Scheme for generating distillation-favorable continuous-variable entanglement via three concurrent parametric down-conversions in a single &#x003C7;<sup>(2)</sup> nonlinear photonic crystal

Yan-Xiao Gong; ShengLi Zhang; P. Xu; S. N. Zhu

doi:10.1364/OE.24.006402

1. Introduction

Continuous-variable (CV) quantum entangled states of light have wide applications in CV quantum information tasks [1–3 ] as well as high-precision phase measurement beyond the shot noise limit [4]. However, the performance of these applications is often limited due to the inevitable noises and losses in generating and transmitting the CV entangled states, which leave the states only weakly entangled. These limitations can be combatted by entanglement distillation [5], a procedure in which highly entangled states can be produced from a larger number of less entangled states by means of local operations and classical communication. Entanglement distillation is an essential protocol for long-distance quantum communication [6].

In all the CV developments, Gaussian states and Gaussian operations are commonly addressed and employed, since they are readily available in many laboratories [3]. The Gaussian states are the states with Gaussian Wigner function, such as vacuum, thermal, and coherent states. A typical Gaussian entangled state is the two-mode squeezed vacuum (TMSV) states, known as EPR states, which can be obtained by parametric-down conversion (PDC) [7,8]. The Gaussian operations are the operations that can preserve the Gaussian nature of a state, including the routine passive linear optical elements, squeezers, and homodyne detection. However, entangled Gaussian states cannot be distilled by local Gaussian operations and classical communication [9–11 ]. One has to resort to non-Gaussian operations to increase Gaussian entanglement [12–14 ]. Photon subtraction (PS) is one of the most experimentally feasible non-Gaussian operations proposed for distilling Gaussian entanglement [12]. Both nonlocal [15] and local [16,17] PS from the TMSV states have been experimentally demonstrated to increase entanglement.

Although Gaussian operations cannot distill Gaussian entanglement, recent studies have revealed that local displacement [18–20 ] or squeezing [21] prior to PS can improve the performance of the PS-based entanglement distillation. However, in experiment, performing on-line local squeezing or displacement is not straightforward but requires rigorous phase-locking and mode-matching. To this point, one possible solution would be moving the local operation to offline state preparation procedure. However, high-quality interferometry is still required to merge different squeezing operations. Explicitly, following from the Bloch-Messiah decomposition of two-mode symplectic transformations, any pure two-mode Gaussian state with zero displacement can be obtained by generating two single-mode squeezed states (which can have different strengths of squeezing and different orientations of squeezing ellipses) and combining them on a (generally unbalanced) beam splitter. As a matter of fact, the Gaussian entangled state in Takahashi et al.’s experiment [18] produced by splitting a single-mode squeezed vacuum state in half at a balanced beam splitter (BS), is a single-mode-squeezing TMSV state, namely the state by performing single-mode squeezing operations on both modes of the TMSV state. This method is easily implemented, however, the squeezing parameters in single-mode squeezing for rotation and in the TMSV state are fixed and hence cannot be tuned for the optimal performance developed in the Ref. [21]. In this work, we make an approach by generating the single-mode-squeezing TMSV states with tunable squeezing parameters directly from a single χ ⁽²⁾ nonlinear photonic crystal (NPC) which enables concurrent single-mode and two-mode squeezing operations. The simultaneous realization of the three squeezing operations releases the requirement of high-quality interferometry.

The χ ⁽²⁾ NPC, first introduced by Berger [22], is specified as the two-dimensional quasi-phase-matching (QPM) material where the second-order nonlinear susceptibility χ ⁽²⁾ is spatially modulated. QPM materials [23,24] have been extensively applied in nonlinear optics due to its advantages over birefringence phase-matching (BPM), such as higher efficiency and enabling flexible frequency-tunable nonlinear processes. In quantum optics field, QPM PDC has been employed to build efficient sources for producing photon pairs [25] and CV squeezed states of light [26]. Compared with one-dimensional QPM materials, the χ ⁽²⁾ NPC can provide more flexible multiple-beam or multiple-frequency nonlinear processes [27–35 ] and thus has aroused great interest. Recently, its applications have been explored in quantum optics field. Schemes for generating photonic polarization [36] or path [37, 38] entangled states from concurrent multiple QPM PDCs in the χ ⁽²⁾ NPC have been proposed and some of them have been demonstrated in experiment [39–41 ]. In this paper, we extend the application of NPC to the CV field. We propose a scheme for generating the single-mode-squeezing TMSV states via the NPC designed in our previous work [37] to provide QPM conditions for three concurrent PDCs, corresponding to two single-mode and one two-mode squeezing operations, respectively. The ratio of squeezing parameters is tunable in experiment. The simultaneous realization of three PDCs in a single crystal greatly reduces the experimental complexity in merging different squeezing operations, and opens up a way to compact CV squeezed light sources.

