1. Introduction

Corona discharges are low-current electrical discharges between two electrodes with a high-voltage gradient. Since practical uses of such discharges extend from industry to ecology, intensive research in this area focuses on the distributions of the electric field and charge density between the two electrodes. In recent studies, the distribution of charge density was estimated by numerical modeling [1,2]. Other investigations on the corona discharge effects such as ionic wind [3], or the drag [4], or the decomposition of volatile organic compounds [5], were also recently reported. Among numerous corona discharge applications, we emphasize the development of a radon trap device [6] and the synthesis of carbon nanotubes [7].

For measuring the optical density of the corona discharge, interferometric techniques are well suited. However, in most cases only small refraction index gradients occur and thus the sensitivity of classical interferometers is insufficient for registering the corona discharge density distribution. In classical holographic interferometry, wavefront changes are recorded and reconstructed optically: in such cases, the measurement sensitivity can be increased or reduced through various experimental approaches such as multipass setup, superposition of different diffraction orders, two-wavelength interferometry, nonlinear effects, or resonance interferometry [8]. On the other hand, the fringes may also be manipulated post-recording by various digital image manipulations like multiplication, subtraction, or squared modulus. For example, by multiplying a fringe pattern with its negative or by performing a squaring operation, fringes can be doubled or multiplied [9]. Same result is obtained by shifting a fringe pattern, subtracting it from the original one, and then taking the absolute values of the subtracted image [10].

Numerical procedures are especially useful in digital holography (DH). For example, the subtraction of two stochastically changed speckled digital holograms (primary fringe patterns) will suppress the zero-order disturbance in the off-axis DH [11]. In digital holographic microscopy, it was shown that by computing the numerical parametric lenses with the automatic fitting procedures can suppress the phase aberrations and image distortion [12]. In time-averaged DH, movies are synthesized to demonstrate a dynamic characterization of musical instruments [13] and mode beating between two distant vibration frequencies [14]. Wavefront changes are evaluated numerically in digital holographic interferometry systems [15–17]. Higher sensitivity was demonstrated experimentally by the use of a shorter wavelength [18]. Furthermore, a numerical method for adjustable contouring sensitivity using digital holography was proposed [19]. The method relies on the interference of the object complex data and a virtual reference wave with arbitrary curvature in the object reconstruction stage. The sensitivity limitations in time-averaged DH are also discussed recently [20].

In this work, a new sensitivity enhancement method is proposed that can be used in digital holographic interferometry. This method is described theoretically in Sec. 2. Experimental details are presented in Sec. 3 and the results for an opaque object and a DC corona discharge are given in Sec. 4.

2. Theory

2.1 Summary of Digital holography principles

Consider a quasi-Fourier setup with an off-axis reference point source and an object (opaque or transparent), both defining the input plane. A CCD sensor is located at a distance d from the input plane. The coordinates in the input and detector (hologram) planes are denoted by (x, y) and (u, v), respectively. If the input plane is described as

where the point source is located at (X,Y) and s(x,y)=|s(x,y)|exp[iφ(x,y)] is the object wavefront, then the diffracted field at the detector plane is expressed according to the Fresnel approximation by

where F denotes the Fourier-transform operator, ζ(x,y)=(π/λd)(x ²+y ²), λ is the wavelength, and where constant terms are omitted. The exposure E(u,v) (interference primary fringes) captured by an array photo-detector represent a digital hologram. Such hologram can be easily turned to the reconstruction image by performing an inverse Fourier transform to obtain an image composed of the zero-order term plus two identical object reconstructions located symmetrically around the DC point. This reconstruction procedure can be described by

To summarize in a simplified notation, one exposure term,

leads to the object reconstruction intensity (x,y)=I _s(x,y) where I_s(x,y)∝|s(x,y)|².

2.2 Digital holographic interferometry

A phase change of the object wavefront s′(x,y)=|s(x,y)exp[iϕ(x,y)+iΔϕ(x,y)] leads to the exposure term

Standard holographic interferometry fringes are obtained by performing the reconstruction procedure on the sum of two exposures E(u,v) and E′(u,v), i. e.

where C ₀+ is a real-valued constant describing the overall image brightness. I ₀+(x,y) describes the image of the object modulated by the fringes with the standard sensitivity.

