We report an analytic model for quantitatively calculating the transmission attenuation of single-wall hollow-core anti-resonant fibers. Our calculations unveil the light leakage dependences on azimuthal angle, polarization, and geometrical shape and have been examined in a variety of fiber geometries. Based on our model, a simple and clear picture about light guidance in hollow-core lattice fibers is presented. Formation of equiphase surface at fiber’s outermost boundary and light emission ruled by Helmholtz equation in transverse plane constitute the basis of this picture. Using this picture, we explain how the geometrical shape of a single-wall hollow-core fiber influences its transmission properties.
© 2014 Optical Society of America
Hollow core micro-structured optical fibers (HC-MOFs), consisting of air core surrounded by arrangement of micron-scaled silica webs, have recently been proposed as unique hosts for both photons and vaporous gases. Numerous applications in fiber optics and light-matter interactions, such as optical frequency comb generation , optical soliton pulse shaping , high-power laser delivery , and Terahertz wave transmission , to mention just a few, severely rely on efficient light guidance in such fibers whose most important parameters are transmission attenuation and bandwidth. According to different waveguiding mechanisms, HC-MOFs can be categorized to two classes with the first one, i.e. the hollow-core photonic bandgap fiber (HC-PBGF) , being result of photonic bandgap (PBG) effects in fiber’s cladding area and the second one, i.e. hollow-core anti-resonant fiber (HC-ARF) , for example Kagome-lattice photonic crystal fiber , being result of anti-resonant light reflections occurring at core-cladding interface . Opening PBG leads to annihilation of propagating optical states in fiber’s cladding, which enables light confinement inside the air core. In such kind of fibers, loss figure as low as 1.2dB/km has been demonstrated , and the PBG-induced light confinement becomes stronger as the number of cladding layer increases . In contrast, HC-ARFs only have spectral regions with low density of optical states (DOS) . Adding cladding layer has minor and vague effects on transmission attenuation [12–14]. However, in aspect of transmission bandwidth, the HC-ARF substantially surpasses the HC-PBGF. At the expense of attenuation coefficient (two orders of magnitude worse), the HC-ARF can provide much broader transmission band (one order of magnitude wider) than the HC-PBGF. A widely accepted explanation to this tradeoff is that opening a full two- dimensional photonic bandgap in fiber’s cladding area benefits light confinement in core but imposes additional restrictions on the operating wavelength . Obtaining advantages in both aspects seems difficult.
However, this knowledge about HC-MOF is now gradually changing with the appearance of low-loss HC-ARFs, especially when several groups reported that a hypocycloid shape , or negative curvature , core-surround can efficiently lower the attenuation coefficient while maintaining the transmission bandwidth. These findings revive the investigations on the guidance mechanism of HC-ARF. But, previous efforts, such as low DOS in fiber’s cladding area , radial light confinement induced by concentric glass rings , and spatial power overlap between core mode and cladding glass , only qualitatively explain part of the transmission attenuation properties. A quantitative calculation of the influence of geometrical shape to the attenuation spectrum is urgently needed. In this paper, we present an analytic model for quantitatively calculating attenuation spectra of single-wall HC-ARFs. We relate the attenuation coefficient to the integral of leaked energies in all the transverse directions. Our analytic model is inspired by the leaky mode solution in one-dimensional (1D) slab waveguide [18–20]. By using a proper geometry transformation, our analytical treatments can be extended from 1D slab geometry to 2D circular ring fiber and then to 2D polygon shape fibers with gradually reduced structural symmetry. In principle, our analytic model can be employed in arbitrary single-wall HC-ARFs.
