Phase-only shaping algorithm for Gaussian-apodized Bessel beams

Charles G. Durfee; John Gemmer; Jerome V. Moloney

doi:10.1364/OE.21.015777

1. Introduction

Bessel beams have been explored for a number of years as a means to localize the intensity of a laser beam over a distance much longer than the Rayleigh diffraction range corresponding to the focal spot size. Since an ideal conical wave with a constant approach angle to the optical axis will have the same spot size independent of propagation distance, these beams are often called diffraction-free. Moreover, in principle, linear superpositions of Bessel beams can be used to engineer beams with desired profiles [1]. However, ideal Bessel beams require an infinite amount of energy and thus in practice Bessel beams are realized through truncation of the ideal wave. These apodized Bessel-beams accurately approximate the ideal Bessel beam in a “Bessel-zone” of finite width [2] and transition to a ring-beam in the far field [3, 4].

There are numerous applications of Bessel beams in a wide range of fields. On low energy systems, Bessel beams have been used for large depth of focus microscopic imaging systems [5, 6], optical tweezers [7], and optical coherence tomography [8]. They also have applications in laser micro-machining [9], laser electron acceleration [10], and filament formation [11]. In addition, to the best of our knowledge, the Bessel beam is the most efficient means of producing a high-intensity line focus, much more efficient than a cylindrical lens focus. The Bessel-Gauss beams in particular are useful for projecting a line focus to a position at some distance from the optic, especially for intense beams.

The two common methods for creating apodized Bessel beams are diffractive optical elements and axicons. The advantage of diffractive elements is that there is more flexibility in engineering the beam properties, but they are lossy and exhibit strong chromatic dispersion. Refractive and reflective axicons are more energy efficient but are difficult to fabricate and the Bessel-zone of such beams is localized near the tip of the axicon. In particular, with uniform input, the axial intensity profile from an axicon grows linearly with distance near the tip; with Gaussian beam input, the output axial profile has a nominal profile of zexp (−z²/w²). As will be shown below, the axial profile of a Bessel-Gauss focus has a Gaussian shape. For many applications it is advantageous to create an axial intensity profile that is nearly uniform in the Bessel-zone. The nonlinear material interactions necessary for micro-machining or plasma filament formation in air can take place within a sharp threshold and a uniform intensity profile will lead to well-controlled uniform plasma and the associated material modification. A number of methods have been proposed to engineer the radial phase profile of the optic to produce a uniform line focus. One of these is the logarithmic axicon which has been used to generate super-Gaussian profiles on the optical axis [12, 13]. Improvements in aspheric optical element production techniques have allowed the fabrication of refractive versions of the logarithmic axicon for constant intensity profile [13] and also for a profile that produces constant axial intensity in the presence of linear absorption [14]. A two-step method to first convert a Gaussian beam to a ring beam with a controlled amplitude profile, then to redirect that beam to a flat top axial line focus was proposed by Honkanen and Turunen [15]. Fresnel refractive axicons have also been attempted [16]. A second technique to control the axial profile is with holographic diffractive elements. The first demonstration of this type was by Sochacki et al [17]; the holographic technique has recently been improved by using a liquid-crystal spatial light modulator (SLM) to create the desired axial profiles [18]. While the holographic method can produce high-fidelity profiles, they exhibit low efficiency (∼5–7%), since the fidelity in the target region is achieved by allowing power to scatter away from the area of interest [19]. It is also important to note that in the logarithmic axicon design the profile of the beam on the optical axis has the same functional form as the radial cross-section at the front of the axicon. That is, ring-beams with a Gaussian or super-Gaussian cross section passed through the axicon will form Gaussian or super-Gaussian profiles on the optical axis [12].

