# Resonant Transmission Line Method for Unconventional Fibers

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Mathematical Equivalence of Homogeneous Circular Cylindrical Layers to Electric Transmission Lines

#### 2.1. Decoupling the Transmission Line Equations

#### 2.2. Equivalent Circuits for Cylindrical Layers, Boundary Conditions, and Birefringence

_{M},I

_{M}) are the TM modes, while the modes (V

_{E},I

_{E}) are the TE modes. For $l$ > 0, the modes (V

_{M},I

_{M}) are the HE modes, while the modes (V

_{E},I

_{E}) are their HE birefringence modes. For $l$ <0 the modes (V

_{M},I

_{M}) are the EH modes, while the modes (V

_{E},I

_{E}) are their EH birefringence modes. For any given $l$, using the resonance technique the β values of the two birefringence modes can be calculated. The Equation (16) is given as a MATLAB function in Appendix A.

_{1}= 1.54, n

_{2}= 1.47 the V

_{M}, V

_{E}, fundamental modes for V = 3.3, can be calculated and their β/k

_{0}values are respectively 1.518934962534846 and 1.518340184686295, hence their birefringence is equal to 0.0004947 or 0.0391%. The β/k

_{0}value for the equivalent mode V

_{eq}was also calculated and was equal to 1.518638548412019 (that is approximately equal to the mean value of the previous β/k

_{0}values), while the β/k

_{0}value calculated classically by Bessel functions is equal to 1.518642063686336. These β values are very close differing only 0.0002315%.

_{1}= 1.54, n

_{2}= 1.47, and of n

_{1}= 1.475, n

_{2}= 1.47 as functions of V are shown.

^{2}= (n

_{1}− n

_{2})

^{2}, thus the birefringence of step-index fibers of very small Δn is negligible. For instance, for a value of V = 2.4, and Δn = 1.54 − 1.47 = 0.07, the birefringence is found to be 0.168 × 0.0049 = 0.0008232 or ~0.055% on the average β, while for Δn = 1.475 − 1.47 = 0.005, the birefringence becomes 0.168 × 0.000025 = 0.000042 or ~0.0028% on the average β. What is remarkable is that our method is sensitive and calculates it.

#### 2.3. Calculating “Voltages” V_{M}, V_{E} and “Currents” I_{M}, I_{E} and Resulting Fields

_{M}= 1 at the center point of the fiber (r = 0), the respective value of I

_{M}at the same point can be calculated by the respective terminal impedance. Using the matrix relations between input–output for the equivalent successive T-circuits, the values of ${V}_{M}$ and ${I}_{M}$ at the rest thin cylindrical layers can be calculated. In fact, from the general theory of the telegrapher’s equation we know that the inputs and outputs are associated via a transfer matrix as follows

_{i}. Finally, we obtain the actual fields via the relations

## 3. Unconventional Fibers

#### 3.1. UOF with Non-Circular, Non-Symmetric, or Eccentric Cores

_{k}of the function $n{\left(\phi \right)}^{2}$ are negligible in comparison to its steady component ${n}^{2}$ and can be omitted.

^{2}is the average value of the η

^{2}(φ) of each layer along φ in the [0, 2π] interval.

#### 3.2. Application to Elliptic Core Fibers

_{1}= 1.54, and a cladding value of n

_{2}= 1.47 (Figure 5) for various wavelengths (defined by various V factor values $V=2\pi b\sqrt{{n}_{1}^{2}-{n}_{2}^{2}}$) and four ellipticity ratios a/b = 1.1, 1.3, 1.5, and 2.0. Results are presented in tabulated format (Table 1, Table 2, Table 3 and Table 4) compared with previous results calculated with Mathieu functions together with differences and relative differences showing a deviation which goes as only 0.01/0.123% on the average. Note the results in Table 1, Table 2, Table 3 and Table 4 are not normalised the same way as in Figure 4, they are simply the odd modes, b

_{11}= β

_{o}/k

_{o}. We can see that for small ellipticities especially, the results compare very well for all V values quite well with the Mathieu Functions results. As the ellipticity becomes very large as in Table 4, the results begin to differ. However, the accuracy of the Mathieu Functions used in this case is not known precisely, so the trend is correct and the actual difference could be debated.

_{1}for r < b, n

_{2}for r > a, and as $({n}_{1}{\phi}_{1}+{n}_{2}{\phi}_{2})/\pi $ when b < r < b, where φ

_{1}, φ

_{2}are the arcs of the circle of radius r, inside and outside the ellipse in the upper semi ellipse.

#### 3.3. Application to a Rectangular Core Fiber

_{1}= 1.54, and a cladding of refractive index n

_{2}= 1.47, for various wavelengths, defined by various V factor values $V=bb\xb7{k}_{0}\xb7\sqrt{{n}_{1}^{2}-{n}_{2}^{2}}$ and four ratios a/b = 1.1, 1.3, 1.5, and 2.0.

