Experimental observation of aberration cancellation in entangled two-photon beams

Luísa. A. P. Filpi; M. V. da Cunha Pereira; C. H. Monken

doi:10.1364/OE.23.003841

1. Introduction

Optical imaging systems may cause undesirable distortions on the wavefront of light, the so called aberrations. Aberrations may arise, for example, from the spherical shape of a lens (spherical aberrations) which causes non-paraxial rays to focus closer to the lens than paraxial ones. Oblique incidence may cause light to focus at different positions on a plane transverse to the optical axis (coma) or at two separate planes (astigmatism) [1].

Traditionally, aberrations have been corrected, or at least minimized, through the use of aspheric lenses, aperture stops or lenses associations, as in the case of eyeglasses. More complex imaging systems employ the techniques of adaptive optics, which consists of using deformable mirrors to compensate for known aberrations. This technique is used in astronomy to correct distortions induced by atmospheric turbulence [2] and in microscopy [3] to compensate for both optical system and sample-induced aberrations.

Since the mid 1990s, the strong correlations present in light generated via spontaneous parametric down-conversion (SPDC) allowed for new techniques of image formation, such as ghost imaging [4, 5] and correlation-beam imaging [6, 7, 8]. In both cases, the image of an object is hidden in the spatial fourth-order (photon-photon) correlations of the down-converted field, while the object is illuminated either by the down-converted field itself (ghost imaging) or by the laser that pumps the nonlinear SPDC crystal (correlation-beam imaging). More recently, spatial correlation effects in SPDC light have been exploited to produce dispersion and aberration cancellation effects: [9, 10, 11, 12, 13]. Later, it was shown that the dispersion cancellation could be achieved using classical light [14, 15, 16, 17]. In [12], the authors also presented a proposal to achieve aberration cancellation using classical light.

In this work we experimentally demonstrate the cancellation of odd-order aberrations in a correlation-beam imaging system without the use of deformable mirrors. The aberrations are induced in a controllable way by a spatial light modulator on entangled photons generated via SPDC. Our method of cancellation relies on a coordinate inversion of one of the photons, as explored on a recent paper describing the cancellation of atmospheric turbulence effects [18]. The conditions under which the effect takes place are also discussed.

2. Theory

Let us consider a simple optical system as represented in Fig. 1. A two-photon light source is provided by an SPDC nonlinear crystal pumped by a laser whose transverse field profile is the optical Fourier transform of an object placed on the focal plane of the lens L₁ (object plane). The exit pupil is a spatial light modulator (SLM) playing the role of a concave mirror. The system is designed such that the lens L₂ creates an image of the SPDC source on the SLM plane. Due to the transfer of angular spectrum from from the pump beam to the SPDC two-photon state [6], the image of the object is recovered in coincidence detection at the focal plane of the SLM (detection plane). Convenient coordinate transformations are performed inside the dotted rectangle, as described below.

Fig. 1 Basic geometry. A simple optical system is composed by a two-photon light source provided by spontaneous parametric down-conversion (SPDC) in a nonlinear crystal pumped by a laser whose transverse profile is the Fourier transform of an object on the focal plane of the lens L₁. The exit pupil is a spatial light modulator (SLM) playing the role of a concave mirror. The system is designed such that the lens L₂ creates an image of the SPDC source on the SLM plane, in order to maximize the spatial correlation of the twin photons on the exit pupil. The image of the object is recovered by fourth-order (coincidence) detection at the focal plane of the SLM. Convenient coordinate transformations described in the text are performed inside the dotted rectangle.

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The two-photon state generated by SPDC, neglecting birefringence effects introduced by the nonlinear crystal centered at the origin of coordinates, in the monochromatic and paraxial approximations can be written as [19]

| SPDC 〉 \propto \int d^{2} q_{s} \int d^{2} q_{i} ℰ (q_{s} + q_{i}) \sin c (\frac{τ}{4 k} | q_{s} - q_{i} |^{2}) | q_{s}, σ_{s} 〉 | q_{i}, σ_{i} 〉,