In this paper we present an efficient method for controlling the axial intensity profile of apodized Bessel-beams through a single step of phase-only shaping. The method we present in this paper is more general than the logarithmic axicon approach in that it allows different initial beam radial profiles to be shaped into a desired on-axis profile. Specifically, we consider starting with a Gaussian ring-beam and adjusting its wavefront to produce a desired axial intensity profile. Such a ring beam can be formed, for example, by passing a Gaussian beam through a diffraction grating with concentric grooves [20]. While both phase and amplitude shaping will perform the task, it is either lossy or requires two separate shaping steps. Our phase shaping scheme is inspired by work that has been performed to shape the focal spot of conventional Gaussian beams. An example is shaping the wavefront of a Gaussian beam to create a super-Gaussian beam profile [21] at the focal plane of a lens. The technique we use in this paper to determine the optimum wavefront is the iterative Gerchberg-Saxton (GS) algorithm [22]. The GS algorithm was originally introduced for phase retrieval in electron microscopes [22] and has since found applications in astronomy [23], radial beam shaping [24], pulse shaping [25] and image reconstruction [26]. While it is well-known that the focal plane of a lens is the Fourier transform of the input field, it is less appreciated that there is a Fourier-like mapping of the input radial profile to the axial profile. This allows us to reduce the two dimensional problem to one. In this work we employ a modified version of the GS algorithm with rapid FFT iteration between two domains to optimize the axial profile to a super-Gaussian. We find that the phase shaping can be simplified by controlling the initial divergence of the input beam before the ring Gaussian is formed.

2. Bessel-Gauss beams

As mentioned in the Introduction, all physical Bessel beams must be of finite extent. One attractive representation of a finite conical wave is the Gaussian-apodized Bessel beam, also commonly called the Bessel-Gauss (BG) beam [3]. By starting with an expression for a Gaussian beam tilted with an angle γ₀ to the z-axis it has been shown within the paraxial approximation that these beams can be expressed analytically by:

E (r, z) = E_{0} \frac{w_{0}}{w (z)} exp [i (k z - Φ (z)) - i \frac{β^{2}}{2 k} z] exp [i \frac{k}{2} q (z) (r^{2} + {(\frac{β z}{k})}^{2})] J_{0} [β r (1 - z q (z))],

where E₀ is the peak field at the origin (z = 0), k is the wavenumber in free space, β = (ω/c)sinγ₀ is the projection of the wavenumber on the optical axis, w₀ is the characteristic width of the input Gaussian and we have suppressed the exp(ikz − iω₀t) carrier wave. Departing slightly from the notation used in [3], we have expressed the Gaussian beamlet profile in terms of the familiar Gouy phase Φ(z) and complex beam parameter q(z), which in turn is defined as below in terms of the z–dependent beam radius w(z) and radius of curvature R(z):

\begin{array}{l} q (z) = \frac{1}{R (z)} + i \frac{2}{k w^{2} (z)}, w (z) = w_{0} \sqrt{1 + z^{2} / z_{R}^{2}}, \\ R (z) = z (1 + z_{R}^{2} / z^{2}), Φ (z) = {tan}^{- 1} (z / z_{R}) . \end{array}

The Rayleigh range for the Gaussian beam is modified from its usual form by the tilt:

z_{R} = \frac{π w_{0}^{2}}{λ} cos γ_{0}

. The axial (r = 0 in Eq. (1)) intensity profile takes the form

I (z) = \frac{w_{0}^{2}}{w^{2} (z)} exp [- 2 \frac{z^{2}}{w^{2} (z) / {sin}^{2} γ_{0}}] .

The leading ratio is the well-known Lorentzian axial profile of the focused Gaussian beam-let. This modulation of the axial intensity is often negligible. In the geometric limit where the diffraction of the Gaussian ring can be neglected, w is constant and 2w/ sinγ₀ is the axial projection of the spot diameter. The opposite limit is where there is considerable diffraction of the Gaussian beamlet. In this case, if there is to be a ring beam in the far field, γ₀ must be at least twice the divergence angle of the beamlet, w₀/z_R. In this case, the axial intensity profile can be well-approximated by Eq. (3) if we set w(z) = w₀.

Bagini et al extended the analytic formulation of the BG beams to create “generalized” Bessel-Gauss beams [4]. In particular, it was shown that the profile given by Eq. (1) can be created remotely by propagating a ring-beam of the form

E (r, 0) = A_{0} exp (- \frac{(r^{2} + a^{2})}{w_{0}^{2}}) I_{0} (\frac{2 a r}{w_{0}^{2}} - i β r),

where I₀ is the zero order modified Bessel function of the first kind and a is the the radius of the ring-beam. The Bessel-zone for these beams is centered at the point z_d = ak/β and the width of the Bessel-zone can be accurately approximated by z_w ≈ kw₀/β. Moreoever, if a/w₀ ≫ 1 the initial intensity is well approximated by

{| E (r, 0) |}^{2} \approx A_{0}^{2} \frac{w_{0}^{2}}{4 π a r} exp (- \frac{2 {(r - a)}^{2}}{w_{0}^{2}}) .

