Dispersion in chiral optical fibres

R.C. Qiu I.-T.Lu

Indexing terms: Optical fibres, Chirality, Modal dispersion, Chromatic dispersion

Abstract: Through an example, for the first time it is theoretically shown that introducing chirality to the core of an optical fibre can change its modal dispersion, including its waveguide dispersion characteristics, drastically, and thus reduce the overall chromatic dispersion in a certain wavelength range. This may provide a novel approach to modifying and managing the chromatic dispersion of a fibre.

1 Introduction

Fibre technology has been revolutionised by the erbium-doped fibre amplifier (EDFA) [ 11. Regular replacement of EDFAs can effectively compensate for the signal power loss along the transmission fibre. The amplified spontaneous emission (ASE) noise generated by the EDFAs, and the chromatic dispersion and non- linearities in the transrnission fibre determine the maxi- mum bit rate that can be transmitted over a given distance. Thus the management of chromatic dispersion [2, 31, including waveguide dispersion and material dis- persion, has become increasingly essential because of the introduction of ETIFAS.

Recently, chirality, or optical activity, has received considerable interest both theoretically and experimen- tally [4-91. In optics, It has applications such as mode conversion [4, 61, fibre optical switches [8], and optical fibre transmission [9]. Chirality, or optical activity, introduces many new features, such as a low reflection return and broad bandwidth in scattering, mode bifur- cation and asymmetric field distribution in loaded waveguides, and polarisation flexibility and high effi- ciency in antennas [4, 51. It is also found to change the waveguide dispersion and the mode distribution in optical fibres [3, 91. Chirality or optical activity, known in optics after Argo (1811) and Pasteur (1848), is a geo- metric notion. It seems that the terminology of ‘optical activity’ is widely used in optics while ‘chirality’ is used in electromagnetic engineering. Optical active or chiral media are found to cause a rotation of the plane of polarisation of linearly polarised light or waves passing

0 IEE, 1998 IEE Proceedings online no. 1’2982081 Paper first received 30th Jariuary 1996 and in revised form 5th August 1997 R.C Qiu is with Bell Labs., Lucent Technologies, Mt. Olive, NJ 07828, USA I.-T. Lu is with the Department of Electrical Engineering, Weber Research Institute, Polytechnic University, Route 110, Farmingdale, NY 11735, USA

through them. In optics, many substances are now known to exhibit optical activity or chirality. Among these are quartz, cinnabar, sodium chlorate, turpentine, sugar, strychnine sulphate, tellurium, selenium and sil- ver thiogallate [lo].

loo c E Y E . 4

E 0 ._

CL .- 1

_-__________---------

0.001 0.0008 0.0006

0.0004

0.0002 Cc=O

-1 -50 00 1 i .o 1.5 2.0 2.5 3.0

wavelength h. ,prn Fig. 1 Comparison between waveguide dispersion D of a single-mode chiral fibre and the material dispersion of pure fused silica Fibre parameters: n, = n2 (1 + A), n2 = 1.46, A = 0.00815, NA = 0.1876, a = 2 . 4 ~ Material parameters: pure fused silica (SiOz). Note that the zero dispersion wavelength of pure fused silica (SO2) is 1 . 2 7 ~ and the curve 5 = 0 denotes the conventional singlemode optical fibre

In our previous work [3, 91, we found that adding chirality to the optical fibre results in unique features, and can modify fibre properties efficiently and conven- iently [9]. Before going into the technical details, we give some physical explanation as to why chirality sup- presses the chromatic dispersion in a certain wave- length range (but not in all). The fundamental reason is that the sign of waveguide dispersion (positive in the high chirality case and negative with low chirality at some long wavelengths) displayed by the chiral fibre (e.g. chiral core and achiral cladding) is different from the sign of the material dispersion (always negative in our work) in some wavelength ranges. Thus the overall chromatic dispersion is reduced dramatically in some wavelength ranges, due to the cancellation of the waveguide dispersion and material dispersion (see Fig. 1 for details). Chirality always amplifies the waveguide dispersion in the wavelength range we have studied. So the overall chromatic dispersion is a func- tion of chirality, and in a particular wavelength range could be reduced to zero by adjusting the chirality instead of fibre geometric configurations or material properties associated with them. The physical insights into this phenomenon are as follows: for the case of an achiral core, as the wavelength varies so the portion of the power travelling in the cladding changes, and thus the average refractive index experienced by the wave