where q_s and q_i are the transverse wave vector components of the down-converted photons, signal and idler, σ stands for the polarization, ℰ is the pump beam plane wave spectrum, τ is the crystal thickness and k is the pump wavenumber. This state gives rise to a two-photon probability amplitude for detection at points r_s = (ρ_s,z), r_i = (ρ_i,z)

\begin{array}{l} A (r_{s}, r_{i}) \propto \int d^{2} ρ_{s}^{″} \int d^{2} ρ_{i}^{″} U (\frac{ρ_{s}^{″} + ρ_{i}^{″}}{2}) V (\frac{ρ_{s}^{″} - ρ_{i}^{″}}{2}) \\ \times \exp [\frac{i k}{4 z} (| ρ_{s} - ρ_{s}^{″} |^{2} + | ρ_{i} - ρ_{i}^{″} |^{2})], \end{array}

where U is the pump field transverse profile at the central plane of the crystal (z = 0) and

V \propto 1 - (2 / π) Si (k | ρ_{s}^{″} - ρ_{i}^{″} |^{2} / 4 τ)

is the Fourier transform of sinc(τ|q_s−q_i|²/4k), with Si representing the sine integral function. The exponential term is the Fresnel propagator, which accounts for free propagation of the two-photon beam through a distance z from the crystal. The integrals are performed on the z = 0 plane.

Consider that the z = 0 plane of the crystal is imaged on the active surface of a spatial light modulator (SLM), which introduces a phase modulation ϕ(ρ) in each photon of the pair. In this case, Eq. (2) takes on the form:

\begin{array}{l} A (r_{s}, r_{i}) \propto \int d^{2} ρ_{s}^{'} \int d^{2} ρ_{i}^{'} U^{(i)} (\frac{ρ_{s}^{'} + ρ_{i}^{'}}{2}) V^{(i)} (\frac{ρ_{s}^{'} - ρ_{i}^{'}}{2}) \\ \times \exp [i ϕ (ρ_{s}^{'}) + i ϕ (ρ_{i}^{'}) + \frac{i k}{4 z^{'}} (| ρ_{s} - ρ_{s}^{'} |^{2} + | ρ_{i} - ρ_{s}^{'} |^{2})], \end{array}

where U⁽ⁱ⁾ and V⁽ⁱ⁾ are the images of the U and V and z′ is the distance from the SLM to the detection plane. The integrals are now performed on the SLM plane.

Now, assume that the transverse coordinates of just one of the photons of each pair (the idler, say) are inverted before it reaches the SLM. The amplitude $U^{(i)} (ρ_{s}^{'}, ρ_{i}^{'}) V^{(i)} (ρ_{s}^{'}, ρ_{i}^{'})$ will be replaced by $U^{(i)} (ρ_{s}^{'}, - ρ_{i}^{'}) V^{(i)} (ρ_{s}^{'}, - ρ_{i}^{'})$ , that is, the photo-detection probability amplitude can now be written as:

\begin{array}{l} A (r_{s}, r_{i}) \propto \int d^{2} ρ_{s}^{'} \int d^{2} ρ_{i}^{'} U^{(i)} (\frac{ρ_{s}^{'} - ρ_{i}^{'}}{2}) V^{(i)} (\frac{ρ_{s}^{'} + ρ_{i}^{'}}{2}) \\ \times \exp [i ϕ (ρ_{s}^{'}) + i ϕ (ρ_{i}^{'}) + \frac{i k}{4 z^{'}} (| ρ_{s} - ρ_{s}^{'} |^{2} + | ρ_{i} - ρ_{s}^{'} |^{2})] . \end{array}

If the phase ϕ(ρ′) is set to ϕ(ρ′) = −k|ρ′|²/(4f), the SLM will behave as a concave mirror, creating a fourth-order (photon coincidence) image of the object on the plane z′ = f. This fact can be easily seen if one makes R = (ρ_s − ρ_i)/2 and S = (ρ_s + ρ_i)/2:

A (R, S) \propto [e^{i \frac{k}{2 f} S^{2}} \int d^{2} S^{'} V^{(i)} (S') e^{- \frac{i k}{f} S' \cdot S}] [e^{i \frac{k}{2 f} R^{2}} \int d^{2} R^{'} U^{(i)} (R') e^{- \frac{i k}{f} R' \cdot R}] .