This paper is organized as follows. In section 2, a wave equation and its solution of leaky mode in an M-type 1D slab waveguide are presented. The relationship between the attenuation coefficient and the field amplitude at the outermost boundary is derived under a picture of oblique plane wave radiation. In section 3, the equivalence of 2D circular ring fiber and 1D slab waveguide is discussed. A geometry transformation and an analytic calculation approach are presented. In section 4, our model is tested in different shapes of single-wall HC-ARFs, i.e. the regular triangle, square, hexagon, octagon, and hypocycloid. Electric fields and attenuation spectra are calculated and are compared with numerically simulated results. The light leakage dependences on azimuthal angle, polarization, and geometrical shape are analyzed. In the last section, fundamental principles and further developments of our model are summarized before conclusions are drawn.
2. Leaky mode in M-type slab waveguide
Our studies begin with calculating propagation constant of leaky mode in an M-type slab waveguide . We limit our investigations in fundamental core modes, whose field profiles in the core area are peaked at the central axis.
As shown in Fig. 1(a), the core (with the thickness of 2ain) and the surrounding areas of the slab waveguide are filled with air (n1 = 1), while the cladding consists of single layer of silica with the refractive index n2 = 1.45 and the thickness t = 0.67 μm. From the Helmholtz equation, , the electric field distributions in this slab can be written as ,Fig. 1(a). The propagation constant in the z direction, β, equals to with k0 the propagation constant in vacuum and neff the modal index. The transverse wave-vector number kxj is defined as . The complex field amplitude A, the phase ζ, and the eigenvalue equation relevant to β can be derived from boundary continuity conditions . In the rest of this paper, the eigenvalue equation of β, which can be found in many textbooks, will be frequently mentioned and belongs to one part of our analytic calculation approach.
As to the leaky mode character of Eq. (1), we stress that the field in the surrounding area only contains outward-propagating wave component, whereas the fields in the core and the cladding areas are standing waves. Additionally, the transverse wave-vector number in the surrounding area, kx1, has a non-zero imaginary part, leading to an exponential growth of field amplitude as . Strictly speaking, for a dielectric waveguide, a finite number of discrete guided modes and a continuum of radiation modes constitute the complete set of orthogonal basis. Leaky modes, which are mathematical solutions under the assumption that no inward-propagating wave component exists in the outermost layer, are not members of this orthogonal basis. However, detailed analyses have verified the equivalence between this mathematical solution of leaky mode and the realistic physical process of energy diffusion in the radiation mode continuum .
Figure 1(b) plots the field amplitudes and phases of the leaky modes calculated in a slab waveguide. For the p-polarization, only the transverse electric field, Ex, is presented. The longitudinal field component, Ez, is weak and ignored. Note that, in these calculations, the modal index, neff, is regarded as a complex number with its imaginary part proportional to the attenuation coefficient, . Moreover, the complex nature of neff can also be seen in the field amplitude profiles, which grow exponentially in the direction transverse to the propagation as illustrated by the blue dashed line in Fig. 1(b).
The field distribution of the leaky mode, i.e. Equation (1), can also be understood with the energy conservation. In a lossless dielectric waveguide, energy decrease in the longitudinal direction, due to attenuation, should be equal to energy leakage in the transverse directions. As illustrated in the insert of Fig. 2, two outward-inclined plane waves lie in the surrounding area and satisfy the phase-matching condition in the longitudinal direction. The oblique angle of these plane waves can be approximately estimated, and a formula relating the complex modal index and the field amplitude at the outermost boundary can be derived,Eq. (2), we read out the field amplitudes at the outermost boundary from Fig. 1(b) (marked by the blue circles), estimate , and then derive imaginary parts of the modal indices for the s- and p-polarization waves to be 1.08 × 10−4 and 4.77 × 10−4 respectively, which agree well with the precisely calculated results based on the eigenvalue equation [Im(neff) = 1.08 × 10−4 and 4.73 × 10−4 respectively]. More importantly, by reversely using Eq. (2), we can also estimate the field amplitude at outermost boundary of the waveguide from the complex modal index. Below, this method will be frequently used. In contrast, obtaining full field distribution by solving Eq. (1) requires explicit geometrical information besides the knowledge of modal index.
