Optical fiber design with orbital angular momentum light purity higher than 99.9%

Zhishen Zhang; Jiulin Gan; Xiaobo Heng; Yuqing Wu; Qingyu Li; Qi Qian; Dongdan Chen; Zhongmin Yang

doi:10.1364/OE.23.029331

1. Introduction

Orbital-angular-momentum (OAM) light has gained much interest for the potential application of greatly increasing the capacity of communication recently [1–3 ]. Its inherent orthogonality provides an additional degree of freedom for the multiplex technique. Compared with the free space telecommunication, the fiber optic network is more suitable for ultralong distance communication. Thus the OAM propagation in the fiber with low channel crosstalk is the primary consideration and fundamental research. Due to the confinement of the optical fiber waveguide, the OAM light no longer propagates as a whole. Instead, it is considered as the superposition of the eigenstates with the same orders ( $even mode \pm j \cdot odd mode$ [4]) and only can be the circular polarization state. In common fiber, some eigenstates for different OAM light are degeneracy in the weakly guiding waveguide, interactively couple along with light propagation and these coupling will cause severe crosstalk problem. S. Ramachandran et al. have indicated that a waveguide whose profile mirrors the electric field distributions of the OAM light, the annular structure, would be more suitable for breaking the near-degeneracy modes [5]. To our knowledge, till now most of schemes adopt the ring-fiber for supporting the OAM light [6–11 ]. Further increasing the refractive index difference (from 10⁻³ to 10⁻¹) is also conducive to separating the near-degeneracy modes and increasing the number of OAM modes supported by a fiber core. For example, C. Brunet et al. have used an annular index profile and an air core to excite 9 orders of the OAM light (36 modes) [10].

Due to the large index difference, the OAM light synthetic formula ( $even mode \pm j \cdot odd mode$ ) is calculated without the weakly guiding approximation and the detailed deduction of the formula is presented in Section 2. Consider only the plus situation (or the minus) of the synthetic formula, the exact solution is no longer a pure OAM light with single charge number. Instead, two OAM light beams are synthesized at one time with different circularly polarizations and different charge numbers, as show in Fig. 1(a) . These two OAM light beams will simultaneously propagate along the fiber with the fixed ratio. In a weakly guiding fiber, the weight of one synthesized OAM light is nearly to zero and the other is almost 100% pure. However, both the synthesized OAM lights are no longer at extreme proportion (0% and 100%) in a large index-difference fiber. S. Golowich et al. found the same phenomenon in the strong spin-orbit coupling [12] fiber with a high-contrast interface in the refractive index profile. So when injected into the fiber, a pure OAM light will excite the corresponding fiber mode and finally split into two different OAM lights, as shown in Fig. 1(b). Thus the intrinsic crosstalk is already induced in this process. Because the encoding and decoding of signals are based on the modulation parameters of the OAM light (such as intensity, polarization or charge number), this intrinsic crosstalk will greatly deteriorate the communication quality. From the above we can see that the large-index-difference fibers will significantly increase the intrinsic crosstalk of the OAM optical fiber communication networks. S. Ramachandran et al. thought it may be a fundamentally insurmountable challenge [13] for the ring-fiber used for the OAM optical fiber communication system.

Fig. 1 (a) The OAM light synthetic formula in fibers. The dash arrows represent the circularly polarizations (left or right); the spiral sectors represent the charge numbers( $+ l - 1$ , $+ l + 1$ , $- l + 1$ , and $- l - 1$ , where l is the order number of the fiber mode). (b) The intrinsic crosstalk of the OAM optical fiber. LCP: the left-handed circularly polarized light; RCP: the right handed circularly polarized light; $l \pm 1$ represent the charge numbers.

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The enhancement of the purity of the synthesized OAM light benefits to solving this intrinsic crosstalk problem caused by the fiber structure. Here the purity is defined as the power weight of the major OAM light in the all synthesized OAM lights and the intrinsic crosstalk is directly determined by the lowest purity of all the synthesized OAM lights in fiber. This intrinsic crosstalk is a theoretical limit of the practical crosstalk of the OAM optical fiber communication networks. If the target intrinsic crosstalk level is −30 dB, the needed lowest purity of the synthesized OAM light is higher than 99.9%.