For values of the crystal thickness of the order of a few millimeters, the function V ∝ 1 $(2 / π) Si (k | ρ_{s}^{″} - ρ_{i}^{″} |^{2} / 4 τ)$ is very narrow. So it is reasonable to approximate V⁽ⁱ⁾(S′) by a delta function δ²(S′) in Eq. (5). Within this approximation, one can disregard the coordinate S and the amplitude A is determined by the Fourier transform of U⁽ⁱ⁾. Reminding that U⁽ⁱ⁾ is a two-photon field that maps the Fourier transform of the object, then, apart from a quadratic phase factor, the two-photon detection probability amplitude A(R) maps the object field at the focal plane of the SLM.

If aberrations occur, they will be accounted for as an aberration function ψ(ρ′) added to ϕ(ρ′), that is, ϕ(ρ′) → −k|ρ′|²/(4f) + ψ(ρ′). Now, Eq. (4) is no longer separable in R and S as in Eq. (5) and takes the form

\begin{array}{l} A (R, S) \propto e^{i \frac{k}{2 f} (S^{2} + R^{2})} \int d^{2} S^{'} \int d^{2} R^{'} V^{(i)} (S') U^{(i)} (R') \\ \times \exp [i ψ (S' + R') + i ψ (S' - R') - \frac{i k}{f} (S' \cdot S + R' \cdot R)] . \end{array}

In the limit V⁽ⁱ⁾(S′) → δ²(S′), Eq. (6) becomes

A (R) \propto e^{i \frac{k}{2 f} R^{2}} \int d^{2} R^{'} U^{(i)} (R') e^{i \tilde{ψ} (R')} e^{- \frac{i k}{f} R' \cdot R},

where

\tilde{ψ} (R') = ψ (R') + ψ (- R')

. The exact form of ψ(R′) depends on the object profile. Regardless its form, ψ(R′), can always be written as a sum of symmetric and an antisymmetric functions, ψ_s(R′) = ψ_s(−R′) and ψ_a(R′) = −ψ_a(−R′). The expression of

\tilde{ψ}

shows that antisymmetric aberrations should disappear in coincidence detection when a coordinate inversion is made in one of the photons.

3. Experiment

In order to measure the cancellation effect described in the previous section, we set up the following experiment. Light produced by a Ti:sapphire laser is frequency doubled on a lithium triborate (LBO) crystal, generating a vertically polarized pulsed beam centered at the wavelength of 413 nm. This beam is used to pump a type II β−barium borate (BBO) crystal of 5 mm length, generating via SPDC pairs of orthogonally polarized entangled photons with central wavelength of 826 nm in a collinear configuration. A dichroic mirror (DM) is used to eliminate the pump beam. Instead of an object mask, we use the Gaussian profile of the pump beam itself, which is as good as any other field profile for our purposes. A lens (L) of 40 cm focal length placed after the crystal creates an 1.8× magnified image of the central plane of the crystal on the plane of a spatial light modulator (SLM). After the lens, the down-converted photons are split by a polarizing beam splitter P₁ as shown in Fig. 2, in such a way that horizontally polarized photons go through path 1 while vertically polarized ones can go either trough paths 2 or 3, depending on the angle of the half wave plate HW₁. When HW₁ is oriented at 0°, the photons are reflected by P₂, going through path 2. A quarter wave plate (QW) oriented at 45° in a double pass configuration rotates the polarization by 90°, so that photons reach P₃ horizontally polarized and with an extra reflection with respect to their entangled companions. This causes a coordinate inversion on the horizontal plane and we will refer to this as the situation with inversion. When HW₁ is oriented at 45°, the photons that leave P₁ are transmitted by P₂ and follow path 3. HW₂ rotates the polarization by 90°, so that photons reach P₃ vertically polarized and are then reflected. In this case, they suffer an even number of reflections, as do their twins, and no coordinate inversion takes place. We will refer to it as the situation without inversion. When path 2 is selected, the angle of HW₃ is set to 0° so that the polarization of photons leaving P₃ remains unchanged. On the other hand, when path 3 is chosen, the angle of HW₃ is set to 45° to rotate the polarization by 90°. This is necessary due to a restriction of the SLM, which only modulates horizontally polarized light. The three path lengths are made to be approximately the same. Signal and idler transverse profiles overlap only at the SLM plane, so that both photons experience the same aberrations. The SLM is a liquid crystal on silicon device by Hamamatsu, which modulates phase only. It has a pixel size of 20 μm and can achieve a phase modulation of more than 2π. It was programmed to create a parabolic phase profile ϕ(x) = αx² with polynomial aberrations of the form ψ_n(x) = β_nxⁿ added, where n ranges from 0 to 4. In the present context, we identify n as the aberration order. Note that this definition differs from the usual in aberration theory. For n = 0 (no aberration) the SLM acts as a concave mirror of focal length f = 38 cm. After reflection on the SLM, the photons are detected in coincidence on its focal plane by two photon counting detectors, equipped with 30 nm bandwidth interference filters, centered at 826 nm, and 200 μm slits. Scanning one of the detectors (detector 1 hereafter), we measured the intensity profile of the coincidence beam on the horizontal x direction for each value of the aberration order, both with and without the coordinate inversion.