155 IEE Proc-Optoelectron., Vol. 145, No. 3, June 1998

also changes. For the case of a chiral core [3]; the intro- duction of chirality in the core changes the power dis- tribution within a fibre, since a chiral-guided structure supports two bulk waves k, = .t. upi& + [k12 + (mpl,92]112 (parameters as defined in Section 2) called right and left circularly polarised waves, respectively. The interaction between the two bulk waves results in the overall guided wave modes (singlemode or multi- mode) and thus the average refractive index experi- enced by the wave. This interaction changes with variations of chirality and wavelength, and thus leads to the varying of the average refractive indexes and the derived waveguide dispersion.

In this paper, we continue to theoretically examine important fibre properties caused by the introduction of chirality into the core of a fibre, such as the normal- ised propagation constant, the normalised group delay, the normalised waveguide dispersion, etc. which have practical and important potential applications. In our work we assume that the total chromatic dispersion is obtained to a first approximation by the algebraic addition of the material dispersion and the waveguide dispersion, although the relationship between material dispersion and waveguide dispersion is quite complex, especially with the introduction of chirality in the fibre. We also assume that introducing chirality will not change the material dispersion. We have no motivation here to study the practical aspects of fabrication of such a fibre, which is itself a complicated issue. How- ever, with the richness of chirality displayed by rich substances and the rapid advances in material technol- ogy, we are confident that such a chiral fibre will be justified if sufficient special features are reported. In this spirit, this work will continue to contribute to the collection of special features. It is still an open issue whether or not the chiraIity we assumed in our example is too high. This is a value strongly dependent on the- state-of-the-art of the material technology. As argued in our previous work [3, 91, this chirality is not too high in theory. For example, as in Fig. 1, even for a large chirality of eC = 0.001 with S‘,,max = 0.00424 (assumed throughout our work), it only accounts for 23.56% of its maximum value. However, the waveguide dispersion can be strong enough to cancel the material dispersion in a certain wavelength range even for a small chirality of Ec = 0.0002, i.e. 4.7% of its maximum value. But many open issues associated with the achievable chiral- ity values still exist.

2 Formulation

Consider a step-index chiral fibre with a core including chiral material and conventjonal achiral cladding. The core has a radius ‘a’ and is characterised by permeabil- ity pl, permittivity and chiral impedance EC. The cladding is characterised by permeability p2 and permittivity E ~ . The wavenumbers k, are defined as m(p

and the refraction index ni as k, ko where I = 0, 1 and 2 correspond to the free space, the chiral core and the cladding, respectively. In the chiral core, the consti- tutive relations are coupled

D = EIE + j&B; IEI J&E + (I/pi)B where E, B, D and H are electromagnetic field vectors. Owing to this coupling relation, the chiral core sup- ports two wave species with wavenumbers k , = +- o.y1& + [kI2 + (~op~E,-)~]~’~. The corresponding refraction indexes n, are defined as kJko [4]. To find the propaga-

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tion characteristics of the chiral fibre, the coupled con- stitutive relations of the chiral materials and the Maxwell equations are first applied, leading to the cou- pled vector wave equations. Since the vector wave equations are separable for waveguides with circular cylindrical symmetry, they can be transformed into coupled scalar wave equations. The waveguide propa- gation and attenuation constants are then obtained by solving the corresponding eigenvalue equations [9]. Fol- lowing the conventional fibre notations, the normalised frequency V is defined as koa(NA), and the normalised propagation constant b is defined as [(/3//~,)~ - n22]/ (NA)2 where N A = (nL2 - n22)1’2 is the numerical aper- ture. The advantage of using normalised parameters is that the obtained results will be valid for a class of chi- ral fibres and will not be limited to a specific set of geometric and material parameters.