To exhibit spectral properties of this M-type slab waveguide, Fig. 2 plots the attenuation spectra of the fundamental core modes for both the s- and p-polarizations. It is seen that the two high-loss wavelength regions, determined by the anti-crossings between the fundamental core modes and different order glass confined modes, appear at λ0 ≈0.7 and 1.4 μm. The m-th order resonant wavelength, [8,19], has little relationship with the polarization. However, inside each transmission band, the p-polarization wave exhibits much worse light confinement than the s-polarization, implying that, in the case of a hybrid polarization wave, the p-polarization component will play the primary role in light leakage. This anti-resonant reflecting optical waveguide (ARROW) mechanism has been well studied in the context of planar waveguides  and fiber geometries . Figure 2 also plots the phases of the electric fields at the outermost boundary. Surprisingly, these phases seem only determined by the anti-resonant order of the transmission band, irrelevant to polarization and geometrical dimension (data not shown). This phase locking effect is very important and will be used in the following treatment.
Briefly speaking, all the information about the electric field at the outermost boundary of an M-type slab waveguide can be quickly obtained once the complex modal index of the leaky mode is known. The field amplitude can be estimated from Eq. (2), and its phase is fixed to a constant value only relevant to the anti-resonant order of the transmission band.
3. Analogy between single-wall circular ring fiber and M-type slab waveguide
Comparing with slab waveguide, a circular ring fiber loses the continuous translation symmetry in the y direction. The continuous rotational symmetry of the geometry is also impaired by the linear polarization of the fundamental core mode. To solve this problem, a reasonable geometry transformation from 2D geometry to 1D slab may be helpful.
Figure 3(a) depicts cross section of a circular ring fiber (with the inner radius ain and the glass thickness t) in a cylindrical coordinate. The radial light leakage through the glass wall at any azimuthal angle ϕ [denoted by the green arrow in Fig. 3(a)] is supposed to be equivalent to that in a slab waveguide having the inner radius r’ and the glass thickness t. We assume and rotate the glass segment to parallel to the y axis. The first procedure is based on the consideration that, as a result of geometry transformation, the transverse light confinement of the core mode will be relaxed from the x-y plane (2D) to the x direction (1D). The second procedure will be accompanied with a rotation of polarization direction having the same degree. The whole geometry transformation is depicted in the insert of Fig. 3(a).
After this geometry transformation, the complex modal indices of the leaky modes, , and the field amplitudes at fiber’s outermost boundary, , can be quickly solved in the equivalent slab structure, whose inner radius and glass thickness are and t respectively.Equation (3) is equivalent to Eq. (2). The additional factor is because of the geometry transformation from 2D to 1D.
Taking into account the proportions of s- and p-polarization components in different angular segments (cos2ϕ and sin2ϕ, respectively), the overall modal index of the whole fiber can be expressed as an arithmetical average,
In order to obtain the overall attenuation coefficient, we exploit the phase-matching condition in the longitudinal direction and the leaky mode characteristics in the surrounding area. According to the phase-matching condition, the 2D Helmholtz equation in the transverse plane can be written as . And, according to the leaky mode characteristics, the far-field radiation at any azimuthal angle, ξ [the pink arrow in Fig. 3(a)], can be obtained by integrating the electric field along fiber’s outermost boundary. The amplitudes of these electric fields are products of Eq. (3) and cosϕ or sinϕ, depending on different polarizations [depicted in Fig. 3(a)], and the phases of these fields are fixed (Fig. 2). Therefore, the overall attenuation coefficient of the circular ring fiber can be expressed as,Equation (5a) represents the energy conservation in a leaky mode, Eq. (5b) averages the contributions from all the radiation directions and from both the s/p-polarizations, and the integral of Eq. (5c) takes into account the angle between x/y-axis and s/p-polarization (ϕ), the inclination angle of each glass segment relative to the radiation direction (ξ - ϕ), and the phase delay in the radiation direction (ξ). The transverse wave-vector, , utilizes the result of Eq. (4). A closed loop integral in Eq. (5c) is implemented along fiber’s outermost boundary and can be understood by Green’s theorem. Detailed derivation of Eq. (5) is given in the Appendix.