_{1}, φ

_{2}being the arcs of the circle of radius r, inside and outside the rectangle in the upper half plane.

_{0}, for the rectangular and elliptical waveguides. We can see the elliptical waveguide has greater birefringence in all cases of ellipticity and for all wavelengths (represented by the V value).

#### 3.4. The PCF Case

_{1}− r

_{2}. We can then approximate n(r,φ) as n(φ) for the average r <r> = r + δr/2. The refractive index can be written as a Fourier series, i.e., as $n{\left(\phi \right)}^{2}={\langle n\rangle}^{2}+{{\displaystyle \sum}}_{-\infty}^{+\infty}{N}_{k}\mathrm{exp}\left(jl\phi \right)$. Taking into account the properties of the Fourier Transform we see that $FT(\mathrm{exp}(jl\varphi )\cdot f(\phi ))=f(l+{l}^{\prime})$ so that the expressions in the second terms of Equation (18) spread around a spectrum of harmonics. This is also to be understood as a result of successive scatterings from the bored air holes. We can now use the natural geometry of the usual hexagonal lattice to see that for each set of holes we can have either 6k harmonics. For the fundamental harmonic of ${l}^{\prime}$ = 1, the derived harmonics passing through a layer of 6k holes should be 6k + 1. Thus, for the fundamental wave crossing the successive layers it “sees” a different set of periodic rectangle functions that will be shown rigorously to contribute a different number of harmonics (7, 13, 19, …).

_{0}is cut while moving clockwise along the large circle.

_{1}for the silica refractive index, n

_{2}≃ minimum refractive index = 1.123, Λ for the reduced air hole distance, d for the reduced air hole diameter, and $\Lambda -d/2$ = reduced inner core of PCF.

## 4. Conclusions

## Funding

## Conflicts of Interest

## Appendix A

## Appendix B

## References

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**Figure 1.**Example of the alternating character of the local index value from a thin shell radial discretization.

**Figure 2.**Schematic depiction of the alternating step index resulting from radial discretization in a standard photonic crystal fiber (PCF).

**Figure 4.**Normalized birefringence of two step-index fibers with different refractive indexes as functions of their parameters V.

**Figure 5.**Elliptic fiber with three indicative elliptic thin layers. Inside the ellipse r < b (n = n

_{1}), outside the ellipse r > a (n = n

_{2}) and partly outside b < r < a (n

_{1}> n > n

_{2}).

**Figure 6.**β-V diagram of the even fundamental mode an elliptic fiber of semi axis ratio aa/bb = 2 and core refractive index 1.54 and cladding index 1.47.

**Figure 7.**Rectangular core fiber of semi-sides aa and bb, for r < bb, n = n

_{1}, for r > aa, n = n

_{2}, for bb < r < aa, n

_{2}< n < n

_{1}.

**Figure 8.**Average refractive indexes of circular thin layers of elliptic and rectangular core fibers.

**Figure 10.**Average electric field of the even fundamental mode along the radial distance of the PCF.

a/b = 1.1 | Mathieu | RTL | Differences | Relative Diff. (0/00) |
---|---|---|---|---|