3 Results and discussions

The nature of Figs. 2-4 is of general significance in the sense that the normalised parameters are specified. We focus on reporting the facts associated with changing a normalised frequency V. In Fig. 1, we try to give a spe- cial case based on Fig. 4. The normalised propagation constants of the fundamental mode HElI and the first three high-order modes HEol, HEo*, HEl2 in the step- index chiral fibre are shown in Fig. 2 as functions of the normalised frequency. This plot is valid for a group of step-index chiral fibres with a normalised frequency V. It is worth pointing out that the modes here are due to the overall interaction of two wave species k , and k- in the guided structure, instead of only associating with one individual wave species. The relation between the propagation constants of these modes and the two wave species k, and IC has been obtained in the eigen- value equation in our previous work [3, 91. It is obvious that adding chirality considerably increases the normal- ised propagation constant b. It is true that increasing the ordinary dielectric constant of the core will increase the normalised propagation constant b, but some situa- tions exist where the ordinary dielectric constant is fixed due to some specific constraints, such as the waveguide dispersion requirement. As in the case of chirality increasing the group delay and waveguide dis- persion in Figs. 1, 3 and 4, we have been trying to report the facts associated with changing a fibre chiral- ity. We do not have a particular application, such as dispersion compensation, in mind. The slope of the b curve of the chiral fibre is much larger than that of its conventional counterpart near the cutoff range. Since the parameter b represents the fraction of the total mode field within the core of the fibre, b = 0 (/3 = n2ko) specifies the cutoff condition of fibre modes. This is because the fraction of field within the core tends to zero as the mode is cut off. When the normalised fre- quency V increases, b increases. In a conventional achi- ral fibre h must be less than one because the /3 of a propagating mode is less than nlko. In the chiral core fibre, the permissible range of j3 of a propagating mode is between n2ko and the larger of n+ko, and n_ko. Thus, b will not be bounded by 1. It is also interesting to note that the cutoff frequency of the fundamental mode decreases as V increases. Fig. 3 shows the normalised group delay d( Vb)ldV as a function of the V parameter for the fundamental mode HE11 of the chiral fibre. This plot is valid for a group of step-index chiral fibres with a normalised frequency V. The chirality greatly

IEE Proc -0ptoslectran , Vol 145 No 3, June 1998

increases the normalised group delay. For instance, for a conventional fibre, d(Vb)/dV is less than one for V < 2.0, but in a chiral fibre, d( Vb)/dV is as large as over 30 for chirality 5, = 0.0008. For characterising pulse prop- agation in a singlemode chiral fibre, the dispersion determines the pulse spreading. For some applications such as fibre equalisers and sensors, a large group delay is favoured. Fig. 4 shows the effect of chirality on the normalised waveguide dispersion Vu?( Vb)IdV2 as a function of the V parameter for the fundamental mode HE,,. This plot is valid for a group of step-index chiral fibres with a normalised frequency V. It is very interesting to notice that for a large enough chirality, say & = 0.0004, V&(Vb)/dV2 of a chiral fibre is oppo- site in sign to that of its conventional counterpart. Also the absolute value of the normalised waveguide disper- sion V&(Vb)/dp is much larger than that of its con- ventional achiral counterpart. The waveguide dispersion D is defined as d2/3/dcodA where A = 2dk0 is the wavelength in free space. The fact that chirality increases the normalised waveguide dispersion Vu?( Vb)l dP is not necessarily a disadvantage or an advantage, this depends on the particular application. For exam- ple, in the case of Fig. 1, this large waveguide disper- sion can be an advantage.

f) 4.2

m E 2.0 0 C .- I

P e 0. Q

HE1 1

HE01

HE1 2

0.0 0.0 1.0 2.0 3.0 4.0 5.0 6.0

normalised frequency V Fig.2 Effect of chirality ::c on the normalisedpropagation constant h of the funhmentul mode HE and the first three hi h order modes HE,,, HE,,, and HEIz us a funcii& of the Vparameter oft& chiral fibre The dotted line denotes the conventional optical fibre (5, = 0) and the solid lines denote the chiral optical fibre with chiral admittance 5, = 0.0004

0.0008

0.0006 20

0.0004

0.0002

lo t 1 .o 1.5 2.0 2.5 3.0 0

normalised frequency,V Fig . 3 funcrion o h h e Vparameter for the fundamental mode HE,,

IEE Proc-Optoelectron., Vol. 145, No. 3, June 1998

Efect of chirality on the normulised group delay Vd( Vb)/dV as a

0.0002

-5.0 1 .o 1.5 2.0 2.5 3.0

normalised frequency V Fi .4 Effect of chirality on the normalised waveguide dispersion Vj(Vbj/dP us a function of the Vpavameter,for the fundamental mode HE, 1

