Abstract

We investigate the properties of optical fibres made from chiral materials, in which a contrast in only optical rotation forms the waveguide, rather than a contrast in the refractive index; we refer to such structures as pure chiral fibres. We present a mathematical formulation for solving the modes of circularly symmetric examples of such fibres and examine the guidance and polarisation properties of pure chiral step-index, Bragg and photonic crystal fibre designs. Their behaviour is shown to differ for left-and right-hand circular polarisation, allowing circular polarisations to be isolated and/or guided by different mechanisms, as well as differing from equivalent non-chiral fibres. The strength of optical rotation required in each case is quantified.

© 2011 Optical Society of America

1. Introduction

Chirality is a geometrical concept that describes the inability of an object and its mirror image to be superimposed solely through translations and rotations (e.g. like one’s hands, from which the word is derived). This asymmetry of chiral molecules gives rise to optical rotation, which is the ability to rotate the plane of linearly polarised light propagating through a chiral medium. A chiral molecule can have two possible structures known as enantiomers, which are each others’ mirror image. By convention, the enantiomers are identified by their effect on linearly polarised light of 589 nm wavelength. When the observer faces the light source, the dextrorotary or (+) enantiomer rotates the plane of linearly polarised light clockwise, and the levorotary or (−) enantiomer anticlockwise. An equal mixture of the two – a racemic mixture – causes no rotation [1, 2, 3]. Although chiral materials are described by their behaviour at 589 nm, the amount and direction of rotation are wavelength dependent, with one enantiomer always having the opposite effect compared to the other.

The phenomenon of optical rotation is an example of circular birefringence, with the material possessing a different refractive index for right-hand circularly polarised (RHCP) and left-hand circularly polarised (LHCP) light. Linearly polarised light entering the material is decomposed into RHCP and LHCP components, and its rotation is a result of these components traveling in the material at different phase velocities:

vLCP=cn(1+δ),vRCP=cn(1δ),
where n is the average refractive index, c is the speed of light and δ is a dimensionless measure of chirality. The circular birefringence is equal to the difference in the corresponding refractive indices, given by
B=n1δn1+δ=2nδ1δ2,
which is approximately 2 for small δ. The most common way to quantify the optical rotation is the amount of rotation of the polarisation axis per unit length α (here we will use units of radians per metre), which typically has an inverse square dependence on wavelength, and is related to the other quantities and the wavenumber k as follows:
α=12kBknδ.

Circular birefringence in optical fibres has received little attention compared to linear birefringence [4], owing to the difficulty with which it is achieved. Thus far, it has been induced through “macroscopic” geometrical effects (on the scale of the core of the fibre or larger) by spinning a fibre with a non-circularly symmetric structure, resulting in a helix. The helical structure is itself chiral, and gives rise to some circular birefringence that is dependent on the properties of the helix (see [5, 6] and references therein). Clearly, the use of chiral materials in optical fibres will also give rise to circular birefringence but without the need for these macroscopic geometrical effects. It has been suggested that the use of chiral material in optical fibres may be studied using polymer optical fibres (POF) [7, 8], as most chiral materials are organic and compatible with polymers, but would be destroyed at the processing temperatures of silica and soft glasses. The use of chiral materials could lead to the manipulation of the polarisation state of light as it propagates in the fibre. Several studies have considered the addition of chirality to an existing (non-chiral) fibre and reported on its effect on the fibre’s guidance, modal cut-offs and dispersion [9, 10, 11, 12].

In this paper, we go one step further and consider structures where there is no contrast in the average refractive index and any guidance must therefore occur purely from differences in chirality. We define a pure chiral guide as one where there is no average refractive index variation, and the only material property that varies to form the waveguide is the chirality. We present a formulation for modelling such structures and show that the left and right circular polarisations are not coupled at any of the interfaces in the structure and thus the modes are all either pure left- or right-circular polarised fields. Furthermore, such structures may have the property that one circular polarisation (e.g. RHCP) will see a guiding structure while the opposite circular polarisation (LHCP) will see an anti-guiding structure. Thus, such structures can be used as circularly polarising fibres. We will consider a step-chiral-index structure that guides one circular polarisation only by total internal reflection (TIR), and pure chiral Bragg fibre and pure chiral photonic crystal (PCF) fibre structures that simultaneously exhibit guidance by total internal reflection for one circular polarisation and by Bragg reflection or photonic bandgap guidance for the other. A schematic of these designs is shown in Fig. 1.

 

Fig. 1 Schematic of the pure chiral optical fibres considered in this work: (a) step-chiral-index, (b) pure chiral Bragg fibre and (c) pure chiral photonic crystal fibre. The refractive index profiles seen by the two circular polarisations are shown, as well as the modes that may be supported through the various mechanisms. Typical mode effective indices are indicated by the red dashed lines.

Download Full Size | PPT Slide | PDF

2. Wave equations

Electromagnetic fields in chiral media can be described by combining Maxwell’s equations with the Drude-Born-Fedorov constitutive equations [13, 3]

D=ɛ(E+γ×E),
B=μ(H+γ×H),
where the chirality parameter γ has the dimensions of length and is taken as a constant for a given material, independent of wavelength. Note that the presence of curls in the constitutive equations is responsible for the breaking of inversion symmetry. Positive and negative values of γ correspond to rotating the plane of linearly polarised light clockwise and anticlockwise, respectively, i.e. to the right-handed (dextrorotary) and left-handed (levorotary) chiral media respectively for a wavelength of 589 nm.

For time-harmonic problems all fields are assumed to have an eiωt dependence with frequency ω. In such situations, a convenient dimensionless variable is

δ=knγ,
where k = ω/c is the free space wavenumber and n=cμɛ is the average refractive index.

Incorporating the constitutive equations into the time-harmonic Maxwell equations leads to

(1δ2)×E=iωμH+knδE,
(1δ2)×H=iωɛE+knδH.
Note that in the limit of δ → 0 these equations reduce to the conventional equations for non-chiral media.

Maxwell’s equations can be decoupled by defining the combinations

Q±=EiZH,
where Z=μ/ɛ is the wave impedance for both materials. Thus, in a homogeneous medium
×Q+=+kn1+δQ+,
×Q=kn1δQ.
The curls of these equations lead directly to the chiral wave equations
2Q++k2n2(1+δ)2Q+=0,
2Q+k2n2(1δ)2Q=0.

For notational convenience, we introduce a polarisation index p which takes on the values ±1 or equivalently stands for ± when used in a subscript or superscript. Thus, the circular polarisation corresponding to Qp propagates as if in a medium with index

npδ=n1+pδ
and we refer to n as a chiral index to distinguish it from the average index n. Thus, we can write both wave equations as
2Qp+k2npδ2Qp=0.