Consequently, focused ring-beams with Gaussian radial profiles will transition from a ring-beam in a ring-zone to a BG beam in the Bessel-zone and back to a ring-beam in another ring-zone (see Fig. 1).

Fig. 1 Illustration of the propagation of the generalized Bessel-Gauss beam with color used to represent intensity. To highlight the structure of the low intensity rings, the cube root of the intensity is plotted. The black lines show the geometric path of the rings ztanγ₀, and the red lines mark the full width at half maximum of the axial intensity profile.

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3. Phase-only shaping of axial intensity profile

3.1. Mapping E(r, 0) to E(0, z)

As seen above, a Gaussian ring-beam focused with a lens or axicon will produce a BG beam with a Gaussian axial intensity profile. To control the shape of the axial intensity profile we will make modifications to the radial input field. Rather than computing the full Fresnel transform, the input field E(r, 0) can be transformed directly to the axial field E(0, z) [27]. (Note that from this point forward we will treat z = 0 as the input plane). Under the assumption of azimuthal symmetry the Fresnel propagation integral reduces to

F (z) = E (0, z) = - \frac{i k}{z} \int_{0}^{\infty} E (r, 0) exp [i \frac{k r^{2}}{2 z}] r d r .

For an input ring-beam with negligible field strength on axis, the integral can be extended to the entire real line. Therefore, if we make the change of variables s = r² then

G (Ω_{z}) = - 2 i Ω_{z} \int_{- \infty}^{\infty} g (s) exp [i Ω_{z} s] d s,

where

g (s) = {\begin{array}{l} E (\sqrt{s}, 0) & s \geq 0 \\ 0 & s < 0 \end{array},

Ω_z = k/2z and G(Ω_z) = F(k/(2Ω_z)). Equation 7 is in the form of a Fourier transform with Ω_z taking on the role of the frequency variable which allows for rapid numerical evaluation using the fast Fourier transform (FFT).

While the mapping from the radial input field to the axial output field described above does not make any assumptions about the propagation distance z except that the Fresnel transform may be used, we are especially interested in the regime where the distance between the Bessel-zone is large compared to the (effective) Rayleigh range of the envelope on the Bessel function in the center of the Bessel-zone. Therefore, we will consider an input ring-beam of modulus E₀f(r) with a modified radial phase factor ϕ(r²) focused by a lens of focal length z_d:

E (r, 0) = E_{0} f (r) exp [i ϕ (r^{2})] exp [- i \frac{k r^{2}}{2 z_{d}}]

and we define

h (s) = {\begin{array}{l} f (\sqrt{s}) & s \geq 0 \\ 0 & s < 0 \end{array} .

In particular, in this paper we assume that

f (r) = exp [- \frac{{(r - a)}^{2}}{w_{0}^{2}}],

which is a Gaussian ring-beam of radius a and characteristic width w₀ and by Eq. (5) adequately approximates Eq. (4) when a ≫ w_in.

Note that the lens phase can also be written in Ω_z-space as exp [−iΩ_{z_d}s], where Ω_zd = k/2z_d. Therefore we can separate the lens phase from the input function, g(s) and the transform of the ring-beam with the shaper phase ϕ(s) will be a function of the shifted conjugate variable

Ω_{z}^{'} = Ω_{z} - Ω_{z_{d}} = \frac{k}{2} (\frac{1}{z} - \frac{1}{z_{d}}) .

With this change of variables, the Bessel zone is now centered at Ω′_z = 0. Finally, we stress that even though a lens may be used, and there is a Fourier-transform-like relationship between the starting and target domains, the mapping we are using is correct to within the Fresnel, rather than the Fraunhofer limit.