To demonstrate a special application of dispersion in a chiral fibre, we studied how the waveguide dispersion can cancel the material dispersion and reduce the over- all chromatic dispersion in a certain wavelength range, depending on specific materials considered. Fig. 1 shows the waveguide dispersion D as a function of the wavelength A of a certain chiral optical fibre with vari- ous chiral admittance values. Without loss of general- ity, we just use pure fused silica as an example illustrating the underlying idea that chirality can mod- ify the total chromatic dispersion in a certain wave- length range. We could use other materials, such as a Ge-doped fibre, where fibre photosensitivity was used to induce chirality. But we will report that elsewhere. The material dispersion of pure fused silica is shown by the curve running from bottom left to top right cross- ing the other curves in Fig. 1. The total chromatic dis- persion is then obtained by algebraically adding the waveguide dispersion and the material dispersion. For the cases of nonzero chiral admittance, the waveguide dispersion and the material dispersion are opposite in sign and cancel each other when the operating wave- length d is less than 1.27pm. They are identical in sign and intensify each other when the operating wavelength A is greater than 1 . 2 7 ~ . Here, the wavelength of 1 . 2 7 ~ is the zero chromatic dispersion wavelength of this particular step-index chiral fibre. This value has no special significance to our study, and could be another number if other materials were considered. We can tune the chiral admittance to design transmission links with zero dispersion at wavelengths less than 1 . 2 7 ~ .

4 Conclusions

Based on conventions in an achiral fibre, to determine the complete dispersion in a fibre, we have studied b ( V , its slope d(Vb)/dV, and the rate of change of the slope V&(Vb)/dV2. The conclusions drawn work for a group of fibres satisfying the normalised frequency V. There are two free parameters V and chirality Ec instead of only V in a chiral case. Some interesting fea- tures are observed in the step-index fibre with a chiral core and achiral cladding. Adding chirality to a con- ventional fibre can modify its dispersion properties, such as the mode distribution, the propagation con- stant, the cutoff wavelength, the effective group index

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and the waveguide dispersion. The role of chirality is to make the chiral fibre an effective bulk medium with a larger group index. With respect to a single mode, its effect is similar to increasing the numerical aperture NA or decreasing the radius of the core of a conven- tional achiral fibre. The normalised propagation con- stant can be larger than one. The normalised waveguide dispersion of a chiral fibre with sufficiently large chirality is opposite in sign to that of its conven- tional counterpart without chirality. For some specific chiral fibres, when the operating wavelength h is less than 1 . 2 7 , ~ ~ the waveguide dispersion and the mate- rial dispersion cancel each other, and when A is greater than 1.27,um, the waveguide dispersion and the mate- rial dispersion are identical in sign and intensify each other. For the specific example shown in this paper, we can tune the chiral admittance to design fibre transmis- sion links with zero dispersion at wavelengths less than 1 . 2 7 ~ . With other materials used, we could obtain a zero dispersion wavelength diCferent from 1.27 pn, depending on our need. The conclusions drawn in this paper may still be valid for fibres with more compli- cated refractive index profiles that introduce extra waveguide dispersion, called profile dispersion. The relation between the profile dispersion and the normal- ised parameters such as b(V), d(V6)IdV and V@(Vb)l dV2 is a future study topic.

Introducing chirality into a fibre can modify the chromatic dispersion dramatically. This fact may have an impact on some future subjects of study [l], such as solitons, phase conjugation and wavelength division multiplexing, as chromatic dispersion is involved in these systems. Of particular relevance is the potential application of this fact in such areas as chromatic dis- persion compensation in coherent optical communica- tions [2], where waveguide dispersion of equalisers is utilised to cancel conventional fibre chromatic disper- sion. Finally, the new features displayed by a chiral

fibre may have potential applications in many fields, such as long-distance telecommunications, optical pulse shape equalisers, fibre dispersion shifts, optical switches, couplers, sensors, etc. Further discussions on the issues of potential applications are beyond the scope of this short paper.

5 Acknowledgments

The authors thank one of the reviewers for pointing out the necessity for a physical explanation of why chi- rality suppresses dispersion.

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