It is most convenient to solve the wave equations in cylindrical coordinates (r, ϕ, z). For any waveguide structure with cylindrical symmetry, the azimuthal dependence of the field can be chosen to be eimϕ where m is the azimuthal index, and the longitudinal dependence is eiβz where β is the propagation constant and β = kneff where neff is the modal index. The longitudinal and azimuthal components of the electric and magnetic fields are continuous for any structure with cylindrical symmetry. Furthermore, in a pure chiral structure the impedance Z is uniform and, thus, the longitudinal and azimuthal components of Qp are continuous at all interfaces. The absence of a step in the average index n combined with the uniformity of the wave impedence Z throughout the entire structure gives the key result that different circular polarisation modes are not coupled at any interfaces. This property is not true in general for other types of chiral fibres. The modes will thus be purely one circular polarisation or the other.

The radial dependence of the longitudinal fields Qzp then satisfies a Bessel equation

[d2dr2+1rddr+m2r2upδ2]Qzp(r)=0,
where u is the transverse wavenumber given by
upδ=kneff2npδ2.
When the transverse wavenumber is pure imaginary it is convenient to define instead an evanescent parameter
wpδ=knpδ2neff2.
The radial component of the fields can be eliminated from Eq. (8) and then the azimuthal components can be determined from the longitudinal components as follows
Qϕp(r)=1upδ2[mkneffr+pknpδddr]Qzp(r)=1upδ2𝒟Qzp(r),
where the symbol 𝒟 is introduce as a convenient abbreviation for the radial operator in the above equation. Finally, we introduce some matrix notation that allows us to write the fields as a superposition of Bessel functions in alternative ways
(Qzp(r)Qϕp(r))=𝒥m,p(upδ,r)(AB)=m,p(upδ,r)(CD)=𝒦m,p(wpδ,r)(EF),
where
𝒥mp(u,r)=(Jm(ur)Ym(ur)1u2𝒟Jm(ur)1u2𝒟Ym(ur))
if we wish to use standing waves (for Bessel functions of the 1st and 2nd kind J and Y, respectively), and
m,p(u,r)=(Hm(1)(ur)Hm(2)(ur)1u2𝒟Hm(1)(ur)1u2𝒟Hm(2)(ur))
if we wish to use inward and outward travelling waves (for Hankel functions H), and
𝒦m,p(w,r)=(Km(wr)Im(wr)1w2𝒟Km(wr)1w2𝒟Im(wr))
if we wish to use evanescent waves (for modified Bessel functions K and I). The expansion coefficients A-F in (16) are determined by boundary or continuity conditions at each layer.

3. Transfer-Matrix Formulation

The transfer matrices that connect the field amplitudes at opposite ends of each layer are then specified by

(Qz(rn+1)Qϕ(rn+1))=𝒯mp(un,rn,rn+1)(Qz(rn)Qϕ(rn)),
where un is the transverse wavenumber for the layer corresponding to rn < r < rn+1 and
𝒯m,p(u,rn,rn+1)=m,p(u,rn+1)m,p(u,rn)1=𝒥m,p(u,rn+1)𝒥mp(u,rn)1
using either travelling wave or standing wave representations.

For the core we can write

(Qz(r)Qϕ(r))=A𝒥m,p(uco,r)(10)
and for the outermost cladding we can write
(Qz(r)Qϕ(r))=Cm,p(ucl,r)(10)
where A and C are relevant normalisation constants. Equivalently, Eq. (23) which represents the absence of inward travelling waves can also be expressed by the orthogonality condition
(01)m,p(ucl,r)1(Qz(r)Qϕ(r))=0.

Multiplying together all the relevant matrices leads to a single scalar eigenvalue equation for neff given by

(01)m,p(ucl,rN)1[i=1N1𝒯m,p(ri,ri+1,ui)]𝒥m,p(uco,r1)(10)=0,
where N is the number of interfaces. For bound modes, it is more convenient to write the condition in the form
(01)𝒦m,p(wcl,rN)1[i=1N1𝒯m,p(ri,ri+1,ui)]𝒥m,p(uco,r1)(10)=0.

4. Step-chiral-index Fibre

A step-chiral-index fibre can be considered as a special case of a Bragg fibre with N = 1 interfaces. Thus, we can use Eq. (25) in the simplified form

(01)𝒦m,p(wcl,a)1𝒥m,p(uco,a)(10)=0.
where a is the core radius and
uco=knco2neff2,wcl=kneff2ncl2.
We assume that the core and cladding have opposite chiralities and consider the polarisation state which sees a higher index in the core. For this polarisation state, the chiral indices satisfy
nco=n1|δ|>n1+|δ|=ncl
and guidance by total internal reflection is possible. For the opposite polarisation, the chiral indices satisfy nco < ncl and there are no guided modes. This is illustrated in Fig. 1(a).

Multiplying out the matrices in Eq. (27) gives

1wcl2Jm(ucor)𝒟Km(wclr)+1uco2Km(wclr)𝒟Jm(ucor)=0
evaluated at r = a. We define dimensionless parameters U, V and W in a way analogous to that for conventional waveguides:
U=kanco2neff2,V=kanco2ncl2,W=kaneff2ncl2
and using these three parameters the eigenvalue condition can be written in the form
Jm(U)UJm(U)+nclncoKm(W)WKm(W)=mpneffncoV2U2W2.

We can compare the above dispersion relation to the analogous result for a conventional step-index fibre [14]

(Jm(U)UJm(U)+Km(W)WKm(W))(Jm(U)UJm(U)+ncl2nco2Km(W)WKm(W))=m2neff2nco2(V2U2W2)2.

For a conventional step-index fibre, modes with non-zero m come in degenerate pairs: this is related to the symmetry under m → −m for Eq. (33). On the other hand, the circular polarisation destroys this symmetry in the chiral case and the modes obtained from Eq. (32) are non-degenerate for all m.

We calculate the dispersion curves for a chirality δ = 0.25 and compare them to the analogous modes of a conventional fibre with nclnco=0.6. These values are chosen to correspond to the parameters used in Fig. 12-4 in [14]. The dispersion curves are shown in Fig. 2. The chiral modes for m = 0 have effective indices intermediate between the analogous TE and TM modes. When m and p are the same sign, the chiral modes are analogous to conventional HE modes and when m and p have opposite signs the chiral modes are analogous to conventional EH modes.

 

Fig. 2 Comparing dispersion curves for step-chiral-index and conventional step-index fibres. The chiral modes are shown with solid curves and the conventional modes are shown with dashed curves. The colours correspond to different values of the azimuthal index: m = 0 (black), mp = 1 (red), mp = −1 (blue), mp = 2 (green),and mp = −2 (magenta). The chiral modes are all of the one circular polarisation that is guided, as the other will see a lower core index compared to the cladding and not be guided. The chiral modes are solutions of Eq. (32), and the conventional modes of Eq. (33 ).