Fig. 2 Experimental setup used to measure the aberration cancellation. A 413 nm pulsed laser beam pumps a β−barium borate (BBO) nonlinear crystal of 5 mm length, generating via type II SPDC pairs of orthogonally polarized entangled photons with central wavelength of 826 nm in collinear configuration. A dichroic mirror (DM) is used to eliminate the pump beam. Instead of an object mask, we use the Gaussian profile of the pump beam itself. A lens (L) of 40 cm focal length placed after the crystal creates an 1.8× magnified image of the central plane of the crystal on the plane of a spatial light modulator (SLM). After the lens, the down-converted photons are split by a polarizing beam splitter P₁ in such a way that horizontally polarized photons go through path 1 while vertically polarized ones can go either trough paths 2 or 3, depending on the angle of the half wave plate HW₁. When HW₁ is oriented at 0°, the photons are reflected by P₂, going through path 2. A quarter wave plate (QW) oriented at 45° in a double pass configuration rotates the polarization by 90°, so that photons reach P₃ horizontally polarized and with an extra reflection with respect to their entangled companions. This causes a coordinate inversion on the horizontal plane. When HW₁ is oriented at 45°, the photons that leave P₁ are transmitted by P₂ and follow path 3. HW₂ rotates the polarization by 90°, so that photons reach P₃ vertically polarized and are then reflected. In this case, they suffer an even number of reflections, as do their twins, and no coordinate inversion takes place. When path 2 is selected, the angle of HW₃ is set to 0° so that the polarization of photons leaving P₃ remains unchanged. When path 3 is chosen, the angle of HW₃ is set to 45° to rotate the polarization by 90°. This is necessary due to a restriction of the SLM, which only modulates horizontally polarized light. The three path lengths are approximately the same. The SLM is a liquid crystal on silicon device by Hamamatsu, which modulates phase only. It was programmed to act as a concave mirror of 38 cm focal length by inducing a parabolic phase profile ϕ(x) = αx². Polynomial aberrations of the form ψ_n(x) = β_nxⁿ (n = 0,1,2,3,4) were added to ϕ(x). After reflection on the SLM, the photons are detected in coincidence on its focal plane by two photon counting detectors, equipped with 30 nm bandwidth interference filters, centered at 826 nm, and 200 μm slits. Scanning one of the detectors, we measured the intensity profile of the coincidence beam on the horizontal x direction for each value of the aberration order, both with and without the coordinate inversion.

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4. Experimental results and discussion

The measured normalized coincidence profiles are shown in Fig. 3, for aberration orders n = 1 to 4. Plotted together are the data corresponding to the situations with and without inversion (red squares and the blue circles, respectively) for each aberration order and the reference data (green diamonds), corresponding to no applied aberrations. In all cases, single and coincidence photon counts were taken in sampling times of 10 s with coincidence resolving time of 5 ns. Single count rates remained approximately constant around 3×10⁴ counts per second, whereas coincidence counts varied from approximately 10 (background) to 100 coincidences per second. Single and coincidence count rates are listed for each value of n in Table 1.