Therefore, it is critical to enhance the purity when designing the OAM supporting fiber. In this paper, we first theoretically explore the relationship between the fiber structure and the purity of synthesized OAM light. We find the graded-index fiber structure has higher purity than the step-index fiber and the annular OAM fiber. A graded-index fiber is designed for supporting 16 modes (10 OAM modes), all the purities of synthesized OAM lights are higher than 99.9%, and the intrinsic crosstalk has been greatly alleviated to be lower than −30 dB.

2. Theory

In the axially symmetric optical fiber, E_z and H_z satisfy the following equations in a cylindrical coordinate system [14]:

\begin{array}{l} \frac{\partial^{2} E_{z}}{\partial r^{2}} + \frac{1}{r} \frac{\partial E_{z}}{\partial r} + \frac{1}{r^{2}} \frac{\partial^{2} E_{z}}{\partial φ^{2}} + [k^{2} n {(r)}^{2} - β^{2}] E_{z} = 0 \\ \frac{\partial^{2} H_{z}}{\partial r^{2}} + \frac{1}{r} \frac{\partial H_{z}}{\partial r} + \frac{1}{r^{2}} \frac{\partial^{2} H_{z}}{\partial φ^{2}} + [k^{2} n {(r)}^{2} - β^{2}] H_{z} = 0 \end{array}

where k, n(r) and β are the wave number in vacuum, the optical fiber refractive index profile and the propagation constant of the fiber, respectively. Then the transverse electromagnetic fields are related to E_z and H_z as follows:

\begin{array}{l} E_{r} = - \frac{j}{[k^{2} n {(r)}^{2} - β^{2}]} [β \frac{\partial E_{z}}{\partial r} + \frac{ω μ_{0}}{r} \frac{\partial H_{z}}{\partial φ}] \\ E_{φ} = - \frac{j}{[k^{2} n {(r)}^{2} - β^{2}]} [\frac{β}{r} \frac{\partial E_{z}}{\partial φ} - ω μ_{0} \frac{\partial H_{z}}{\partial r}] \\ H_{r} = - \frac{j}{[k^{2} n {(r)}^{2} - β^{2}]} [β \frac{\partial H_{z}}{\partial r} - \frac{ω μ_{0} n {(r)}^{2}}{r} \frac{\partial E_{z}}{\partial φ}] \\ H_{φ} = - \frac{j}{[k^{2} n {(r)}^{2} - β^{2}]} [\frac{β}{r} \frac{\partial H_{z}}{\partial φ} + ω ε_{0} n {(r)}^{2} \frac{\partial E_{z}}{\partial r}] \end{array}

where ω, ε₀ and μ₀ are the angular frequency, the permittivity and the permeability of vacuum. From Eq. (2), the

\partial E_{z} / \partial r

and

\partial H_{z} / \partial φ

(or

\partial H_{z} / \partial r

and

\partial E_{z} / \partial φ

) have the same φ dependencies. Thus E_z and H_z are expressed by

\begin{array}{l} E_{z} (r) = A F_{l z} (r) f (l φ) \\ H_{z} (r) = C F_{l z} (r) g (l φ) \end{array}

where

f (l φ) = {\begin{matrix} \cos (l φ), even modes \\ \sin (l φ), odd modes \end{matrix}; g (l φ) = {\begin{matrix} - \sin (l φ), even modes \\ \cos (l φ), odd modes \end{matrix}

[15]. A and C are the undetermined coefficients. F_lz(r) is the radial dependence of the field profile and l is the mode order.

In optical fibers, the OAM light is synthesized by $even mode \pm j \cdot odd mode$ . Thus the transverse electric field components of the OAM light are obtained.

\begin{array}{l} V_{r}^{\pm} \equiv E_{r}^{e} \pm j E_{r}^{o} \\ = - \frac{j}{[k^{2} n {(r)}^{2} - β^{2}]} [β A \frac{\partial F_{l z} (r)}{\partial r} - \frac{ω μ_{0}}{r} C l F_{l z} (r)] e^{\pm j l φ} \end{array}

\begin{array}{l} V_{φ}^{\pm} \equiv E_{φ}^{e} \pm j E_{φ}^{o} \\ = - \frac{j}{[k^{2} n {(r)}^{2} - β^{2}]} [\frac{β}{r} A l F_{l z} (r) - ω μ_{0} C \frac{\partial F_{l z} (r)}{\partial r}] \cdot (\pm e^{\pm j l φ}) \end{array}