Download Full Size | PPT Slide | PDF

Furthermore, in the weak guidance limit, Eq. (33) tends to the square of Eq. (32) and, thus, apart from the important differences in the polarisation state and degeneracy, the modes for pure chiral waveguides are essentially the same as for conventional waveguides. Practical chiral materials have values of δ ≪ 1 and the weak-guidance approximation will be extremely accurate.

The equivalent numerical aperture for a pure chiral fibre can be defined as follows

NA=nco2ncl2=2nδ1δ22nδ.
Likewise, the equivalent V-parameter becomes
V=kanco2ncl22kanδ.
We can write the optical rotation in terms of V using
αknδkn(V2kan)2=V24kna2.
This analysis can be used to infer the strength of the chirality required to form a waveguide, as shown in Fig. 3. Values of V > 1 are considered necessary for sufficiently robust guidance, as otherwise any defects in or bending of the fibre will cause unacceptably high losses [14]. For n ≈ 1.5, wavelengths in the blue (0.4 μm) and a core radius of 10 μm, this gives a minimum feasible optical rotation of 1.06 ×10−4 radians per μm, or approximately 60 °/cm [15]. This analysis is complicated by the inverse square wavelength dependence of α, which is the quantity most commonly used to describe optical rotation. If the material is chosen as a constant in the analysis, it would correspond to a constant γ, and not a constant α. For this reason, contours corresponding to constant γ are also shown in Fig. 3, and show that a particular fibre with a given γ will reach V < 1 at shorter wavelengths than anticipated by considering α as the constant. Taking the inverse square wavelength dependence of α into account gives Vλ−3/2.

 

Fig. 3 Contours showing the V-parameter for a step-chiral-index fibre as a function of wavelength and optical rotation α; remaining parameters as in the text. The grey area corresponds to V < 1 where the fibre is deemed to not guide effectively. Contours corresponding to constant γ are indicated in red and labelled with the value of α they correspond to at 589 nm.

Download Full Size | PPT Slide | PDF

5. Pure chiral Bragg fibre

A pure chiral Bragg fibre consisting of layers of alternating chirality will be considered, as shown in Fig. 1(b). The preliminary design modelled here has a core radius of 10 μm and alternating chiral layers of 5 μm each. The chirality was chosen to give δ = 0.001 at a wavelength of 1 μm. A total of 9 pairs of layers is sufficient to get confinement losses of the most well confined Bragg guided mode close to a target value of 0.1 dB/m. The dispersion curves for m = 1 and m = 0 modes are shown in Fig. 4. The sign of the chirality is chosen so that the left-circularly polarised modes see a lower index in the core and, contrariwise, the right-circularly polarised modes see a higher index in the core (Fig. 1(b)). The dashed black lines show the high and low indices as seen by both circular polarisations. The red, green and blue curves group modes into bands and reveal which loss curves go with which dispersion curves. The dashed sections of dispersion curves are leaky modes.

 

Fig. 4 Dispersion curves for m = 1 and m = 0. Top and bottom graphs in each pair give the effective modal indices and the loss, respectively. Left and right graphs give the results for left- and right-circular polarisation respectively.

Download Full Size | PPT Slide | PDF

5.1. Short-wavelength behaviour

At very short wavelength and high contrasts the modal fields tend to be confined to the individual high-index layers with little overlap between layers. Each layer will then behave approximately like an independent planar waveguide (if we ignore the curvature of the annuli) with full-width 5 μm. For a wavelength of λ = 0.6 μm we obtain V = 2.574 and expect to find two guided modes with neff ≈ 1.44934 and 1.45104. Indeed the bands of dispersion curves in all cases converge on the short wavelength end of each diagram to these approximate values. For longer wavelengths the fields from adjacent layers have significant overlap, and thus interact strongly and the bands broaden, as seen on the right of each figure.

In addition, right-circular polarisation modes can be guided in the core. For a circular core of radius 10 μm and a wavelength of 0.6 μm we obtain V = 10.296 and expect to see additional modes as follows. For m = 0 the effective indices are 1.4513, 1.4504 and 1.4491, as confirmed by the short-wavelength limits of the red dispersion curves on the bottom right diagram of Fig. 4. For m = 1 the HE modes have effective indices close to 1.4509 and 1.4484, once again confirmed by the isolated red curves on the top right diagram of Fig. 4. The EH modes do not appear on this diagram since they correspond to m = −1. For longer wavelengths the fields in the core begin to interact with the layers in the Bragg structure.

5.2. Anti-guiding core (left-circular polarisation)

For m = 1 and p = −1, the dark blue and green curves in the upper left diagram of Fig. 4 show the dispersion curves for two bands of modes which are guided within the annular layers, and the dashed parts of the curve represent leaky modes beyond cutoff. The modes in the first band (blue) have cutoffs above 1.25 μm, while the modes in the second band (green) have cutoffs between 0.7 and 0.85 μm. In the gap between these bands of modes a mode confined by Bragg reflection exists for left-circular polarisation only and is shown in red. This mode exhibits very strong wavelength dependence, particularly in its loss, since its guidance is highly dependent on a Bragg anti-resonance condition. For the 5 μm periodicity, Bragg reflection occurs for a transverse wavelength of 29 μm which is consistent with a wavelength of 1 μm. For 9 layers the minimum loss is of order 0.1 dB/m and this can be reduced further by including more layers or increasing the core size [16]. For m=0 and left-circular polarisation, similar bands of modes (in dark blue and green curves) can be seen in the lower left of Fig. 4 but in this case there is no Bragg-guided mode.

5.3. Guiding core (right-circular polarisation)

The right-circularly polarised modes have a high effective index in the core and so are potentially guided by total internal reflection; however, the external cladding is also high index (Fig. 1(b)) and thus there is always some leakage for all such modes and none are strictly bound. In this case, there is no possibility of purely Bragg-guidance given the high index core. Modes guided by total internal reflection will be evanescent in the low index layers and no Bragg reflection condition can be devised. This results in loss curves that do not have the wavelength dependence seen for the Bragg-guided left circular polarisation m = 1 mode. Once again for very short wavelengths, the annular layers act like independent planar waveguides of width 5 μm and a tight binding argument gives the narrowing of the bands to these highly localised modes at the left of each diagram. Bragg reflection may still occur for modes of low effective index that are below cut-off, and hence not evanescent in the low index layers, but this was not observed for the parameter space considered in Fig. 4.

6. Implications for pure chiral photonic crystal fibres

A final consideration is a more complex fibre design such as a pure chiral photonic crystal fibre consisting of an array of rods of one handedness, embedded in a background of the other handednes, shown in Fig. 1(c). Although this cannot be modelled directly using the method above, we can infer much about its behaviour from the analysis of the step-chiral-index fibre and the well-known properties of non-chiral PCF (see [20, 21] and references therein). Qualitatively, the behaviour will be the same as the pure chiral Bragg fibres.