3.2. Optimization of the input radial phase with the Gerchberg-Saxton algorithm

To construct a phase ϕ(r²) that will generate a desired on-axis intensity |H_T(Ω′_z)|² we consider the following variational problem

min_{ϕ, E_{0}} J [ϕ] = \int_{- \infty}^{\infty} {(| H (Ω_{z}^{'}) | - | H_{T} (Ω_{z}^{'}) |)}^{2} d Ω_{z}^{'},

where H(Ω′_z) = ℱ[E₀ (h(s)exp(iϕ(s)))] and ℱ denotes the Fourier transform. In phase retrieval, the minimum of (13) is zero since it is assumed that there exists ϕ^* such that H_T and g(s) = h(s)exp(iϕ^*(s)) are Fourier transform pairs but in our case it is not true in general that the minimum of (13) is indeed zero. However, by the reverse triangle inequality and Plancheral’s theorem it follows that for all ϕ:

{[{(\int_{- \infty}^{\infty} {| H_{T} (Ω_{z}^{'}) |}^{2} d Ω_{z}^{'})}^{1 / 2} - \sqrt{2 π} E_{0} {(\int_{- \infty}^{\infty} {| h (s) |}^{2} d s)}^{1 / 2}]}^{2} \leq J [ϕ] .

Therefore, a necessary condition for the minimum value of Eq. (13) to be approximately zero is that

E_{0} \approx \frac{\int_{- \infty}^{\infty} {| H_{T} (Ω_{z}^{'}) |}^{2} d Ω_{z}^{'}}{2 π \int_{- \infty}^{\infty} {| h (s) |}^{2} d s} .

Similar to what has been used in pulse shaping applications, for example Rundquist et al [25], to optimize J[ϕ(s)] we use a modified GS algorithm. In this algorithm, the functions h(s) and H_T (Ω_z′) are uniformly discretized at the points s_i and (Ω′_z)_i respectively and following Fienup’s notation [28] the GS routine is

\begin{matrix} H^{(j)} ({Ω_{s}}^{'}) = ℱ {h (s) exp [ϕ^{(j - 1)} (s)]} \\ h^{(j)} (s) = ℱ^{- 1} {H_{T} ({Ω_{z}}^{'}) exp [Φ^{(j)} ({Ω_{s}}^{'})]} \end{matrix},

where it is understood that ℱ is now the discrete Fourier transform. The algorithm is initiated with a random radial phase and convergence can be assessed by the error function

η_{j} = \frac{\sum_{i} {(| H^{(j)} ({(Ω_{z}^{'})}_{i}) | - | H_{T} ({({Ω^{'}}_{z})}_{i}) |)}^{2}}{\sum_{i} {| H_{T}^{(j)} ({(Ω_{z}^{'})}_{i}) |}^{2}} .

Since we are using the GS algorithm as an error reduction algorithm, i.e. η_j₊₁ < η_j, it follows that the sequence η_j will converge to a fixed value [28]. However, this is not the same as convergence of the algorithm and in particular the sequence ϕ^(j) may not be converging to a phase that is the global minimum of Eq. (13). This lack of convergence manifests itself in the form of stagnation of the algorithm at local minima. If the algorithm gets stuck before getting close to the target function, we use a modification of the GS algorithm called the input-output algorithm [28]. Here a stronger adjustment is applied: |H^(j+1) (Ω_s′)| = H_T (Ω_s′) − α (|H^(j) (Ω_s′)| − |H_T (Ω_s′)|). A typical choice for α is 0.15. We find best convergence if we set α = 0 after the axial function approaches the target. If the optimization converges, ϕ(s) is the required shaper phase.

3.3. Optimization for a super-Gaussian axial profile in the Bessel zone

To create a profile with a uniform intensity we choose the following target field function as a test of our algorithm:

F_{T} (z) = E_{T} exp [- \frac{{(z - z_{d})}^{8}}{{w_{T}}^{8}}] .

If a/w₀ ≫ 1 it follows by Eq. (15) that in order for the the minimum value of J to be as small as possible that

E_{T}^{2} \approx 4.56 E_{0}^{2} \frac{k_{0} r_{0} w_{0}}{w_{T}},

which gives us an approximate relationship as to how the various length scales contribute to the mapping between the peak radial and on-axis intensities. In particular, the peak intensity is inversely proportional to the width of the desired target (w_T), since the available energy is spread out over a volume that is proportional to w_T.