In order to facilitate the analysis, we define $s \equiv \frac{ω μ_{0} C}{β A}$ , $P (r) \equiv \frac{1}{2} [\frac{\partial F_{l z} (r)}{\partial r} + \frac{l}{r} F_{l z} (r)]$ , and $Q (r) \equiv \frac{1}{2} [\frac{\partial F_{l z} (r)}{\partial r} - \frac{l}{r} F_{l z} (r)]$ . Here, note that s is the constant for a specific optical fiber refractive index profile. $V_{r}^{\pm}$ and $V_{φ}^{\pm}$ have been transformed into $V_{x}^{\pm}$ and $V_{y}^{\pm}$ in the following equations because the Cartesian coordinate provides a much more indicative view of the vector superposition.

\begin{array}{l} V_{x}^{\pm} = V_{r}^{\pm} \cos φ - V_{φ}^{\pm} \sin φ \\ = - \frac{j β A}{[k^{2} n {(r)}^{2} - β^{2}]} [(1 + s) P (r) e^{\pm j (l - 1) φ} + (1 - s) Q (r) e^{\pm j (l + 1) φ}] \end{array}

\begin{array}{l} V_{y}^{\pm} = V_{r}^{\pm} \sin φ + V_{φ}^{\pm} \cos φ \\ = \pm \frac{β A}{[k^{2} n {(r)}^{2} - β^{2}]} [(1 + s) P (r) e^{\pm j (l - 1) φ} - (1 - s) Q (r) e^{\pm j (l + 1) φ}] \end{array}

Thus the expression for the OAM light is written as

\begin{array}{l} V^{\pm} = \frac{β A}{[k^{2} n {(r)}^{2} - β^{2}]} \times \\ [(\pm \vec{y} - j \vec{x}) (1 + s) P (r) e^{\pm j (l + 1) φ} + (\mp \vec{y} - j \vec{x}) (1 - s) Q (r) e^{\pm j (l - 1) φ}] \end{array}

where

\vec{x}

and

\vec{y}

are two orthogonal unit vectors of Cartesian coordinate.

From Eq. (6), it is easy to see that the synthesized OAM light in fibers consists of two parts: the left-handed circularly polarized (LCP) light and the right handed circularly polarized (RCP) light. The difference between their charge numbers is 2. If $s = \pm 1$ , the synthesized OAM light only has a single component with a single charge number, which means the purity of 100%. The nearer s² gets close to 1, the higher the purity of the synthesized OAM light is.

Considering an arbitrary point r at the cross-section of the fiber, the left limit and right limit of refractive index at point r are n(r^-) and n(r⁺). The difference between n(r^-) and n(r⁺) means the change of the refractive index at point r. E_z should be continuous [15], i.e.

E_{z} (r^{+}) = E_{z} (r^{-})

By substituting Eq. (3) into Eq. (7),

F_{l z} (r^{+}) = F_{l z} (r^{-})

Similarly, E_φ and H_φ should be continuous [15], i.e.

E_{φ} (r^{+}) = E_{φ} (r^{-})

H_{φ} (r^{+}) = H_{φ} (r^{-})

From Eq. (9-a), we have

\begin{array}{l} β A l {\frac{1}{[k^{2} n {(r^{+})}^{2} - β^{2}]} - \frac{1}{[k^{2} n {(r^{-})}^{2} - β^{2}]}} \\ = ω μ_{0} C {\frac{1}{[k^{2} n {(r^{+})}^{2} - β^{2}]} \frac{\partial F_{l z} (r^{+})}{\partial r} \frac{r}{F_{l z} (r^{+})} - \frac{1}{[k^{2} n {(r^{-})}^{2} - β^{2}]} \frac{\partial F_{l z} (r^{-})}{\partial r} \frac{r}{F_{l z} (r^{+})}} \end{array}

Thus the expression of s² is obtained.