For one circular polarisation the structure of these fibres would appear as low-index rods in a high-index background. The low-index of the rods would result in an “average” lower index cladding, which may result in guidance by modified total internal reflection (MTIR) for this circular polarisation if a sufficiently high V-parameter is achieved.

Conversely, for the other circular polarisation the structure will appear as high-index rods in a low-index background. Such a structure may support photonic bandgaps, and hence this circular polarisation may be guided by means of photonic bandgap guidance. In this case the bandgaps are known to occur between the cut-offs of modes supported by one high-index rod in isolation [22], i.e. a step-chiral-index fibre. (An effective cut-off for the fundamental mode is taken near V ≈ 1.2.) Figure 2 shows that for step-chiral-index fibres the cut-offs occur at essentially the same values as for conventional step-index fibres. Thus, a pure chiral photonic crystal fibre will similarly display bandgaps between values of V ≈ 1.2 and 2.4, 2.4 and 3.9 etc. Using Eq. (36) and/or Fig. 3 indicates that a value of α > 90 °/cm is required to place V = 1.2 at 400 nm, whilst α = 350 °/cm is required to place V = 2.4 at the same wavelength. A value of α in that range would be required for such a fibre to display the longest-wavelength photonic bandgap at visible wavelengths.

Similar to the pure chiral Bragg fibre, such pure chiral PCF may guide one circular polarisa-tion by modified total internal reflection and the other circular polarisation by photonic bandgap guidance. The confinement loss may again be decreased by increasing the width of the photonic crystal cladding, but unlike the Bragg fibre case, the minimum V-parameters or α must be met in order for the bandgaps to form in the first place. In addition to the polarisation aspect, the bandgaps of the pure chiral PCF would differ from those of otherwise equivalent non-chiral fibres in their width. Although the range of V corresponding to the bandgaps is the same, for conventional step-index fibres Vλ−1 whilst for step-chiral-index fibres Vλ −3/2. This difference in the wavelength dependence of V results in bandgaps narrower in wavelength for the pure chiral case.

7. Discussion

Pure chiral structures are interesting both from a conceptual and practical viewpoint. In many ways, a pure chiral structure acts like a conventional waveguide with the circular birefringence playing the role of index difference (or numerical aperture). However, the non-degenerate nature of all the modes of a pure chiral structure is a special feature of using chirality. Further, the two circular polarisation states act in very different ways, and even when both states are guided, the guidance mechanism are different for each polarisation.

The case of guiding circularly polarised modes is interesting in its own right. In conventional fibres a circular polarisation may be achieved by combining pairs of degenerate orthogonally polarised modes (the HE11). However, such combinations are highly susceptible to perturbations which cause the degenerate modes to split into near-linearly polarised non-degenerate modes. In contrast, for the fibres considered here circularly polarised modes are not degenerate, and the circular polarisation states and circular birefringence do not rely on degeneracies or the fibre’s geometrical design, but are intrinsic properties arising from the chirality of the materials.

The case of isolating a single circular polarisation – as in the step-chiral-index fibre or the pure chiral Bragg or PCF away from the Bragg reflection/bandgaps – is also conceptually interesting. It gives the opportunity of isolating a single non-degenerate mode whilst retaining a circularly symmetric mode and fibre. Such fibres would differ from other reported single-mode single-polarisation fibres which rely on linearly birefringent or polarising fibres [4, 17], where one linear polarisation mode is isolated. The linear polarisation and the necessary 2-fold symmetric geometry of those fibres means there is still a preferred direction. Alternative designs for isolating a single non-degenerate mode have been proposed whereby the circularly-symmetric azimuthally polarised TE mode is isolated [18, 19], however, these would require specific launch conditions to excite the TE mode which would be difficult to achieve, compared to using circularly polarised light.

Although this work is theoretical in nature, we briefly consider the practicality of fabricating the designs presented. We have assumed the use of a material available in both (+) and (−) forms so as to give a contrast in chirality with no contrast in the average refractive index, although this could also be achieved with a combination of different (+) and (−) material whose average refractive index was matched. In Section 4 and Fig. 3 we considered the strength of the chirality required to achieve guidance in a step-chiral-index fibre, i.e. to result in a V > 1. The circular birefringence is strongest at shorter wavelengths, and a minimum of α ≈ 60 °/cm at 400 nm was indicated. This corresponds to α ≈ 28 °/cm at 589 nm where α is more commonly quoted [15]. Such a value is high in terms of common material properties, but organic materials exist with such or higher optical activities [2]. It is therefore possible that such materials may be incorporated into polymers for use in polymer optical fibres based on this design. For the pure chiral PCF the values of required optical rotation are larger, becoming increasingly difficult to reach using available materials.

For a pure chiral Bragg fibre, the confinement via Bragg reflection depends not only on the chiral index contrast between the layers, but also on the number of layers and the size of the core [16]. Thus, a decrease in chiral index contrast may be compensated by increasing the number of layers or enlarging the core to achieve the same loss at a given wavelength. An analysis similar to that presented in Section 5 indicates that for a constant core size the number of layers must approximately double to preserve the same loss if the contrast is decreased by a factor of 10. The parameters used for Fig. 4 were 9 pairs of layers and a chirality corresponding to an optical rotation of α ≈ 5000 °/cm at 1 μm, an extremely high value.

8. Conclusion

We have presented an analysis of pure chiral fibres, in which a contrast in the chirality forms the waveguide, with no other refractive index variation. The modes supported by such fibres are shown to be non-degenerate in all cases, with left- and right-hand circularly polarised fields decoupled. We considered three examples of such fibres. Firstly, the step-chiral-index will guide only one circular polarisation, appearing as an antiguide to the other, and give the opportunity to form a single-mode single-polarisation fibre which retains circular symmetry. Secondly, a pure chiral Bragg fibre may guide one circular polarisation by TIR and the other by Bragg reflection and similarly a pure chiral PCF will guide one circular polarisation by MTIR and other by photonic bandgap guidance. Altought this is a theoretical analysis investigating these fibres on a conceptual level, such fibres may potentially find application in polarisation control. The strength of the optical rotation required was quantified in each case, and based on the reported properties of materials the step-chiral-index fibre is the most feasible, requiring the lowest optical rotation. Such fibres may then be fabricated using polymers into which appropriate chiral materials have been incorporated.

Acknowledgments

This work was funded by the Australian Research Council through Discovery Project grant no. DP0880882.

References and links

1. D. B. Amabilino, Chirality at the nanoscale: Nanoparticles, surfaces, materials and more, (Wiley-VCH, Weinheim2009).

2. T. M. Lowry, Optical rotatory power, (Dover Publications, New York1964).

3. K. V. Varadan, V. V. Varadan, and A. Lakhtakia, Time-Harmonic Electromagnetic Fields in Chiral Media, (Springer-Verlag, 1989).