2. Overview of the NPC structure

Figure 1(a) depicts the 2D rectangularly poled NPC structure designed in our previous work [37]. The circularly inverted domains (with −χ ⁽²⁾) distribute rectangularly on a +χ ⁽²⁾ background, with the periods along x and y axes represented by Λ_x and Λ_y, respectively. Figure 1(b) illustrates the corresponding reciprocal lattice, with the reciprocal vectors ${\vec{G}}_{m, n}$ given by

{\vec{G}}_{m, n} = m {\vec{e}}_{1} + n {\vec{e}}_{2}, | {\vec{e}}_{1} | = \frac{2 π}{Λ_{x}}, | {\vec{e}}_{2} | = \frac{2 π}{Λ_{y}},

where m and n are integers. We can write the nonlinear susceptibility χ ⁽²⁾ on the 2D spatial coordinate

\vec{r} = (x, y)

as a Fourier series [22],

χ^{(2)} (\vec{r}) = d \sum_{m, n} G_{m, n} e^{- i {\vec{G}}_{m, n} \cdot \vec{r}},

Fig. 1 (a) Schematic of the 2D rectangularly poled NPC structure. (b) Reciprocal lattice of the crystal. Thin orange arrows represent reciprocal vectors of the crystal. (c) and (d) are two QPM geometries for three concurrent PDC processes. Thick blue arrows represent the pump (p) wave vector. Thick red arrows denote the signal (s), and idler (i) wave vectors. (e) Transverse pattern sketch of the parametric light in the Fourier plane.

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where d is the effective nonlinear coefficient. The Fourier coefficients $G_{m, n}$ depending on the first order Bessel functions, are given by

G_{m, n} = \frac{2 R}{\sqrt{{(m Λ_{y})}^{2} + {(n Λ_{x})}^{2}}} J_{1} [2 π R \sqrt{{(\frac{m}{Λ_{x}})}^{2} + {(\frac{n}{Λ_{y}})}^{2}}],

where R is the radius of the round domain-inverted area.

From Eqs. (1) and (2) we can see multiple reciprocal vectors may lead to multiple QPM geometries in nonlinear optical processes. Here we only consider three most prominent reciprocal vectors ${\vec{G}}_{1, 0}$ , ${\vec{G}}_{1, 1}$ , and ${\vec{G}}_{1, - 1}$ . Considering a degenerate PDC process with the pump wave vector in the x direction, we can choose right wavelength and temperature to design the crystal for satisfying the following QPM conditions simultaneously,

{\vec{k}}_{s} + {\vec{k}}_{i} + {\vec{G}}_{1, 0} - {\vec{k}}_{p} = 0,,

{\vec{k}}_{s 1} + {\vec{k}}_{i 1} + {\vec{G}}_{1, - 1} - {\vec{k}}_{p} = 0,,

{\vec{k}}_{s 2} + {\vec{k}}_{i 2} + {\vec{G}}_{1, 1} - {\vec{k}}_{p} = 0,,

where p, s, i represent the pump, signal, idler fields, respectively. The QPM geometry of Eq. (4) is shown in Fig. 1(c), with the resulting signal and idler light emitted noncollinearly as a cone. The other two QPM geometries corresponding to Eqs. (5) and (6) are depicted in Fig. 1(d), and in these two cases, the signal and idler light fields transmit in a collinear and beam-like way. Figure 1(e) shows schematically the transverse distribution of the PDC light. If selecting the light along A and B which are the overlapping positions of the PDC light satisfying Eq. (4) with that satisfying Eq. (5) and that satisfying Eqs. (6), respectively, using single-mode fibers and narrow-band filters, we can make single-mode approximation and write the operator for the three concurrent PDC processes as (for detail see Ref. [37])

\hat{U} = \exp {χ [r ({\hat{a}}_{A}^{† 2} + {\hat{a}}_{B}^{† 2}) + {\hat{a}}_{A}^{†} {\hat{a}}_{B}^{†}] - H . c .} .