In separated form, the intensity of the object wavefront, I_s(x, y) and the corresponding phase shift, Δϕ(x, y), can be found by

Equation (6) as well as the Eqs. (7) and (8) contain the essential interferometric information which, however, could not be properly visualized if phase difference is less than π (see Fig. 8, middle).

2.3 Sensitivity enhancement

The sensitivity can be enhanced by the following procedure:

(i) Subtracting the two fringe patterns E(u,v) and E′(u,v) and performing the reconstruction procedure to get

where C ₀ and C _0- are real-valued constants, C _0- originate similarly as C ₀₊ and C ₀=C ₀₊/C _0- is used to equalize the total intensity of the signal window.

(ii) Subtracting the Eqs. (6) and (9), followed by the squaring operation to get

$I_{1+}(x,y)={[I_{0+}(x,y)-I_{0-}(x,y)]}^2$

$=I_s^2(x,y)\{\begin{array}{c}{\left(C_{0+}-C_0{C}_{0-}\right)}^2+2\left(C_{0+}^2-C_0^2{C}_{0-}^2\right)\cos \left[\Delta \phi (x,y)\right]\\ +(1⁄2){\left(C_{0+}+C_0{C}_{0-}\right)}^2\left(1+\cos \left[2\Delta \phi (x,y)\right]\right)\end{array}\}.$

Since C ₀ C _0-=C ₀₊, it follows

(iii) Multiplying the Eqs. (6) and (9), yielding

where C ₁ and C _1- are real-valued constants, defined by analogue to the constants C ₀ and C _0-.

(iv) Subtracting the Eqs. (10) and (11), followed by the squaring operation to get

The previous procedure increases the sensitivity by two times [compare I ₁₊(x,y) against I ₀₊(x,y)] and four times [compare I ₂₊(x,y) against I ₀₊(x,y)]. Following the same pattern, a general relation can be obtained:

where the numerical increase in sensitivity, equal to 2^k, is limited mostly by the speckle noise which reduce the fringe contrast.

2.4 Refractive index variations: corona discharge

For a phase object, the phase difference between the waves passing through (in the z-direction) in its two density states can be described by Δφ(x,y)=(2π/λ)ΔL(x,y). The optical path difference ΔL(x,y) is generally described as a line integral of the refractive index variation ΔL(x,y)=∫Δn(x,y,z)dz, where Δn(x,y,z)=[n(x,y,z)-n ₀] and the refractive indexes, initial and changed, are denoted by n ₀ and n(x,y,z), respectively. Once obtained, Δn(x,y,z) can be further used for calculating the density distribution.

We consider the point-plane configuration of the corona discharge system with the arrangements of the electrodes along the y axis. For radially symmetric case and if only macroscopic and static properties are of interest, we can use the Abel transform approach. Thus, for every cross-section defined by the plane (Z,X) at the value y, y∈(0,D), where D denotes the distance between the electrodes, the path difference satisfies the relation:

in which the integral is the known Abel transform of the distribution Δn(r,y). Clearly, the values of ΔL(x,y) are different from zero in the region between two electrodes, the region being narrow at the point electrode and wide at the plane electrode. Once the distribution of the ΔL(x,y) is experimentally determined, the distribution of the refractive index variation Δn(r,y) can be calculated by applying the inversion methods to solve the Abel transform.

The real-valued distribution of fringe orders R_k(x, y) can be easily extracted from Eq. (13), i.e. R_k(x, y)=2^kΔL(x,y)/λ, from where follows:

Equation (15) clearly shows that, for a given wavelength, the same optical path difference can be calculated from the experimental data with different iteration parameter k. For physical systems with low optical densities, i.e. ΔL(x,y)<λ, the detection sensitivity for ΔL(x,y) can be enhanced by increasing the parameter k. In other words, an appearance of fringes in the cases of sub-wavelength optical path differences can significantly improve the quality of the experimental data.