4. J. Noda, K. Okamoto, and Y. Sasaki, “Polarization-maintaining fibers and their applications,” J. Lightwave Technol. 4, 10711089 (1986). [CrossRef]  

5. I. Bassett, “Design principle for a circularly birefringent optical fiber,” Opt. Lett. 13, 844–6 (1988). [CrossRef]   [PubMed]  

6. A. Argyros, J. Pla, F. Ladouceur, and L. Poladian, “Circular and elliptical birefringence in spun microstructured optical fibres,” Opt. Express 17, 15983–15990 (2009). [CrossRef]   [PubMed]  

7. N. S. Pujari, M. R. Kulkarni, M. C. J. Large, I. M. Bassett, and S. Ponrathnam, “Transparent chiral polymers for optical applications,” J. Appl. Pol. Science 98, 58–65 (2005). [CrossRef]  

8. A. Argyros, M. Straton, A. Docherty, E. H. Min, Z. Ge, K. H. Wong, F. Ladouceur, and L. Poladian, “Consideration of chiral optical fibres,” Frontiers Optoelectron. China 3, 67–70 (2010). [CrossRef]  

9. A. K. Singh, K. S. Singh, P. Khastgir, S. P. Ojha, and O. N. Singh, “Modal cutoff condition of an optical chiral fiber with different chiralities in the core and the cladding,” J. Opt. Soc. Am. B 11, 1283–1287 (1994). [CrossRef]  

10. R. C. Qiu and I. T. Lu, “Dispersion in chiral optical fibres,” IEE Proc.: Optoelectron. 145, 155–158 (1998). [CrossRef]  

11. F. M. Janeiro, C. R. Paiva, and A. L. Topa, “Guidance and leakage properties of chiral optical fibers,” J. Opt. Soc. Am. B 19, 2558–2566 (2002). [CrossRef]  

12. P. K. Choudhury and T. Yoshino, “Characterization of the optical power confinement in a simple chirofiber,” Optik 113, 89–95 (2002). [CrossRef]  

13. F.I. Fedorov, “Contribution to the theory of the optical activity of crystals .1. The law of conservation of energy and tensors of optical activity,” Optika I Spektroskopiya 6, 85–93 (1959).

14. A. W. Snyder and J. D. Love, Optical Waveguide Theory, (Chapman and Hall, London1983).

15. The optical activity of a material is commonly quoted as the rotation α, in degrees per decimetre, at 589 nm, at a specified temperature and concentration. Note we use different units in this paper.

16. A. Argyros, “Guided modes and loss in Bragg fibres,” Opt. Express 10, 1411–1417 (2002). [PubMed]  

17. H. Kubota, S. Kawanishi, S. Koyanagi, M. Tanaka, and S. Yamaguchi, “Absolutely single polarization photonic crystal fiber,” IEEE Photon. Technol. Lett. 16, 182184 (2004).

18. I. Bassett and A. Argyros, “Elimination of polarisation degeneracy in round waveguides,” Opt. Express 10, 1342–1346 (2002). [PubMed]  

19. A. Argyros, I. M. Bassett, M. A. van Eijkelenborg, and M. C. J. Large, “Microstructured optical fiber for single-polarization air-guidance,” Opt. Lett. 29, 20–22 (2004). [CrossRef]   [PubMed]  

20. P. St. J. Russell, “Photonic-cystal fibers,” J. Lightwave Technol. 24, 47294749 (2006). [CrossRef]  

21. A. Argyros, “Microstructured polymer optical fibers,” J. Lightwave Technol. 27, 1571–1579 (2009). [CrossRef]  

22. N. M. Litchinitser, A. K. Abeeluck, C. Headley, and B. J. Eggleton, “Antiresonant reflecting photonic crystal optical waveguides,” Opt. Lett. 27, 1592–1594 (2002). [CrossRef]  

References

  • View by:
  • |
  • |
  • |

  1. D. B. Amabilino, Chirality at the nanoscale: Nanoparticles, surfaces, materials and more, (Wiley-VCH, Weinheim 2009).
  2. T. M. Lowry, Optical rotatory power, (Dover Publications, New York 1964).
  3. K. V. Varadan, V. V. Varadan, and A. Lakhtakia, Time-Harmonic Electromagnetic Fields in Chiral Media, (Springer-Verlag, 1989).
  4. J. Noda, K. Okamoto, and Y. Sasaki, “Polarization-maintaining fibers and their applications,” J. Lightwave Technol. 4, 10711089 (1986).
    [CrossRef]
  5. I. Bassett, “Design principle for a circularly birefringent optical fiber,” Opt. Lett. 13, 844–846 (1988).
    [CrossRef] [PubMed]
  6. A. Argyros, J. Pla, F. Ladouceur, and L. Poladian, “Circular and elliptical birefringence in spun microstructured optical fibres,” Opt. Express 17, 15983–15990 (2009).
    [CrossRef] [PubMed]
  7. N. S. Pujari, M. R. Kulkarni, M. C. J. Large, I. M. Bassett, and S. Ponrathnam, “Transparent chiral polymers for optical applications,” J. Appl. Polym. Sci. 98, 58–65 (2005).
    [CrossRef]
  8. A. Argyros, M. Straton, A. Docherty, E. H. Min, Z. Ge, K. H. Wong, F. Ladouceur, and L. Poladian, “Consideration of chiral optical fibres,” Front. Optoelectron. China 3, 67–70 (2010).
    [CrossRef]
  9. A. K. Singh, K. S. Singh, P. Khastgir, S. P. Ojha, and O. N. Singh, “Modal cutoff condition of an optical chiral fiber with different chiralities in the core and the cladding,” J. Opt. Soc. Am. B 11, 1283–1287 (1994).
    [CrossRef]
  10. R. C. Qiu, and I. T. Lu, “Dispersion in chiral optical fibres,” IEE Proc., Optoelectron. 145, 155–158 (1998).
    [CrossRef]
  11. F. M. Janeiro, C. R. Paiva, and A. L. Topa, “Guidance and leakage properties of chiral optical fibers,” J. Opt. Soc. Am. B 19, 2558–2566 (2002).
    [CrossRef]
  12. P. K. Choudhury, and T. Yoshino, “Characterization of the optical power confinement in a simple chirofiber,” Optik (Stuttg.) 113, 89–95 (2002).
    [CrossRef]
  13. F. I. Fedorov, “Contribution to the theory of the optical activity of crystals. 1. The law of conservation of energy and tensors of optical activity,” Opt. I Spektrosk. 6, 85–93 (1959).
  14. A. W. Snyder, and J. D. Love, Optical Waveguide Theory, (Chapman and Hall, London 1983).
  15. The optical activity of a material is commonly quoted as the rotation α, in degrees per decimetre, at 589 nm, at a specified temperature and concentration. Note we use different units in this paper.
  16. A. Argyros, “Guided modes and loss in Bragg fibres,” Opt. Express 10, 1411–1417 (2002).
    [PubMed]
  17. H. Kubota, S. Kawanishi, S. Koyanagi, M. Tanaka, and S. Yamaguchi, “Absolutely single polarization photonic crystal fiber,” IEEE Photon. Technol. Lett. 16, 182184 (2004).
  18. I. Bassett, and A. Argyros, “Elimination of polarisation degeneracy in round waveguides,” Opt. Express 10, 1342–1346 (2002).
    [PubMed]
  19. A. Argyros, I. M. Bassett, M. A. van Eijkelenborg, and M. C. J. Large, “Microstructured optical fiber for singlepolarization air-guidance,” Opt. Lett. 29, 20–22 (2004).
    [CrossRef] [PubMed]
  20. P. St. J. Russell, “Photonic-cystal fibers,” J. Lightwave Technol. 24, 47294749 (2006).
    [CrossRef]
  21. A. Argyros, “Microstructured polymer optical fibers,” J. Lightwave Technol. 27, 1571–1579 (2009).
    [CrossRef]
  22. N. M. Litchinitser, A. K. Abeeluck, C. Headley, and B. J. Eggleton, “Antiresonant reflecting photonic crystal optical waveguides,” Opt. Lett. 27, 1592–1594 (2002).
    [CrossRef]