In principle, r can be any nonzero real number. It depends on R, the ratio of Λ_x to Λ_y, and fiber collection efficiencies. Note that although the fiber collection efficiencies induces loss to the source, such spatial filtering is a common process in most realistic experiments.

3. Generation of single-mode-squeezing TMSV states

In our previous work [37], the NPC structure was proposed to generate multiphoton path-entangled states from weak-regime PDCs seeded by a two-mode coherent state. In this work we further explore the application of the operator $\hat{U}$ given by Eq. (7) in CV field. Due to the commutation relation $[{\hat{a}}_{A}^{† 2} - {\hat{a}}_{A}^{2} + {\hat{a}}_{B}^{† 2} - {\hat{a}}_{B}^{2}, {\hat{a}}_{A}^{†} {\hat{a}}_{B}^{†} - {\hat{a}}_{A} {\hat{a}}_{B}] = 0$ , the operator $\hat{U}$ is actually a concatenation of three Gaussian operations, with corresponding symplectic transformation in phase space given by

S_{\hat{U}} = [S_{A} (2 r χ) \oplus S_{B} (2 r χ)] S_{A B} (χ),

where

S_{A} (2 r χ) = S_{B} (2 r χ) = diag (e^{2 r χ}, e^{- 2 r χ}),

are the single-mode squeezing operations with parameter 2rχ and

S_{A B} (χ) = \cosh (χ) I_{4} + \sinh (χ) (\begin{matrix} 0 & 1 \\ 1 & 0 \end{matrix}) \oplus (\begin{matrix} 1 & 0 \\ 0 & - 1 \end{matrix}),

is the two-mode squeezing operation with parameter χ. Starting from the two-mode vacuum state |0〉_A|0〉_B, the output state from the NPC

\hat{U} {| 0 〉}_{A} {| 0 〉}_{B}

happens to be the bi-side single-mode-squeezing TMSV, which is equivalent to the squeezed TMSV state when moving the on-line single-mode squeezing operations to off-line source preparations in Ref. [21]. As shown in Eq.(8), the parameters for the single-side single-mode squeezing is independent of the squeezing parameter for the two-mode squeezing S_AB. This feature helps to provide much freedom in searching for optimal performance for subsequent entanglement distillation.

4. Performance in PS-based entanglement distillation

In this section we investigate the performance of the single-mode-squeezing TMSV state $\hat{U} {| 0 〉}_{A} {| 0 〉}_{B}$ in PS-based entanglement distillation schemes and show the state is favorable for the distillation schemes compared with the normal TMSV state.

Figure 2 depicts the typical PS-base entanglement distillation scheme, where a BS with transmittance coefficient T and a conventional on/off detector constitute the PS operation in one path, for instance, path B. The success of PS in this path is heralded when the detector clicks. This event physically equivalents to projecting the ongoing D mode to the non-vacuum subspace with the projection operator given by

{\prod^{^}}^{(on)} = I_{\infty} - | 0 〉 〈 0 | = \sum_{n = 1}^{\infty} | n 〉 〈 n | .

Fig. 2 PS-based distillation of the continuous variable entanglement generated from the χ ⁽²⁾ NPC. |0〉 represents the vacuum state. BS denotes the beam splitter.