Our model is then compared with precisely calculated result of single-wall circular ring fiber. The precise calculation is carried out by using Bessel functions and standard transfer matrix technique . In the calculation, the azimuthal number of the Bessel functions is set to be 1, corresponding to the HE11-like leaky core mode, and the surrounding area only contains the outward-propagating Hankel function. Figure 3(b) shows that, with respect to the attenuation coefficient, a very good agreement between our model and precise calculation is achieved in a broad wavelength range. This result verifies that the geometry transformation proposed in Fig. 3(a) is reasonable. Our analytic model not only demonstrates the capability of quantitatively calculating attenuation coefficient of single-wall circular ring fiber but also solves the problem of the symmetry decrease from 1D slab to 2D fiber, which is very useful in the following treatment. Note that the geometrical sizes and spectral range used in Fig. 3 are exactly the same with those in , and our results coincide with theirs as well.
4. Light leakage dependences on azimuthal angle, polarization, and geometrical shape
Our analytic model will be examined in more complicated geometries, i.e. regular polygon and hypocycloid shape single-wall hollow-core fibers. With the loss of cylindrical symmetry of circular ring fiber, precise calculation of attenuation coefficient becomes impossible. We, therefore, use a commercial finite-element mode solver (Comsol Multiphysics) to provide comparison for our model. The precision of our numerical simulation is kept better than 0.5% by choosing appropriate mesh size and perfectly matched layer (PML) configuration .
Figure 4(a) depicts cross section of a regular triangle single-wall fiber, whose inscribed radius is ain. In a cylindrical coordinate, the inner boundary of the fiber is expressed as . In order to construct an equivalent slab waveguide as Fig. 3(a) does, a hypothetical inner radius , a glass thickness t, and a polarization rotation (relative to the y axis) are employed as indicated in Fig. 4(a). Here, θ is a function of the azimuthal angle ϕ and stands for the tangential direction of glass segment. After geometry transformation, a series of equivalent slab waveguides are obtained at different azimuthal angles. Analytical calculation can then be applied to obtain the modal index and to estimate the field amplitude at fiber’s outermost boundary based on Eqs. (1)-(3).
The overall modal index, , is equal to the arithmetic mean of over different angular segments. The proportions of the s- and p-polarization components need to be taken into account.
Based on this overall modal index, the transverse wave-vector number in the surrounding area, , is derived and used in the 2D Helmholtz equation in the x-y plane. Integrating electric field along fiber’s outermost boundary yields the emission energy at any transverse angle ξ [the pink arrow in Fig. 4(a)]. Contributions from x- and y-polarizations should be both taken into account. Then, the overall attenuation coefficient of the fiber can be derived by integrating the radiations over all the transverse directions,Eq. (8c), the integral length element is derived from dL = dl’ = dl/sin(θ-ϕ) [see the insert in Fig. 4(a)], whereas, in Eq. (5c), the length element, dl, is equal to (ain + t)·δϕ.
Figure 4(b) compares our model (red curves) with numerically simulated results (hollow squares) for a triangle single-wall fiber. In terms of transmission attenuation, two calculations agree very well across more than two spectrum octaves (from λ0 = 0.3 μm to 1.7 μm), which explicitly verifies the quantitative calculation capability of our model.
To inspect our model more carefully, Fig. 5 plots the electric field amplitudes and phases at fiber’s outermost boundary [the red dashed lines in Fig. 5(a)]. The field amplitudes along a far-distance circle R (the gray dashed lines) are also plotted. Under two orthogonal polarizations [denoted by the double arrows in Fig. 5(a)], on the aspect of both near fields [Fig. 5(b)] and far fields [Fig. 5(c)], our model (the red curves) agrees well with the numerical simulation (the black curves). The working wavelength is selected to be 680 nm [marked by the blue dot in Fig. 4(b)], which is very close to the edge of one transmission band. Under such a rigorous condition, the quantitative calculation capability of our model is still evident.