2010 (1)

A. Argyros, M. Straton, A. Docherty, E. H. Min, Z. Ge, K. H. Wong, F. Ladouceur, and L. Poladian, “Consideration of chiral optical fibres,” Front. Optoelectron. China 3, 67–70 (2010).
[CrossRef]

2009 (2)

2006 (1)

P. St. J. Russell, “Photonic-cystal fibers,” J. Lightwave Technol. 24, 47294749 (2006).
[CrossRef]

2005 (1)

N. S. Pujari, M. R. Kulkarni, M. C. J. Large, I. M. Bassett, and S. Ponrathnam, “Transparent chiral polymers for optical applications,” J. Appl. Polym. Sci. 98, 58–65 (2005).
[CrossRef]

2004 (2)

A. Argyros, I. M. Bassett, M. A. van Eijkelenborg, and M. C. J. Large, “Microstructured optical fiber for singlepolarization air-guidance,” Opt. Lett. 29, 20–22 (2004).
[CrossRef] [PubMed]

H. Kubota, S. Kawanishi, S. Koyanagi, M. Tanaka, and S. Yamaguchi, “Absolutely single polarization photonic crystal fiber,” IEEE Photon. Technol. Lett. 16, 182184 (2004).

2002 (5)

1998 (1)

R. C. Qiu, and I. T. Lu, “Dispersion in chiral optical fibres,” IEE Proc., Optoelectron. 145, 155–158 (1998).
[CrossRef]

1994 (1)

1988 (1)

1986 (1)

J. Noda, K. Okamoto, and Y. Sasaki, “Polarization-maintaining fibers and their applications,” J. Lightwave Technol. 4, 10711089 (1986).
[CrossRef]

1959 (1)

F. I. Fedorov, “Contribution to the theory of the optical activity of crystals. 1. The law of conservation of energy and tensors of optical activity,” Opt. I Spektrosk. 6, 85–93 (1959).

Abeeluck, A. K.

Argyros, A.

Bassett, I.

Bassett, I. M.

N. S. Pujari, M. R. Kulkarni, M. C. J. Large, I. M. Bassett, and S. Ponrathnam, “Transparent chiral polymers for optical applications,” J. Appl. Polym. Sci. 98, 58–65 (2005).
[CrossRef]

A. Argyros, I. M. Bassett, M. A. van Eijkelenborg, and M. C. J. Large, “Microstructured optical fiber for singlepolarization air-guidance,” Opt. Lett. 29, 20–22 (2004).
[CrossRef] [PubMed]

Choudhury, P. K.

P. K. Choudhury, and T. Yoshino, “Characterization of the optical power confinement in a simple chirofiber,” Optik (Stuttg.) 113, 89–95 (2002).
[CrossRef]

Docherty, A.

A. Argyros, M. Straton, A. Docherty, E. H. Min, Z. Ge, K. H. Wong, F. Ladouceur, and L. Poladian, “Consideration of chiral optical fibres,” Front. Optoelectron. China 3, 67–70 (2010).
[CrossRef]

Eggleton, B. J.

Fedorov, F. I.

F. I. Fedorov, “Contribution to the theory of the optical activity of crystals. 1. The law of conservation of energy and tensors of optical activity,” Opt. I Spektrosk. 6, 85–93 (1959).

Ge, Z.

A. Argyros, M. Straton, A. Docherty, E. H. Min, Z. Ge, K. H. Wong, F. Ladouceur, and L. Poladian, “Consideration of chiral optical fibres,” Front. Optoelectron. China 3, 67–70 (2010).
[CrossRef]

Headley, C.

Janeiro, F. M.

Kawanishi, S.

H. Kubota, S. Kawanishi, S. Koyanagi, M. Tanaka, and S. Yamaguchi, “Absolutely single polarization photonic crystal fiber,” IEEE Photon. Technol. Lett. 16, 182184 (2004).

Khastgir, P.

Koyanagi, S.

H. Kubota, S. Kawanishi, S. Koyanagi, M. Tanaka, and S. Yamaguchi, “Absolutely single polarization photonic crystal fiber,” IEEE Photon. Technol. Lett. 16, 182184 (2004).

Kubota, H.

H. Kubota, S. Kawanishi, S. Koyanagi, M. Tanaka, and S. Yamaguchi, “Absolutely single polarization photonic crystal fiber,” IEEE Photon. Technol. Lett. 16, 182184 (2004).

Kulkarni, M. R.

N. S. Pujari, M. R. Kulkarni, M. C. J. Large, I. M. Bassett, and S. Ponrathnam, “Transparent chiral polymers for optical applications,” J. Appl. Polym. Sci. 98, 58–65 (2005).
[CrossRef]

Ladouceur, F.

A. Argyros, M. Straton, A. Docherty, E. H. Min, Z. Ge, K. H. Wong, F. Ladouceur, and L. Poladian, “Consideration of chiral optical fibres,” Front. Optoelectron. China 3, 67–70 (2010).
[CrossRef]

A. Argyros, J. Pla, F. Ladouceur, and L. Poladian, “Circular and elliptical birefringence in spun microstructured optical fibres,” Opt. Express 17, 15983–15990 (2009).
[CrossRef] [PubMed]

Large, M. C. J.