\begin{array}{l} s^{2} = {\frac{1}{[k^{2} n {(r^{+})}^{2} - β^{2}]} - \frac{1}{[k^{2} n {(r^{-})}^{2} - β^{2}]}}^{2} l^{2} \times \\ {\frac{1}{[k^{2} n {(r^{+})}^{2} - β^{2}]} \frac{\partial F_{l z} (r^{+})}{\partial r} \frac{r}{F_{l z} (r^{+})} - {\frac{1}{[k^{2} n {(r^{-})}^{2} - β^{2}]} \frac{\partial F_{l z} (r^{-})}{\partial r} \frac{r}{F_{l z} (r^{+})}}}^{- 2} \end{array}

By multiplying Eq. (9-a) and Eq. (9-b), we have

\begin{array}{l} β^{2} {\frac{1}{[k^{2} n {(r^{+})}^{2} - β^{2}]} - \frac{1}{[k^{2} n {(r^{-})}^{2} - β^{2}]}}^{2} l^{2} \\ = ω^{2} ε_{0} μ_{0} {n {(r^{+})}^{2} \frac{1}{[k^{2} n {(r^{+})}^{2} - β^{2}]} \frac{\partial F_{l z} (r^{+})}{\partial r} \frac{r}{F_{l z} (r^{+})} - \\ n {(r^{-})}^{2} \frac{1}{[k^{2} n {(r^{-})}^{2} - β^{2}]} \frac{\partial F_{l z} (r^{-})}{\partial r} \frac{r}{F_{l z} (r^{+})}} \times \\ {\frac{1}{[k^{2} n {(r^{+})}^{2} - β^{2}]} \frac{\partial F_{l z} (r^{+})}{\partial r} \frac{r}{F_{l z} (r^{+})} - \frac{1}{[k^{2} n {(r^{-})}^{2} - β^{2}]} \frac{\partial F_{l z} (r^{-})}{\partial r} \frac{r}{F_{l z} (r^{+})}} \end{array}

Here, note that

ω^{2} ε_{0} μ_{0} = k^{2}

\begin{array}{l} \frac{β^{2}}{k^{2}} {\frac{1}{[k^{2} n {(r^{+})}^{2} - β^{2}]} - \frac{1}{[k^{2} n {(r^{-})}^{2} - β^{2}]}} \\ = \frac{{(r^{+})}^{2}}{[k^{2} n {(r^{+})}^{2} - β^{2}]} - \frac{{(r^{-})}^{2}}{[k^{2} n {(r^{-})}^{2} - β^{2}]} \end{array}

So the Eq. (12) is simplified and expressed as

\begin{array}{l} l^{2} {\frac{1}{[k^{2} n {(r^{+})}^{2} - β^{2}]} - {[\frac{n (r^{-})}{n (r^{+})}]}^{2} \frac{1}{[k^{2} n {(r^{-})}^{2} - β^{2}]}} \times \\ {\frac{1}{[k^{2} n {(r^{+})}^{2} - β^{2}]} - \frac{1}{[k^{2} n {(r^{-})}^{2} - β^{2}]}} \\ = {\frac{1}{[k^{2} n {(r^{+})}^{2} - β^{2}]} \frac{\partial F_{l z} (r^{+})}{\partial r} \frac{r}{F_{l z} (r^{+})} - {[\frac{n (r^{-})}{n (r^{+})}]}^{2} \frac{1}{[k^{2} n {(r^{-})}^{2} - β^{2}]} \frac{\partial F_{l z} (r^{-})}{\partial r} \frac{r}{F_{l z} (r^{+})}} \times \\ {\frac{1}{[k^{2} n {(r^{+})}^{2} - β^{2}]} \frac{\partial F_{l z} (r^{+})}{\partial r} \frac{r}{F_{l z} (r^{+})} - \frac{1}{[k^{2} n {(r^{-})}^{2} - β^{2}]} \frac{\partial F_{l z} (r^{-})}{\partial r} \frac{r}{F_{l z} (r^{+})}} \end{array}

If the optical fiber refractive index profile changes slowly, that is $n (r^{-}) \approx n (r^{+})$ , we have