With regard to the electric field at fiber’s outermost boundary, Fig. 5(b) shows that except around corner apexes the phases of the electric fields are fixed to a constant. Additionally, in comparison with the s-polarization, the p-polarization component gives rise to stronger electric fields outside the glass wall. These features are coincident with those shown in 1D slab geometry [Fig. 1(b)]. Figure 5(b) also shows that our model precisely predicts the ratio of the field amplitudes in the central regions of three edges of the triangle. However, when the observation point approaches the corner apexes, the field amplitude predicted by our model is over-estimated. This may be because our approximate model ignores the influences from adjacent glass segments [see the geometry transformation in Fig. 4(a)]. Actually, at the corner apexes, our geometry transformation procedure stops working.
With regard to the far-field properties, Fig. 5(c) plots the field amplitudes at a circle R, whose radius of 60 μm is comparable with Fraunhofer far-field criterion (, where D is the largest dimension of the structure and is the effective transverse wavelength ). In Fig. 5(c), both our model and numerical simulation ensure that the light leakage in the transverse plane is far from isotropic. The radiations pointing at the corner apexes are suppressed, whereas those in the directions vertical to the triangle edges are enhanced. Most radiation energies are emitted out through the triangle edges nearly vertical to the polarization direction, where there exist more p-polarization wave components. With this understanding, it may be possible to design new HC-ARF structures having lower attenuation coefficients. More light confinement structures, for example PBG cladding, could be mainly deployed in the stronger light leaking directions.
We emphasize that the above azimuthal angle dependence of light leakage [Fig. 5(c)] will not cause birefringence for a regular triangle fiber. In calculating the overall modal index and overall attenuation coefficient, our model averages contributions from all the angular segments and all the transverse directions, which eliminates the azimuthal angle dependence of these two parameters.
Before stepping forward, we discuss again the problem of corner apex. In Fig. 6, a single-wall square fiber is compared with its variant, whose corners have been rounded with 4 circles having radius rc = 2.5 μm. Simulated attenuation spectra of these two fibers exhibit one distinct difference. The square fiber shows many spiky features (the black curve), whereas the variant square fiber does not (the green curve). We believe new resonances, rather than those relevant to the uniformly thick glass wall, bring forth these spikes. We attribute them to either back reflection or non-uniformity of the glass thickness occurring at the corner apexes (see the schematic illustrations in Fig. 6). With regard to the second conjecture, we notice Kolyadin et al. have found that touching capillaries in a negative curvature hollow core fiber dramatically degrade transmission performance . In their structure, the back reflection from the touching point can be excluded, so that, the non-uniformity of the glass thickness at the touching point should play an important role. In our case of Fig. 6, the four corner apexes bring about the non-uniformity of glass thickness. Replacing them with rounded curvatures, the variant square fiber shows a much smoother attenuation spectrum, agreeing well with the prediction of our analytic model (the red curve).
Additionally, we cannot find spiky structures in the attenuation spectrum of the triangle fiber [Fig. 4(b)]. One possible explanation is the corner apexes in the triangle fiber lie far away from fiber’s central axis and their influences to the fundamental core mode are weak.