N. S. Pujari, M. R. Kulkarni, M. C. J. Large, I. M. Bassett, and S. Ponrathnam, “Transparent chiral polymers for optical applications,” J. Appl. Polym. Sci. 98, 58–65 (2005).
[CrossRef]

A. Argyros, I. M. Bassett, M. A. van Eijkelenborg, and M. C. J. Large, “Microstructured optical fiber for singlepolarization air-guidance,” Opt. Lett. 29, 20–22 (2004).
[CrossRef] [PubMed]

Litchinitser, N. M.

Lu, I. T.

R. C. Qiu, and I. T. Lu, “Dispersion in chiral optical fibres,” IEE Proc., Optoelectron. 145, 155–158 (1998).
[CrossRef]

Min, E. H.

A. Argyros, M. Straton, A. Docherty, E. H. Min, Z. Ge, K. H. Wong, F. Ladouceur, and L. Poladian, “Consideration of chiral optical fibres,” Front. Optoelectron. China 3, 67–70 (2010).
[CrossRef]

Noda, J.

J. Noda, K. Okamoto, and Y. Sasaki, “Polarization-maintaining fibers and their applications,” J. Lightwave Technol. 4, 10711089 (1986).
[CrossRef]

Ojha, S. P.

Okamoto, K.

J. Noda, K. Okamoto, and Y. Sasaki, “Polarization-maintaining fibers and their applications,” J. Lightwave Technol. 4, 10711089 (1986).
[CrossRef]

Paiva, C. R.

Pla, J.

Poladian, L.

A. Argyros, M. Straton, A. Docherty, E. H. Min, Z. Ge, K. H. Wong, F. Ladouceur, and L. Poladian, “Consideration of chiral optical fibres,” Front. Optoelectron. China 3, 67–70 (2010).
[CrossRef]

A. Argyros, J. Pla, F. Ladouceur, and L. Poladian, “Circular and elliptical birefringence in spun microstructured optical fibres,” Opt. Express 17, 15983–15990 (2009).
[CrossRef] [PubMed]

Ponrathnam, S.

N. S. Pujari, M. R. Kulkarni, M. C. J. Large, I. M. Bassett, and S. Ponrathnam, “Transparent chiral polymers for optical applications,” J. Appl. Polym. Sci. 98, 58–65 (2005).
[CrossRef]

Pujari, N. S.

N. S. Pujari, M. R. Kulkarni, M. C. J. Large, I. M. Bassett, and S. Ponrathnam, “Transparent chiral polymers for optical applications,” J. Appl. Polym. Sci. 98, 58–65 (2005).
[CrossRef]

Qiu, R. C.

R. C. Qiu, and I. T. Lu, “Dispersion in chiral optical fibres,” IEE Proc., Optoelectron. 145, 155–158 (1998).
[CrossRef]

Russell, P. St. J.

P. St. J. Russell, “Photonic-cystal fibers,” J. Lightwave Technol. 24, 47294749 (2006).
[CrossRef]

Sasaki, Y.

J. Noda, K. Okamoto, and Y. Sasaki, “Polarization-maintaining fibers and their applications,” J. Lightwave Technol. 4, 10711089 (1986).
[CrossRef]

Singh, A. K.

Singh, K. S.

Singh, O. N.

Straton, M.

A. Argyros, M. Straton, A. Docherty, E. H. Min, Z. Ge, K. H. Wong, F. Ladouceur, and L. Poladian, “Consideration of chiral optical fibres,” Front. Optoelectron. China 3, 67–70 (2010).
[CrossRef]

Tanaka, M.

H. Kubota, S. Kawanishi, S. Koyanagi, M. Tanaka, and S. Yamaguchi, “Absolutely single polarization photonic crystal fiber,” IEEE Photon. Technol. Lett. 16, 182184 (2004).

Topa, A. L.

van Eijkelenborg, M. A.

Wong, K. H.

A. Argyros, M. Straton, A. Docherty, E. H. Min, Z. Ge, K. H. Wong, F. Ladouceur, and L. Poladian, “Consideration of chiral optical fibres,” Front. Optoelectron. China 3, 67–70 (2010).
[CrossRef]

Yamaguchi, S.

H. Kubota, S. Kawanishi, S. Koyanagi, M. Tanaka, and S. Yamaguchi, “Absolutely single polarization photonic crystal fiber,” IEEE Photon. Technol. Lett. 16, 182184 (2004).

Yoshino, T.

P. K. Choudhury, and T. Yoshino, “Characterization of the optical power confinement in a simple chirofiber,” Optik (Stuttg.) 113, 89–95 (2002).
[CrossRef]

Front. Optoelectron. China (1)

A. Argyros, M. Straton, A. Docherty, E. H. Min, Z. Ge, K. H. Wong, F. Ladouceur, and L. Poladian, “Consideration of chiral optical fibres,” Front. Optoelectron. China 3, 67–70 (2010).
[CrossRef]

IEE Proc., Optoelectron. (1)

R. C. Qiu, and I. T. Lu, “Dispersion in chiral optical fibres,” IEE Proc., Optoelectron. 145, 155–158 (1998).
[CrossRef]

IEEE Photon. Technol. Lett. (1)

H. Kubota, S. Kawanishi, S. Koyanagi, M. Tanaka, and S. Yamaguchi, “Absolutely single polarization photonic crystal fiber,” IEEE Photon. Technol. Lett. 16, 182184 (2004).

J. Appl. Polym. Sci. (1)

N. S. Pujari, M. R. Kulkarni, M. C. J. Large, I. M. Bassett, and S. Ponrathnam, “Transparent chiral polymers for optical applications,” J. Appl. Polym. Sci. 98, 58–65 (2005).
[CrossRef]

J. Lightwave Technol. (3)

J. Noda, K. Okamoto, and Y. Sasaki, “Polarization-maintaining fibers and their applications,” J. Lightwave Technol. 4, 10711089 (1986).
[CrossRef]

P. St. J. Russell, “Photonic-cystal fibers,” J. Lightwave Technol. 24, 47294749 (2006).
[CrossRef]

A. Argyros, “Microstructured polymer optical fibers,” J. Lightwave Technol. 27, 1571–1579 (2009).
[CrossRef]

J. Opt. Soc. Am. B (2)

Opt. Express (3)

Opt. I Spektrosk. (1)

F. I. Fedorov, “Contribution to the theory of the optical activity of crystals. 1. The law of conservation of energy and tensors of optical activity,” Opt. I Spektrosk. 6, 85–93 (1959).

Opt. Lett. (3)

Optik (Stuttg.) (1)

P. K. Choudhury, and T. Yoshino, “Characterization of the optical power confinement in a simple chirofiber,” Optik (Stuttg.) 113, 89–95 (2002).
[CrossRef]

Other (5)

A. W. Snyder, and J. D. Love, Optical Waveguide Theory, (Chapman and Hall, London 1983).

The optical activity of a material is commonly quoted as the rotation α, in degrees per decimetre, at 589 nm, at a specified temperature and concentration. Note we use different units in this paper.