{[\frac{n (r^{-})}{n (r^{+})}]}^{2} \approx 1

By substituting Eq. (16) into Eq. (15), we have

\begin{array}{l} {\frac{1}{[k^{2} n {(r^{+})}^{2} - β^{2}]} - \frac{1}{[k^{2} n {(r^{-})}^{2} - β^{2}]}}^{2} l^{2} \\ = {\frac{1}{[k^{2} n {(r^{+})}^{2} - β^{2}]} \frac{\partial F_{l z} (r^{+})}{\partial r} \frac{r}{F_{l z} (r^{+})} - {\frac{1}{[k^{2} n {(r^{-})}^{2} - β^{2}]} \frac{\partial F_{l z} (r^{-})}{\partial r} \frac{r}{F_{l z} (r^{+})}}}^{2} \end{array}

Comparing the Eq. (17) with Eq. (11), we finally have

s^{2} = 1

Based on Eq. (16) and Eq. (18), the slower the optical fiber refractive index profile changes, the nearer s² approximates to 1. The step-index fiber always have some discontinuous area with relative large index difference, which results in the value of s² no longer approaching to 1 and the purity of synthesized OAM light deteriorating seriously. However, the discontinuous area can be completely avoided by using graded-index fibers with the continuous refractive index profile. So the graded-index fibers can support the OAM light propagation with higher purity and lower intrinsic crosstalk.

3. Fiber design and results

Here the refractive index profile of the graded-index optical fiber is supposed as $n (r) = Δ n \times [1 - {(\frac{r}{r_{0}})}^{α}] + n_{1}$ , where ∆n, r₀, n₁ and α are the maximum refractive index difference, the core radius, the cladding index and the shape factor, respectively. The key of the OAM fiber design is to achieve that the differences among the effective indices of modes are larger than 1 × 10⁻⁴, which will greatly decrease the mode coupling [4]. A simple method is to design a relative large refractive index difference ∆n which obviously breaks the weakly guiding approximation. n₁ is set to 1.460. Then ∆n and r₀ are sweeped to achive enough modes and enough difference between the effective indices of modes for the step-index fiber ( $α = \infty$ ). At last, α is optimized through the chromatic dispersion of the fiber modes and r₀ is refined at the same time. By design, the highest-order mode of this fiber is HE₄₁ because it is difficult to break the degeneracy of LP₁₂ and needs more trade-offs. The final fiber refractive index profile is shown in Fig. 2 .

Fig. 2 The refractive index profile of the graded-index fiber.

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The effective refractive indices of the fiber eigenmodes are calculated at C-band by using simulation software (COMSOL Multiphysics). This fiber supports 9 different order fiber eigenmodes and can support 16 modes (10 OAM modes) to transmit information. As shown in Fig. 3(a) , the minimum difference of the effective refractive index is 1.16 × 10⁻⁴ between TM₀₁ and HE₂₁, which decreases the inter-mode crosstalk to very little. Figure 3(b) shows the chromatic dispersion of the fiber modes. Over the whole C-band, all the mode dispersions are less than 35 ps/(km∙nm), and the difference between two arbitrary modes is less than 40 ps/(km∙nm). Although the dispersion of designed fiber are slightly greater than Corning SMF-28e (less than 18 ps/(km∙nm) at 1550 nm), this dispersion property is nearly the same with the reported few-mode fibers [17] and OAM ring-fibers [9, 10 ]. The dispersion of this designed fiber is suitable for long distance fiber communication using.

Fig. 3 (a) The effective refractive indices and (b) the chromatic dispersion of the fiber modes.