Next, we consider another square fiber geometry in order to illuminate the influence of glass thickness non-uniformity. In Fig. 7, the inscribed radius of the fiber is ain, and the glass thickness in the vertical direction (t2 = 0.6 μm) is slightly thinner than that in the horizontal direction (t1 = 0.67 μm). With the variation of polarization, both the simulated and the modelled attenuation spectra exhibit notable changes. First, two sets of resonant conditions () commonly determine the edges of the transmission bands for both polarizations. Second, the shapes of the attenuation spectra are dramatically changed under two polarizations. Our analytic model (the red and the green curves) agrees very well with the numerical simulation (the black and the blue ones), implying that our model has grasped the physical essence of the light leaking process occuring in such a complex 2D structure. For reference, we also plot attenuation spectra of two normal square fibers, which have uniform glass thicknesses of t = 0.6 μm and 0.67 μm in four sides respectively (denoted by the dashed cyan and pink lines respectively). With uniform glass thickness, the polarization dependence vanishes, but different t’s lead to a shift of transmission band. It is seen that at specific polarizations and wavelengths (e.g. the reddish and the greenish shaded areas) the modified square fiber, which has non-uniform glass thickness, exhibits lower attenuation coefficient than the normal square fiber, which has uniform glass thickness. This result is against the widely accepted viewpoint that a uniform glass thickness across whole lattice fiber leads to better light guidance . The principle underneath this phenomenon may be that most light leakages are caused by the p-polarization components, and the anti-resonance condition of the p-polarization wave is only relevant to the glass thickness vertical to the polarization direction.
To elucidate the influence of geometrical shape on fiber’s transmission properties, Fig. 8 plots attenuation spectra of various single-wall regular polygon fibers with regular triangle, square, hexagon, and octagon. Both numerical simulations and our analytic model are carried out with the inscribed radii ain = 9.67 μm. Apart from some spiky features, our model agrees well with numerical simulation, especially in terms of their variation tendency. As the number of edges of the polygon increases, the polygon fiber becomes more and more alike a circular ring and results in a worse and worse light confinement.
Since a circular ring fiber is not favored for light guidance, we study another HC-ARF structure, which has a hypocycloid shape core-surround [15,16]. Our analytic model is able to elucidate why light guidance is improved in such a fiber. Figure 9 compares two single-wall fibers with one having a square shape and the other having a hypocycloidal shape. Both numerical simulation and analytical model corroborate that the latter one has lower loss. As pointed out above, the electric fields at fiber’s outermost boundaries form equiphase surfaces. A hypocycloidal, or concave, equiphase surface seems good at suppressing the overall light leakage. In Fig. 9, although the decreasing extents of the simulated and the modelled attenuation spectra show some discrepancy, the light leakage suppression effect caused by the hypocycloid core-surround is clearly demonstrated. We believe the small disagreement in Fig. 9 can be overcome with further development of our model.
5. Discussions and conclusions
Shortly speaking, the analytic and quantitative characteristics of our calculation approach have been corroborated. Throughout this paper, except those spiky features, the discrepancy between our model and numerical simulation is roughly less than 0.5 dB, or 10%, in terms of transmission attenuation. The logarithmic scales used in Figs. (3,4,6-9) clearly exhibit this. More importantly, thanks to the following three techniques, our model correctly predicts the change of the shape of the attenuation spectrum and the variation tendency of the loss figure as fiber’s geometrical shape changes.
- ● The phase of electric field at the outermost boundary is only determined by the order of the anti-resonant transmission band, irrespective of polarization, geometrical dimension and working wavelength. This property can be approximately derived in 1D slab geometry [from Eq. (1)]. We apply it in 2D fiber structures. Fortunately, our attempt has been verified by simulation [Fig. 5(b)] and leads to a very simple picture of equiphase surface.
- ● The relationship between the longitudinal and the transverse wave-vector numbers, , combining with the leaky mode character, provide us conveniences to deal with the light leaking problem. First, the real part of the longitudinal wave-vector number, , represents the propagation constant of the fiber, and the real part of the transverse wave-vector number, , defines a 2D Helmholtz wave equation in the x-y plane, . Second, the imaginary part of the longitudinal wave-vector number, , is related to the attenuation coefficient, α, and has relationship with the field amplitude at fiber’s outermost boundary. Third, once knowing the fields at the outermost boundary, rather than the full field distribution in the cross section, the leaked energies in all the transverse directions can be derived from the 2D Helmholtz equation. The shape of the fiber’s outermost boundary influences the overall light leakages and determines the attenuation coefficient.