D. B. Amabilino, Chirality at the nanoscale: Nanoparticles, surfaces, materials and more, (Wiley-VCH, Weinheim 2009).

T. M. Lowry, Optical rotatory power, (Dover Publications, New York 1964).

K. V. Varadan, V. V. Varadan, and A. Lakhtakia, Time-Harmonic Electromagnetic Fields in Chiral Media, (Springer-Verlag, 1989).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (4)

Fig. 1
Fig. 1

Schematic of the pure chiral optical fibres considered in this work: (a) step-chiral-index, (b) pure chiral Bragg fibre and (c) pure chiral photonic crystal fibre. The refractive index profiles seen by the two circular polarisations are shown, as well as the modes that may be supported through the various mechanisms. Typical mode effective indices are indicated by the red dashed lines.

Fig. 2
Fig. 2

Comparing dispersion curves for step-chiral-index and conventional step-index fibres. The chiral modes are shown with solid curves and the conventional modes are shown with dashed curves. The colours correspond to different values of the azimuthal index: m = 0 (black), mp = 1 (red), mp = −1 (blue), mp = 2 (green),and mp = −2 (magenta). The chiral modes are all of the one circular polarisation that is guided, as the other will see a lower core index compared to the cladding and not be guided. The chiral modes are solutions of Eq. (32), and the conventional modes of Eq. (33 ).

Fig. 3
Fig. 3

Contours showing the V-parameter for a step-chiral-index fibre as a function of wavelength and optical rotation α; remaining parameters as in the text. The grey area corresponds to V < 1 where the fibre is deemed to not guide effectively. Contours corresponding to constant γ are indicated in red and labelled with the value of α they correspond to at 589 nm.

Fig. 4
Fig. 4

Dispersion curves for m = 1 and m = 0. Top and bottom graphs in each pair give the effective modal indices and the loss, respectively. Left and right graphs give the results for left- and right-circular polarisation respectively.

Equations (40)

Equations on this page are rendered with MathJax. Learn more.

v LCP = c n ( 1 + δ ) , v RCP = c n ( 1 δ ) ,
B = n 1 δ n 1 + δ = 2 n δ 1 δ 2 ,
α = 1 2 k B k n δ .
D = ɛ ( E + γ × E ) ,
B = μ ( H + γ × H ) ,
δ = k n γ ,
( 1 δ 2 ) × E = i ω μ H + k n δ E ,
( 1 δ 2 ) × H = i ω ɛ E + k n δ H .
Q ± = E i Z H ,
× Q + = + k n 1 + δ Q + ,
× Q = k n 1 δ Q .
2 Q + + k 2 n 2 ( 1 + δ ) 2 Q + = 0 ,
2 Q + k 2 n 2 ( 1 δ ) 2 Q = 0 .
n p δ = n 1 + p δ
2 Q p + k 2 n p δ 2 Q p = 0 .
[ d 2 d r 2 + 1 r d d r + m 2 r 2 u p δ 2 ] Q z p ( r ) = 0 ,
u p δ = k n eff 2 n p δ 2 .
w p δ = k n p δ 2 n eff 2 .
Q ϕ p ( r ) = 1 u p δ 2 [ m k n eff r + p k n p δ d d r ] Q z p ( r ) = 1 u p δ 2 𝒟 Q z p ( r ) ,
( Q z p ( r ) Q ϕ p ( r ) ) = 𝒥 m , p ( u p δ , r ) ( A B ) = m , p ( u p δ , r ) ( C D ) = 𝒦 m , p ( w p δ , r ) ( E F ) ,
𝒥 m p ( u , r ) = ( J m ( u r ) Y m ( u r ) 1 u 2 𝒟 J m ( u r ) 1 u 2 𝒟 Y m ( u r ) )
m , p ( u , r ) = ( H m ( 1 ) ( u r ) H m ( 2 ) ( u r ) 1 u 2 𝒟 H m ( 1 ) ( u r ) 1 u 2 𝒟 H m ( 2 ) ( u r ) )
𝒦 m , p ( w , r ) = ( K m ( w r ) I m ( w r ) 1 w 2 𝒟 K m ( w r ) 1 w 2 𝒟 I m ( w r ) )
( Q z ( r n + 1 ) Q ϕ ( r n + 1 ) ) = 𝒯 m p ( u n , r n , r n + 1 ) ( Q z ( r n ) Q ϕ ( r n ) ) ,
𝒯 m , p ( u , r n , r n + 1 ) = m , p ( u , r n + 1 ) m , p ( u , r n ) 1 = 𝒥 m , p ( u , r n + 1 ) 𝒥 m p ( u , r n ) 1
( Q z ( r ) Q ϕ ( r ) ) = A 𝒥 m , p ( u c o , r ) ( 1 0 )
( Q z ( r ) Q ϕ ( r ) ) = C m , p ( u c l , r ) ( 1 0 )
( 0 1 ) m , p ( u c l , r ) 1 ( Q z ( r ) Q ϕ ( r ) ) = 0 .
( 0 1 ) m , p ( u c l , r N ) 1 [ i = 1 N 1 𝒯 m , p ( r i , r i + 1 , u i ) ] 𝒥 m , p ( u c o , r 1 ) ( 1 0 ) = 0 ,
( 0 1 ) 𝒦 m , p ( w c l , r N ) 1 [ i = 1 N 1 𝒯 m , p ( r i , r i + 1 , u i ) ] 𝒥 m , p ( u c o , r 1 ) ( 1 0 ) = 0 .
( 0 1 ) 𝒦 m , p ( w c l , a ) 1 𝒥 m , p ( u c o , a ) ( 1 0 ) = 0 .
u c o = k n co 2 n eff 2 , w c l = k n eff 2 n cl 2 .
n c o = n 1 | δ | > n 1 + | δ | = n c l
1 w c l 2 J m ( u c o r ) 𝒟 K m ( w c l r ) + 1 u c o 2 K m ( w c l r ) 𝒟 J m ( u c o r ) = 0
U = k a n co 2 n eff 2 , V = k a n co 2 n cl 2 , W = k a n eff 2 n cl 2
J m ( U ) U J m ( U ) + n cl n co K m ( W ) W K m ( W ) = m p n eff n co V 2 U 2 W 2 .
( J m ( U ) U J m ( U ) + K m ( W ) W K m ( W ) ) ( J m ( U ) U J m ( U ) + n cl 2 n co 2 K m ( W ) W K m ( W ) ) = m 2 n eff 2 n co 2 ( V 2 U 2 W 2 ) 2 .
N A = n c o 2 n c l 2 = 2 n δ 1 δ 2 2 n δ .
V = k a n c o 2 n c l 2 2 k a n δ .
α k n δ k n ( V 2 k a n ) 2 = V 2 4 k n a 2 .

Metrics