Starting with choices of a, w₀, k, and z_d, we pick a value for the target width w_T and run the GS algorithm. Figure 2 shows the results of such an optimization for input conditions z_d = 1m, w_in = 5mm, and a = 50mm and a target width of w_T = 7mm. For this set of parameters, γ₀ ≈ 2.9° and the axial width without shaping would be 1.2mm. The optimized axial profile [Fig. 2(a)] is very close to the target super-Gaussian and the error as defined in Eq. (17) is approximately 10⁻⁴. Even more accurate optimization can be obtained with larger w_T (see below in Fig. 4) The shape of the optimized radial phase for this run is shown in the blue curve of Fig. 2(b). There is a strong quadratic component that will be discussed below.

Fig. 2 (a) Lineouts of the target (red, dashed) and optimized (blue, slight vertical offset) axial profiles for w_T = 7mm. The narrow black profile in (a) represents the axial profile of the unshaped Bessel zone. (b) Cross-sectional plots in the input plane. Red: input intensity profile; Blue: optimized radial phase profile for w_T = 7 mm; Black/dashed: optimized phase profile after an initial guess of the focusing phase is added to the input beam.

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To see how the wavefront shaping affects the beam away from the z-axis, we numerically propagated the input Gaussian ring beam with the calculated optimum phase profile with Fresnel integration. Figure 3 shows the intensity profile in the Bessel zone. The Bessel zone radial profile looks as expected for a Gaussian-apodized Bessel beam, except that there is a taper to the diffraction rings. This taper indicates that the effective approach angle is z-dependent, γ(z). For these conditions, the radius to the first zero varies from 5μm to 6μm across the line focus. Figure 5(a) shows a cross-sectional lineout of the intensity profile rI(r) at a position that is approximately 17mm after the focal plane. To illustrate that the strong outwardly-decreasing taper is a profile that is geometrically consistent with a flat top line focus we also plot the function rI(r). Resulting from the initial wavefront shaping, the radial cross-section of the beam evolves towards a profile resembling a flat top sloped away from the optical axis. Without any shaping, the half width of the axial profile for these conditions would be 1.2mm, substantially smaller than this choice of target width, w_T = 7mm. Note that the optimized phase profile is strongly quadratic in the quantity r − r₀ (Fig. 2). This indicates a focusing phase, an idea supported by the image in Fig. 5(b). The first-order phase change to the beam is to defocus the ring so that the beamlets are larger where they cross the z-axis. This change in divergence can work with either sign: from the point of view of intense beam propagation, it is desirable to have the ring focus after the Bessel zone. We took advantage of this effect to improve the optimization convergence by estimating the quadratic component of the focusing phase. A straightforward calculation using the Gaussian beam propagation formula for w(z) (Eq. (2)), projected onto the z-axis is used to predict the effective divergence required to spread the beam to the desired size. After re-optimization, the phase correction excursion is considerably smaller as seen in the black dashed curve in Fig. 2.

Fig. 3 Image of the calculated intensity vs r and z for the optimized axial profile for w_T = 7 mm.

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Fig. 4 Left: Contour plot showing optimized axial intensity profiles as a function of the target width w_T. The inset shows the log of the error in the optimization, calculated with Eq. (17). Right: Same data, but with the intensity scaled with the target width.

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Fig. 5 (a) Red: Radial section of the shaped ring-beam intensity I(r) at approximately 17mm past the focal plane. Blue: same radial section weighted with the radius, rI(r). (b) Image of the calculated shaped field on both sides of the Bessel zone.

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3.4. Scaling of shaped axial profiles with wavelength and focal conditions

To explore the limits of what shaped widths could be found, we chose to scan through a range of values for w_T, using the predicted quadratic phase as the starting phase for the beam. In Fig. 4, we show a contour map of the optimized axial profile as a function of the target super-Gaussian width w_T for the same test conditions as above, z_d = 1m, w_in = 5mm, and a = 50mm. The log of the convergence error is shown in the inset. Provided the target width of the superGaussian is sufficiently large, phase-only shaping yields very good results. For small w_T, there is a overshoot seen at the edge that may be important for some applications. The 3D image on the right illustrates how the decrease in intensity with target width follows the expected inverse dependence.