3. Experimental setup

The experimental setup is schematically shown in Fig. 1. We used a krypton ion laser (647 nm) as light source and a CCD sensor (Kodak Megaplus, 1008×1018 9 µm square pixels) as array photo-detector. A ground glass plate is used to diffusely scatter the beam of the phase object. The corona discharge is obtained in a point-plane configuration of electrodes (Fig. 2) in which the spacing between electrodes was set to 10 mm. High voltage was applied by an adjustable DC power supply (0-10 kV, 0.075-10 mA).

The laser beam is first split into an object and a reference beam. These beams are then steered and expanded by optical elements to finally recombine and form an interference pattern at the detector plane. The object beam is passing through the corona discharge and the ground glass, while the point source is formed by focusing the reference beam onto the plane of the ground glass. The intensity ratio between the reference and object beam is adjusted to 3:1 using neutral density filters.

 

Fig. 1. Experimental setup for recording digital holograms of the corona discharge.

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Fig. 2. Scheme of the corona discharge configuration.

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4. Results

To demonstrate the sensitivity enhancement, the setup shown in Fig. 1 was modified appropriately to illuminate an opaque reflective object (a piece of a steel ruler) instead the corona discharge. Holograms are recorded for two deformation states of the ruler. The basic information was calculated according to Eqs. (7) and (8), see Fig. 3. Performing the proposed procedure, the results for double enhancing sensitivity are given in Fig. 4, where (a) and (b) show the interferometric fringes and (c) and (d) their profiles (averaged over 100 horizontal lines). By comparing profiles, it appears that the sensitivity increase is accompanied with the lowering of the modulation depth of the fringes, as noted at end of Sec. 2.3. Figure 5 demonstrates four times increase in sensitivity for a very weak deformation. In Figs. (4) and (5) only one object reconstruction is shown.

 

Fig. 3. Basic information: the intensity of the reconstructed wavefront (left) and the phase shift (right).

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Fig. 4. Sensitivity enhancement: (a) the standard interferometric fringes of a deformation, (b) the same deformation with double sensitivity, (c) the profile of (a), and (d) the profile of (b).

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Fig. 5. Sensitivity enhancement in the case of a very weak deformation: the reconstructions (left) and profiles (right). (a) the undeformed object, (b) the sub-fringe deformation with the standard holographic sensitivity, (c) the same deformation with double sensitivity, and (d) the same deformation with four times increased sensitivity.

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The results for the corona discharge are shown below. The first example (Fig. 6) is with the strongest current we applied (10 mA). The second shows the case of a scarcely visible discharge (130 µA) which becomes clearly visible by double increase in sensitivity (Fig. 7). The third example (75 µA) demonstrates most clearly the usefulness of the proposed method. The initial interferogram does not indicate the existence of any fringe (Fig. 8, middle), as if there was no corona discharge. However, by applying our method, the fringes start appearing (Fig. 8, right).

 

Fig. 6. Sensitivity enhancement for the corona discharge, 10 kV, 10 mA. Left: the reconstruction without corona, middle: standard interferometric fringes of a deformation, right: the same deformation with double sensitivity.

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Fig. 7. As in Fig. 6, but for 130 µA.

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Fig. 8. As in Fig. 6, but for 75 µA.

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5. Summary

Using the procedure described in this paper, sensitivity of digital holographic interferometry can be increased and in turn experimental measurements achieved with higher accuracy. The procedure has an iterative character in which every step doubles the sensitivity. It is thus especially useful in applications with sub-wavelength wavefront deformations. Once the digital holograms are acquired, the proposed procedure is easy realizable numerically as it requires performing only common image processing.

The procedure is described theoretically and demonstrated experimentally. In the first experiment, the method was applied to a deformation of an opaque object. In the second experiment, a corona discharge was detected even with an extremely low electric current. Thus, we have demonstrated the effective use of information whose existence was not obvious when using standard interferometric procedures.