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In the following, we first consider a single-side PS-based distillation scheme, namely, the PS operation is only in one path, and here without loss of generality, we let the PS be performed on the path B. The state evolution during the whole distillation can be conveniently given in phase space. The covariance matrix for the state of modes A, B and D after BS coupling between modes B and D follows [21]

V_{A B D} = (I_{A} \oplus S_{B D} (T)) [(S_{\hat{U}} \frac{1}{2} I_{A B} S_{\hat{U}}^{T}) \oplus \frac{1}{2} I_{D}] {(I_{A} \oplus S_{B D} (T))}^{T},

where S_BD(T) is the 4 × 4 symplectic matrix for BS. Then after a successful PS operation, the distilled state of modes A and B can be represented as

ρ_{d}^{(1)} = \frac{1}{P_{d}^{(1)}} {Tr}_{D} [ρ_{A B D} (I_{A B} \otimes {\prod^{^}}_{D}^{(on)})],

where

P_{d}^{(1)}

denotes the success probability. In analogy with the calculations developed in Ref. [21], we obtain

P_{d}^{(1)} ρ_{d}^{(1)} = ρ (V_{1}^{(1)}) - \frac{1}{\sqrt{\det (V_{D} + \frac{1}{2} I_{2})}} ρ (V_{2}^{(1)}),

where ρ(V) is the normalized density matrix of a gaussian state with its covariance matrix V and

V_{1}^{(1)} = V_{A B}

,

V_{2}^{(1)} = V_{A B} - σ {(V_{D} + I_{2} / 2)}^{- 1} σ^{T}

. The relevant matrices V_AB,V_D and σ are defined in partitioning the matrix V_ABD given by

V_{A B D} = (\begin{matrix} V_{A B} & σ \\ σ^{T} & V_{D} \end{matrix}),

where V_AB is a 4 × 4 matrix. The distillation probability of P_d is then obtained as

P_{d}^{(1)} = 1 - \frac{1}{\sqrt{\det (V_{D} + \frac{1}{2} I_{2})}} .

We use the logarithmic negativity [42, 43] as a measure to quantify the distilled entanglement, which is defined as $E_{N} (ρ_{A B}) \equiv \log_{2} ‖ ρ_{A B}^{T_{A}} ‖$ for a bipartite state ρ_AB with T_A denoting the partial transposition. In Fig. 3(a), we numerically characterize the logarithmic negativity and investigate the enhancement of distillation with single-mode squeezing S_A(χ) ⊗ S_B(χ), for the values of χ = 0.08, 0.16, 0.24 and choose T = 0.95 that was often used in practical experiments [16]. The distillation enhancement of the single-mode-squeezing TMSV state against the normal TMSV state (r = 0) can be found for small r values. The enhancement decreases as r becomes large. Optimal values r = 0.57, 0.5, 0.43 that maximize the distilled entanglement can be estimated. The corresponding success probabilities are shown in Fig. 3(b).

Fig. 3 (a)The distilled entanglement measured with logarithmic negativity and (b) the success probability (in logarithmic scale) of distilling the NPC-generated entanglement using the single-side PS scheme with T = 0.95.

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We then investigate the performance of the single-mode-squeezing TMSV state in the bi-side PS distillation scheme, where both outgoing modes A and B are performed a PS operation. The success of the distillation is heralded when two detectors fires simultaneously. The distilled entanglement is well determined by a linear combination of four Gaussian states [21],

P_{d}^{(2)} ρ_{d}^{(2)} = \sum_{j = 1}^{4} P_{j} ρ (V_{j}^{(2)}),

where P_j and

V_{j}^{(2)}

are given by Eq.(24) in Ref. [21]. We choose χ = 0.05, 0.1, 0.15, and evaluate the entanglement with the logarithmic negativity as a function of r. The numerical simulation of the entanglement is shown in Fig. 4(a). All numerical simulation is carried out within the photon number subspace spanned by |0〉 |1〉, …, |6〉, the dimension of which is large enough for numerical convergence. Compared with the single-side PS case, we can see a similar trend. The difference is the bi-side PS scheme provides a stronger entanglement due to the A-B symmetry of the state to be distilled. The distillation success probability is simulated in Fig. 4(b) which is substantially smaller compared with its single-side variants.

Fig. 4 (a) Entanglement of the bi-side distilled state measured with the logarithmic negativity and (b) the distillation success probability (in logarithmic scale) with parameter T = 0.95.