Fig. 3 Normalized coincidences as a function of transverse position of the signal detector. Green diamonds: non-aberrated (reference) curve for the case with inversion. Blue circles: aberrated curves without correction (no coordinate inversion). Red squares: corrected curves (with coordinate inversion). Top left: aberration order n = 1. Top right: aberration order n = 2. Bottom left: aberration order n = 3. Bottom right: aberration order n = 4. Statistical error bars lie within the markers.

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Table 1. Average single count rates (R₁, R₂) and coincidence count rates ranges (R₁₂) without (ninv) and with (inv) coordinate inversion. All numbers are in counts per second.

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First order aberration causes a wavefront tilt with the consequent lateral displacement of the curve at the detection plane, as shown by the blue circles in the upper left plot. This tilt is highly suppressed when the coordinate inversion is made (red squares). Second order aberration results are shown on the upper right plot. This aberration causes a defocus, changing the width of the curve at the detection plane. The two situations, with and without inversion, are almost indistinguishable, as expected from the theory. The bottom left plot represents the results for third order aberration. It is responsible for the asymmetric features exhibited by the curve in blue circles. It was expected, according to the theory developed in Sec. 2, total cancellation of this effect. However, the small bumps visible at each side of the curve defined by the red squares show that this is not the case. We assign this to the fact that correlation at the SLM is not a perfect Dirac delta, so the two photons are not subjected to exactly the same aberrations. This will be discussed in more detail in section 5. Fourth order aberration changes the wave front curvature, which causes the Gaussian tails to raise. As an even order aberration, it was not expected to be sensitive to the coordinate inversion, as the bottom right curve shows.

In order to quantify the mismatch between aberration corrected (n = 1,…, 4) and nonaberrated curves (n = 0), we applied the definition of the distance between two probability distributions

D = \frac{1}{2} \sum_{j} | P_{n} (x_{j}) - P_{0} (x_{j}) |,

where P_n is the normalized nth order aberrated curve and P₀ is the reference (non-aberrated) curve. This quantity is bounded below by 0, corresponding to maximum overlap between the two curves and has an upper bound of 1, corresponding to two completely disjoint curves. The results of the calculations are summarized on the Table 2, where D^inv and D^ninv refer to the situations with and without inversion, respectively. For a better comparison of the cancellation effects, it is convenient to define a relative degree of correction as r_c = (D^ninv − D^inv)/D^ninv. This gives us a (89 ± 2)% correction for the first order aberration and (68±2)% for the third order. It can thus be seen that the coordinate inversion actually reduces odd order aberrations, while having no effect on even ones.

Table 2. Deviation of aberrated curves without and with coordinate inversion (D^ninv, D^inv) from the reference and the relative correction factor (r_c) for aberration orders n. The symbol ∼ 0 stands for values within the error bar.

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5. Simulations

To arrive at Eq. (7) we have approximated the function V(S′) by a Dirac delta δ²(S′). In order to investigate the validity of this approximation, that is, dependence of the cancellation effect on the width of V, we calculated numerically the photo-detection probability based on a one-dimensional version of Eq. (6), neglecting quadratic phase factors:

\begin{array}{l} P_{n} (ξ, η) = N | \int d ξ^{'} \int d η^{'} U^{(i)} (ξ^{'}) V (η^{'}) \\ {\times \exp [i ψ_{n} (η^{'} + ξ^{'}) + i ψ_{n} (η^{'} - ξ^{'}) - \frac{i k}{f} (ξ ξ^{'} + η η^{'})] |}^{2}, \end{array}

where ξ = (x_s − x_i)/2, η = (x_s + x_i)/2 and N is a normalization constant. The simulated aberrations had the form ψ_n(x′) = π(x′/a)ⁿ with n = 1,3 and a = 0.33 mm, equivalent to 16.5 SLM pixels, the actual value used in the experiment. The image of the two-photon field at the SLM plane was represented by a Gaussian of the form U⁽ⁱ⁾ = exp(−ξ²/w_p²), with w_p =1 mm. The function V was approximated by the Gaussian exp(−η²/w²). For a central pump wavelength of 413 nm and a crystal length of 5 mm, the values used in the experiment, the best adjusted Gaussian has a half width w =0.017 mm. So if we were correctly imaging the central plane of the crystal, the V half width at the SLM would be equal to this value times the lens magnification, that is, 0.03 mm. However, due to uncertainties in the distances measurements, the image of the central plane of crystal may not coincide with the SLM plane and the transverse correlation length may be larger. To analyze the effect of increasing the transverse correlation length we simulated Eq. (9) for values of w ranging from 0.03 to 0.14 mm. The normalized coincidences are shown on Fig. 4 as a function of x/a, where x is the position of the signal detector for a fixed position of the idler detector. The parameter a corresponds to the position on SLM for which the phase is π and it is thus related to the aberration strength. The curves are plotted for different values of the parameter w/a.