Having determined an optimized shaper phase function, how does this function change when the system parameters are changed? The radial to axial transform in Eq. (6) transforms between the radial variable s = r² and its shifted conjugate variable Ω_z′ (see Eq. (12)). For our case we have optimized for an axial profile that is super-Gaussian axial profile in the variable δz = z − z_d. Provided that w_T/z_d ≪ 1, Eq. (12) can be written to show that $δ z = - 2 z_{d}^{2} {Ω_{z}}^{'} / k_{0}$ . This means that if we have optimized the radial phase profile for a width ΔΩ_s′, the corresponding length of the Bessel zone in z–space can be calculated as $w_{T} = 2 z_{d}^{2} Δ Ω_{s}^{'} / k_{0}$ . Thus the same shaper phase profile will result in a similar axial profile if the wavelength or lens focal length are changed. For example, if we change the lens focal length in our example from 1m to 5m, the target width in Fig. 2(b) would change from 20mm to 500mm.

For broadband pulses this scaling leads to the conclusion that the length of the optimized axial region is proportional to the wavelength. With an axial length of the profile as a function of wavelength, the bandwidth will vary along the Bessel zone. This effect will be important for pulses that are sufficiently short that the fractional bandwidth is appreciable. It has been shown that Bessel beams can be made achromatic provided that the central transverse wavenumber k_T = (ω/c)n(ω)sinγ₀ (ω) can be made independent of frequency [29]. This condition is possible if the ring-beam is formed by normal incidence on a diffraction grating, where the diffracted angle follows k_T = 2π/d, where d is the groove spacing. Concentric gratings of this form have been developed [20].

4. Summary and Discussion

In this paper, we have shown that wavefront shaping of an annular ring-beam with a Gaussian cross-section can result in a line focus along the optical axis of nearly constant intensity. Starting from a Fourier transform integral mapping of the radial input profile to the axial profile, we used a modified Gerchberg-Saxton algorithm to optimize the focused profile using fast one-dimensional FFT operations. Across the input radial profile, the optimized phase excursions for our example are approximately 4π, well within the phase range of liquid crystal spatial light modulators. As mentioned in the introduction, the ring beam can be formed with a concentric diffraction grating, which also allows the radial spectral dispersion to be exploited to produce an achromatic beam. With the circular diffraction grating, a simple lens or curved mirror beam expander can be used to control the input quadratic phase of the beamlet before ring formation. This removes the quadratic component from the shaper, reducing further the phase excursions required over most of the beam profile. In fact, the residual phase seen in the black dashed curve of Fig. 2(b) is mostly quartic. A well-controlled spherical aberration term could remove that component as well. It may even be possible to design a multi element lens with the correct phase profile, as in the lens axicon designed by Burvall et al [30]. At the expense of added complexity, the ring beam can be formed with another phase plate, in which case a two step shaping process can be used [15].

We anticipate this approach will find applications ranging from micro-machining to long-range filament formation in air. In the former case, the goal might be to deliver pulses at an intensity that is a prescribed amount over the damage threshold. In the second application, the line focus can be used to efficiently create an ionized column of desired length projected at a long range. We are currently working on the experimental demonstration of the methods described in this paper. The experimental work as well as the transition to applications will be addressed in subsequent publications.

Acknowledgments

The authors acknowledge funding support under the MURI AFOSR grant FA9550-10-0561 and C.D. acknowledges funding support from AFOSR under the grant FA9550-10-1-0394. The authors wish to thank Ewan Wright for many useful discussions and bringing to our attention reference [4]. C. D. also wishes to thank Daniel Adams for useful input on the convergence of the G. S. algorithms.

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Phase-only shaping algorithm for Gaussian-apodized Bessel beams

Abstract

1. Introduction

2. Bessel-Gauss beams

3. Phase-only shaping of axial intensity profile

3.1. Mapping E(r, 0) to E(0, z)

3.2. Optimization of the input radial phase with the Gerchberg-Saxton algorithm

3.3. Optimization for a super-Gaussian axial profile in the Bessel zone

3.4. Scaling of shaped axial profiles with wavelength and focal conditions

4. Summary and Discussion

Acknowledgments

References and links

Cited By

Figures (5)

Equations (19)

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