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Finally, we further study distillation of the mixed NPC-generated entangled state after bi-side amplitude-damping channel, explicitly, the state is obtained by sending two outgoing modes A and B to two independent lossy bosonic channels. This corresponds to the primary source of errors in current optic quantum communication. The bi-side amplitude-damping channel is a Gaussian channel and in the phase space, we can use a Gaussian map to simulate it. We can simply replace $S_{\hat{U}} \frac{1}{2} I_{A B} S_{\hat{U}}^{T}$ with

S_{\hat{U}} \frac{1}{2} I_{A B} S_{\hat{U}}^{T} \to η (S_{\hat{U}} \frac{1}{2} I_{A B} S_{\hat{U}}^{T}) + \frac{1 - η}{2} I_{A B},

where η is the transmittance efficiency. All the following evaluation during photon subtraction stays unchanged as Eqs. (12) and (16).

In Fig. 5, we numerically evaluate the distillation of mixed entanglement in different amplitude damping conditions for r = 0.2, 1, 3, 5, with η varying from 0.1 to 0.99. The results when r = 0.2 and 1 show that even in the presence of amplitude-damping noise, the single-mode-squeezing TMSV state can also be distilled with PS technique. Moreover, to show the advantage of the state, the distillation of normal TMSV state with the same two-mode squeezing parameter r = 0, χ = 0.1 is also included. It can be easily observed that in the distillation, the single-mode-squeezing TMSV state (marked with circle lines) when r = 0.2 is more favorable than the TMSV (marked with squared lines), for all range of η. However, as shown in the figure, when r increases to large values, the distillation performance may diminish and the entanglement could even be less than the state before distillation.

Fig. 5 Performance of bi-side PS distillation of amplitude-damped single-mode-squeezing TMSV state. We assume that the two-outgoing modes A and B are sent to two independent amplitude-damping channels, with each transmittance efficiency η. Other parameters are chosen as :χ = 0.1, T = 0.95. All quantum states are truncated within the photon number subspace spanned by |0〉, |1〉,⋯, |6〉, whose computation convergence is sufficiently verified.

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It is interesting to compare our scheme where the single-mode squeezing operation is before channel loss with an alternative scenario in Ref. [21], where a normal TMSV state is prepared and transmitted through lossy channels, and the local single-mode squeezing operations are applied only after the transmission. We give a numerical comparison in Fig. 6 based on bi-side distillation scheme. The circled line gives the bi-side PS distillation of the single-mode-squeezing TMSV state, whereas the dotted lines represent the result of squeezing-assisted distillation in Ref. [21]. We choose χ = 0.15 and evaluate the state evolution within the photon number subspace spanned by |0〉, |1〉,⋯, |6〉. Our result shows that the squeezing-assisted scheme in Ref. [21] is actually more robust and can improve distillation for a variety of channel transmittance efficiencies η = 0.95 (Fig. 6(a), (b)), η = 0.5 (Fig. 6(c), (d)), η = 0.1 (Fig. 6(e), (f)). However, the single-mode-squeezing TMSV states are only superior to the normal TMSV state for the bi-side PS distillation scheme in low-loss case shown in Fig. 6(a), (b). While in the high-loss case, as shown in Fig. 6(c), (d), (e), (f), the distilled entanglement monotonously decreases as r increases. The reason is that the transmission loss will damp the effect of local squeezing, and hence the effective distillation operation is loss-damped squeezing-assisted PS distillation. However, in the original scheme in Ref. [21], the single-mode squeezing operation is not disturbed by the loss. Consequently, the scheme in Ref. [21] is superior to the present scheme in lossy channels if one neglects the challenges in on-line squeezing operations.

Fig. 6 Comparison of the bi-side PS distillation performance between the amplitude-damped single-mode-squeezing TMSV state (circled lines) and the amplitude-damped TMSS state with single-mode squeezing operation applied after loss (dotted lines). Other parameter are chosen as χ = 0.15, T = 0.95. All quantum states are truncated within the photon number subspace spanned by |0〉, |1〉,⋯, |6〉, whose computation convergence is sufficiently verified.

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5. Discussions and conclusions

We would like to give a brief discussion on the experimental feasibility. Although as far as we know there is still not an experimental report for producing a CV entangled state from the 2D χ ⁽²⁾ NPC, numerical experiments have reports the optical parametric processes via the 2D χ ⁽²⁾ NPC [29, 31–33 , 35, 41] and efficient CV entanglement sources based on the 1D periodically poled crystals [26, 44]. Furthermore, the concurrent PDC processes of the NPC in our scheme has been observed experimentally in some sense [39–41 ]. We believe that these experimental investigations can guarantee the experimental feasibility of our scheme.

