computationally efficient fashion. Further propagation of the intensity hologram to other planes, as illustrated in Fig. 1(b), show that phase distortions are successfully cancelled.

The RIPPH technique has several important advantages over existing methods like DIH, EFP, PCDIH and IPR. Traditional EFP and DIH experiments do not implement phase cancellation and cannot be used quantitatively in distorted environments with both amplitude and phase objects. Iterative phase retrieval methods require repetitive processing to separate phase objects from amplitude objects. With RIPPH, no quantitative phase information is needed; therefore the numerical correction algorithms are significantly simpler, computationally efficient, and more practical to apply than IPR methods. Only the approximate $z$-location of the distortion is required and phase distortions are automatically removed during intensity calculation. After correction with RIPPH, quantitative image segmentation can be completed with the same algorithms as traditional DIH techniques. Because scattering processes are linear, RIPPH does not require high power lasers, nonlinear optical methods, or four-wave mixing. Therefore RIPPH is significantly simpler to experimentally implement than PCDIH and scales well to higher repetition rates. In addition, light only needs to pass once through the object field and imaging artifacts associated laser light time-of-flight effects no longer need to be considered.

In this paper, we explore the RIPPH technique in several different scenarios where refractive index gradients cause optical distortions. To evaluate the effectiveness of the technique, the theory behind RIPPH is first discussed followed by simulation and numerical distortion cancellation methodologies. RIPPH is then compared to DIH, EFP, and IPR techniques. Several different experimental systems are then studied using both DIH and RIPPH, ranging from low Reynolds number flows driven by thermal gradients to extreme supersonic and explosive environments. Results illustrate how RIPPH can be successfully applied in several systems with phase distortions that were previously difficult to study using DIH. In this paper, we demonstrate successful numerical phase distortion cancellation for the first time for digital holograms. This work is also the first to apply the technique to environments with shock-waves, blast-waves, and density gradients, which are of increasing importance to fields ranging from fluid dynamics to hypersonics to energetic materials.