Fig. 4 Simulated coincidence profiles as a function of x/a, where x is the position of the signal detector with the idler detector position kept fixed. Left plot corresponds to first order aberration and right plot to third order.

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In order to quantify the cancellation, we calculated the distance D^inv between P₀(x_s,x_i) and P₁(x_s,x_i) and between P₀(x_s,x_i) and P₃(x_s,x_i), with x_i fixed. The results are presented in Fig. 5, as functions of w/a.

Fig. 5 Distance between aberration corrected and reference simulated curves, for aberration orders n = 1 and n = 3, for different values of the parameter w/a.

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It can be seen from Figs. 4 and 5 that the cancellation in fact decreases as the width of the correlation at the SLM plane increases. However, first order aberration is much less affected than third order, in agreement with the experimental results presented in Table 2.

6. Conclusion

The strong near field correlation shared by photons generated by SPDC was explored to produce cancellation effects of odd order aberrations induced by a spatial light modulator. This was achieved by preparing the source in such a way that one of the photons of the pair suffered a coordinate inversion before the aberration plane. The cancellation proved more effective for first order aberration than for third order. We assigned this to the fact that higher order aberrations are more sensitive to an increase in the correlation width.

The pure cancellation effect could in principle be mimicked by a conveniently engineered incoherent classical source in the regime of coincidence detection. However, the structure of the correlations present in the entangled photons, unlike those present in thermal light, produce correlation beams, allowing the encoding of information by free manipulation of the pump profile. This effect can be useful in imaging systems where polynomial aberrations of odd order are important, such as imaging through random media.

Acknowledgments

The authors acknowledge the financial support from the Brazilian agencies CNPq, CAPES and FAPEMIG.

References and links

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n	$R_{1}^{ninv}$	$R_{2}^{ninv}$	$R_{12}^{ninv}$	$R_{1}^{inv}$	$R_{2}^{inv}$	$R_{12}^{inv}$

0	31841	30236	10 to 197	30773	27449	8 to 184
1	31610	30345	9 to 207	31662	27974	7 to 194
2	33727	30666	10 to 118	33451	29024	10 to 113
3	20787	23138	3 to 77	21431	21390	4 to 136
4	27665	29340	18 to 96	27748	27492	16 to 85

n	D^ninv	D^inv	r_c (%)

1	0.53±0.01	0.06±0.01	89±2
2	0.25±0.01	0.26±0.01	~0
3	0.65±0.01	0.21±0.01	68±2
4	0.31±0.01	0.32±0.01	~0

n	$R_{1}^{ninv}$	$R_{2}^{ninv}$	$R_{12}^{ninv}$	$R_{1}^{inv}$	$R_{2}^{inv}$	$R_{12}^{inv}$

0	31841	30236	10 to 197	30773	27449	8 to 184
1	31610	30345	9 to 207	31662	27974	7 to 194
2	33727	30666	10 to 118	33451	29024	10 to 113
3	20787	23138	3 to 77	21431	21390	4 to 136
4	27665	29340	18 to 96	27748	27492	16 to 85

n	D^ninv	D^inv	r_c (%)

1	0.53±0.01	0.06±0.01	89±2
2	0.25±0.01	0.26±0.01	~0
3	0.65±0.01	0.21±0.01	68±2
4	0.31±0.01	0.32±0.01	~0

Experimental observation of aberration cancellation in entangled two-photon beams

Abstract

1. Introduction

2. Theory

3. Experiment

4. Experimental results and discussion

5. Simulations

6. Conclusion

Acknowledgments

References and links

Cited By

Figures (5)

Tables (2)

Equations (9)

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