For crystal structure design, here taking the periodically poled lithium tantalate (LiTaO₃) crystal as an example, we give a structure working for wavelengths λ_p = 390nm, λ_s = λ_i = 780nm. Since the fiber collection efficiencies for the three squeezed lights highly depend on the realistic experimental settings including the light emitting angle, lenses, fiber diameter, collection distance, etc. Hence, here we roughly give an estimation by assuming the ratio of the collection efficiency for the two-mode squeezed light to that for the single-mode squeezed light as 1 : 10. Then to obtain a parameter r = 0.25, we can design the crystal structure as Λ_x = 4.479μm, Λ_y = 6.201μm, R = 1.120μm and the working temperature is 80^◦C. Note that in this design we utilize the reciprocal vectors ${\vec{G}}_{1, 0}$ , ${\vec{G}}_{1, 3}$ , and ${\vec{G}}_{1, - 3}$ , with the latter two being equivalent to but higher orders of ${\vec{G}}_{1, 1}$ and ${\vec{G}}_{1, - 3}$ in Eqs. 5 and 6. For the choice of the crystal length, it should be much shorter than the noncollinear length given in Ref. [45] to suppress the PDC spatial walk-off effect. This condition decreases the generation rate compared with the collinear QPM PDC sources. The noncollinear structure also makes the squeezing operated in the way of optical parametric oscillation like the experiment in Ref. [8]. However, using a pulse laser as the pump light for single-pass PDC like the experiment in Ref. [17] can get a strong enough nonlinearity. Compared with standard QPM collinear Type-II PDC process, besides the crystal length limitation due to the walk-off problem, the two-mode squeezed light emits as a cone, and thus only a fraction of the light could be collected. Hence the collinear process would give a higher degree of two-mode squeezing than the filtered conical output of the PDC process in our scheme. However, the standard collinear two-mode squeezing operation cannot be realized concurrently with two single-mode squeezing operations. While with the structure introduced in our work, the three squeezing operation can be realized simultaneously in a single crystal.

In conclusion, we have presented a method for generating the single-mode-squeezing TMSV states via a single χ ⁽²⁾ NPC that was designed to enable three concurrent QPM PDC processes. The three PDC processes play the role of two single-mode squeezing and a two-mode squeezing operations, respectively, and moreover, we showed the squeezing parameters are tunable for optimal requirement. Such states are favorable for existing PS-based entanglement distillation schemes. We demonstrated their improvement on the distilled entanglement and success probability compared with the normal TMSV states, in single- and bi-side PS schemes, respectively. We also showed the improved performance in the amplitude-damping case.

A significant merit of our scheme lies in the concurrent three squeezing operations in a single crystal, which releases the requirement of on-line local squeezing operation and hence greatly reduces the experimental complexity. Our scheme thus opens up a way for CV compact sources, and in particular, if combined with waveguide poling techniques [44] and interferometers [46], our approach may play an important role in active on-chip CV quantum information processing.

To the end, we have demonstrated the advantages of the single-mode-squeezing TMSV states against the normal TMSV states in the applications in the PS-based distillation schemes. It would be straightforward to reveal their improved performance in the related applications, such as quantum teleportation [12, 47] and a loophole-free Bell test [48, 49]. We hope our approach may stimulate more investigations on the single-mode-squeezing TMSV states.

Acknowledgments

This work was supported by the State Key Program for Basic Research of China (Grants No. 2011CBA00205 and 2012CB921802), the National Natural Science Foundations of China (NSFC) (Grants No. 11474050, No. 11204379), and by “the Fundamental Research Funds for the Central Universities” (Grant No. 2242014K40034).

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Scheme for generating distillation-favorable continuous-variable entanglement via three concurrent parametric down-conversions in a single χ ⁽²⁾ nonlinear photonic crystal

Abstract

1. Introduction

2. Overview of the NPC structure

3. Generation of single-mode-squeezing TMSV states

4. Performance in PS-based entanglement distillation

5. Discussions and conclusions

Acknowledgments

References and links

Cited By

Figures (6)

Equations (18)

Optics Express