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The major purpose of this designed fiber is to achieve the high purity OAM light and decrease the channel crosstalk for the OAM optical fiber communication system. Therefore, we first qualitatively analyze the purity of the OAM light through its intensity distribution and polarization characteristics, as show in Fig. 4 . The parameters of the ring-fiber are the same as described in the literature [6], which breaks the weakly guiding approximation with $Δ n = 0.05$ . It is interesting that both the intensity distributions of OAM lights, which are synthesized by the graded-index fiber and the ring fiber, are no longer the uniform circle because the purities are not 100%. Instead, they are cyclical variations in azimuthal direction, as shown in Fig. 4(b) and 4(c). The polarization characteristics of the synthesized OAM light are shown in Fig. 4(e)-(f). The red circles are the core boundary of the fiber, the blue ellipse shape represents the polarization characteristics of the OAM light and the ellipse area means the energy amount in that region. From Eq. (6), the synthesized OAM light in fiber consists of LCP light and the RCP light, which means the purity of the OAM lights is inversely proportional to the ellipticities of polarization structures. In Fig. 4(d), the blue ellipse shapes are nearly the perfect circles, which mean the light synthesized by $H E_{11}^{e} \pm j \cdot H E_{11}^{o}$ in the graded-index fiber is almost just the LCP light (or the RCP light). So the synthesized light is nearly pure and only contains a single charge number. However, the ellipticities of the blue ellipses in Fig. 4(e) are slightly larger, which means the light synthesized by $E H_{11}^{e} \pm j \cdot E H_{11}^{o}$ in the graded-index fiber is composed of the LCP light and the RCP light together. Thus this synthesized light is less pure and contains two different charge numbers. It is easy to see that the ellipticities in Fig. 4(f) is even larger which means the synthesized OAM lights in the ring fiber are less pure and the intrinsic crosstalk is much worse.

Fig. 4 The intensity distributions and polarization characteristics of the synthesized OAM light.

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In order to further quantitatively analyze the purity of the synthesized OAM light, the power weight of the OAM light with two different charge numbers are calculated [17–19 ] through processing the integral of two parts on the right hand side of Eq. (6). As shown in Fig. 5 , all the synthesized OAM light by the graded-index fiber eigenmodes have the purity high than 99.9%, which is equivalent to that the intrinsic crosstalk is lower than −30 dB. The highest and lowest purities of the OAM light, which are respectively synthesized by HE₁₁ and EH₁₁, are 99.9998% and 99.9142%. For comparison, the ring-fiber is calculated with the same method (the parameters of the ring fiber are the same as described in the literature [6]). The lowest purity is 98.4515% synthesized by HE₁₁ and the intrinsic crosstalk of this ring-fiber is −18 dB.

Fig. 5 The purity of the synthesized OAM light in the designed graded-index fiber and the ring-fiber [6].

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From the results mentioned above, this designed graded-index fiber can achieve the low dispersion, high purity and low intrinsic crosstalk. Still there is a main concern of the index contrast ∆n up to 0.1, which is relative high for the existing commercial fiber fabrication. When the refractive index contrast ∆n decreases to 0.05, the designed fiber still owns the intrinsic crosstalk less than −39 dB. However, at this situation the modes number will decrease to 10 (6 OAM modes) and the minimum difference of the effective refractive index is down to 0.65 × 10⁻⁴, which is still enough to break the near-degeneracy modes [11]. If we need to support more OAM modes, the refractive index contrast needs to be adjusted higher than 0.05. Fortunately, there are some materials and schemes which can achieve the high index contrast with the graded-index distribution. For example, the refractive index of the composition of GeO₂-SiO₂ glasses can be adjusted from 1.444 to 1.587 at 1550 nm and can achieve ∆n to 0.14 [20]. Here, the optical loss of the fiber with high index contrast needs to be improved with more research. Finally, we believe that our proposed graded-index OAM supported fiber is a viable candidate for future ultralong distance OAM multiplexing fiber networks.

4. Conclusion

In conclusion, we theoretically prove that the slower the optical fiber refractive index profile changes, the higher purity of the synthesized OAM light is. And then we propose and design the graded-index fiber for the OAM light. A specific designed graded-index optical fiber supports 16 modes (10 OAM modes) to transmit information and the dispersion of all modes are less than 35 ps/(km∙nm). The purity is higher than 99.9% and the intrinsic crosstalk is suppressed to be lower than −30 dB. This designed graded-index fiber supports multiple OAM light propagation with low crosstalk, and can be potentially used for the long-haul OAM optical communication system.

Acknowledgment

This research was supported by the China State 863 Hi-tech Program (2012AA041203), National Natural Science Foundation of China (NSFC) (51302086, 61575064), the Fundamental Research Funds for the Central Universities (2014ZM0033), and National Science Fund for Distinguished Young Scholars of China (61325024).

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Optical fiber design with orbital angular momentum light purity higher than 99.9%

Abstract

1. Introduction

2. Theory

3. Fiber design and results

4. Conclusion

Acknowledgment

References and links

Cited By

Figures (5)

Equations (21)

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