- ● In order to simplify mathematical treatments, we propose a geometry transformation from 2D to 1D. By splitting the cross section of a single-wall fiber to different angular segments and converting them to a series of slab structures, the modal indices and the fields at the outermost boundary can be quickly and analytically solved. For fundamental core mode, the transverse field distribution inside the air core can be approximated to be a linearly polarized Gaussian beam. Each angular segment and the corresponding slab waveguide, therefore, should contribute equally to the overall modal index and light leakage. Besides, in order to further simplify this problem, our model deals with s- and p-polarization wave components separately, and their proportions in each angular segment are determined in the geometry transformation procedure.
We have to admit our model still has many drawbacks. The first one is the geometry transformation procedure. As depicted in Fig. 4(a), each angular segment split in the cross section of the fiber is treated independently. The influences from adjacent segments have not been incorporated, which may explain why the field amplitude in Fig. 5(b) is over-estimated as the observation point approaches the corner apexes. The second drawback is that, at the current stage, our model is only suitable for single-wall hollow core fibers. Although many people believe the core-surround layer plays the most important role in determining the transmission attenuation, it would be better to include the influences from other cladding structures. This work is ongoing now and will be reported later. The third drawback is that our calculations have not considered influences of corner apexes. Although the field amplitude over there is very weak according to simulation [see Fig. 5(b)], appearance of new resonances will bring about spiky features in many spectral regions.
In conclusion, our analytic model exhibits the capability of quantitatively calculating transmission attenuation of single-wall hollow core fibers. This model elucidates the light leakage dependences on azimuthal angle, polarization, and geometrical shape and has been examined in a variety of fiber geometries. A simple and clear physical picture about the light leaking process has been presented. First, an equiphase surface is formed at fiber’s outermost boundary. Then, the light energy is transversely emitted out ruled by a 2D Helmholtz wave equation. Our model not only grasps the physical essences but also simplifies mathematical treatments by introducing many approximations.
Since fiber geometry can influence the light leakage and the attenuation coefficient, many interesting low-loss HC-ARF designs are in prospect.
Below, we present derivation of Eq. (5). Based on the 2D Helmholtz equation in the x-y plane, , and Green’s theorem, the x/y-component of the electric field in the surrounding area can be expressed as ,Fig. 10. The 2D Green’s function, , satisfies , where represents the Dirac delta function and the Hankel function of the first kind of order zero has asymptotic form of as s approaches infinity.
Since fiber’s outermost boundary constitutes an equiphase surface, we hypothesize each segment on this closed loop produces a plane wavelet pointing at its normal direction n, i.e. . Meanwhile, , as s approaches infinity. With r ≈ s, Eq. (9) can be approximately expressed as,Fig. 10) lies in the far-field region. Note that in Eq. (12) both the x- and y-components are taken into account because the 2D scalar Helmholtz equation allows both polarized waves to propagate toward the infinity. The transverse-wave characteristic of a propagating electromagnetic wave is maintained in three-dimensional space. However, in the case of glancing incidence, the transverse k-vector and the transverse electric field vector can be both in the radial direction.
On the other hand, the exponential energy attenuation in the longitudinal direction can be expressed asEq. (12) = Eq. (13), and the definition of E(s,p) [Eq. (3)], we obtain the following expression, i.e. Equation (5),
This work was supported by the National Basic Research Foundation of China (No. 2011CB922002), the National Natural Science Foundation of China (No. 61275044, 61377098, and 11204366), the Key Programs of the Chinese Academy of Sciences (No. YZ201346), the Start-up Funding of Institute of Physics (No. Y1K501DL11), and the Beijing National Science Foundation (No. 4142006).
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