V | b_{11, Nο} | b_{11, Νο} | ||

1.5 | 1.487454917000000 | 1.48753837672558 | −0.00008345972558 | 0.056109079 |

1.7 | 1.493245000000000 | 1.49345098615188 | −0.00020598615188 | 0.137945315 |

1.9 | 1.498457700000000 | 1.49872279558264 | −0.00026509558264 | 0.17691229 |

2.1 | 1.503029750000000 | 1.50331246585078 | −0.00028271585078 | 0.188097309 |

2.3 | 1.506994250000000 | 1.50727053366314 | −0.00027628366314 | 0.183334252 |

2.5 | 1.510418500000000 | 1.51067588097959 | −0.00025738097959 | 0.170403752 |

2.7 | 1.513376830000000 | 1.51360949624267 | −0.00023266624267 | 0.153739794 |

2.9 | 1.515936970000000 | 1.51614477492828 | −0.00020780492828 | 0.13708019 |

3.1 | 1.518160870000000 | 1.51834484072766 | −0.00018397072766 | 0.121179996 |

3.3 | 1.520101500000000 | 1.52026268667918 | −0.000161186679180 | 0.106036787 |

a/b = 1.3 | Mathieu | RTL | ||
---|---|---|---|---|

V | b_{11, Nο} | b_{11, Νο} | Differences | Relative Diff. (0/00) |

1.5 | 1.491188512000000 | 1.491027765079550 | 0.000160746920450 | 0.107797853 |

1.7 | 1.497028990000000 | 1.496897637511800 | 0.000131352488200 | 0.087742114 |

1.9 | 1.502119714000000 | 1.501986592174880 | 0.000133121825120 | 0.088622647 |

2.1 | 1.506471523000000 | 1.506335918376860 | 0.000135604623140 | 0.090014727 |

2.3 | 1.510205927000000 | 1.510039049600890 | 0.000166877399110 | 0.110499764 |

2.5 | 1.513423500000000 | 1.513196002523000 | 0.000227497477000 | 0.150319773 |

2.7 | 1.516170122000000 | 1.515897403413190 | 0.000272718586810 | 0.179873342 |

2.9 | 1.518513480000000 | 1.518220317018870 | 0.000293162981130 | 0.193059189 |

3.1 | 1.520539300000000 | 1.520228501721160 | 0.000310798278840 | 0.204400030 |

3.3 | 1.522298190000000 | 1.521974104375010 | 0.000324085624990 | 0.212892341 |

a/b = 1.5 | Mathieu | RTL | Differences | Relative Diff. (0/00) |
---|---|---|---|---|

V | b_{11, Nο} | b_{11, Νο} | ||

1.5 | 1.494250610000000 | 1.493636836360350 | 0.000613773639650 | 0.410756827 |

1.7 | 1.499922346000000 | 1.499340905733040 | 0.000581440266960 | 0.387646913 |

1.9 | 1.504818670000000 | 1.504203108321470 | 0.000615561678530 | 0.409060368 |

2.1 | 1.509039170000000 | 1.508315227121150 | 0.000723942878850 | 0.479737633 |

2.3 | 1.512533150000000 | 1.511793195339390 | 0.000739954660610 | 0.489215500 |

2.5 | 1.515493100000000 | 1.514745797603970 | 0.000747302396030 | 0.493108412 |

2.7 | 1.518011570000000 | 1.517265908951310 | 0.000745661048690 | 0.491209068 |

2.9 | 1.520166130000000 | 1.519429880525980 | 0.000736249474020 | 0.48432172 |

3.1 | 1.522019120000000 | 1.521299532774000 | 0.000719587226000 | 0.472784617 |

3.3 | 1.523620800000000 | 1.522924688766490 | 0.000696111233510 | 0.456879582 |

a/b = 2 | Mathieu | RTL | Differences | Relative Diff. (0/00) |
---|---|---|---|---|

V | b_{11, Nο} | b_{11, Νο} | ||

1.5 | 1.499390500000000 | 1.497675992184210 | 0.001714507815790 | 1.143469840 |

1.7 | 1.504727250000000 | 1.502885066432660 | 0.001842183567340 | 1.224264110 |

1.9 | 1.509110880000000 | 1.507251064452310 | 0.001859815547690 | 1.232391584 |

2.1 | 1.512712190000000 | 1.510915134944720 | 0.001797055055280 | 1.187968913 |

2.3 | 1.515667864000000 | 1.514006539095880 | 0.001661324904120 | 1.096100896 |

2.5 | 1.518085750000000 | 1.516632675098700 | 0.001453074901300 | 0.957175773 |

2.7 | 1.520047200000000 | 1.518879733445780 | 0.001167466554220 | 0.768046252 |

2.9 | 1.521631209000000 | 1.520816115931370 | 0.000815093068630 | 0.535670578 |

3.1 | 1.522869900000000 | 1.522496060692140 | 0.000373839307860 | 0.245483418 |

3.3 | 1.523799100000000 | 1.523962746538420 | −0.000163646538420 | 0.107393775 |

Rectangular Core | Elliptic Core | ||
---|---|---|---|

a/b = 1.1 | Fundamental Mode Values | Birefringence (TR) | Birefringence (TR) |

V | |||

1.5 | 1.492539945916010 | 0.000346704531850 | 0.000303585935370 |

1.7 | 1.498357447022790 | 0.000513384576600 | 0.000575417316090 |

1.9 | 1.503350570368440 | 0.000576764589930 | 0.000712035787010 |

2.1 | 1.507590076049970 | 0.000580793732230 | 0.000757892695190 |

2.3 | 1.511183551577450 | 0.000554099561380 | 0.000750121835720 |

2.5 | 1.514237341469060 | 0.000513428535340 | 0.000713597265680 |

2.7 | 1.516844499544290 | 0.000468033918970 | 0.000663528284040 |

2.9 | 1.519082616576080 | 0.000422786387590 | 0.000608733773240 |

3.1 | 1.521015087494710 | 0.000380083447450 | 0.000554096605350 |

3.3 | 1.522693317372840 | 0.000340959781570 | 0.000502161305990 |

Rectangular Core | Elliptic Core | ||
---|---|---|---|

a/b = 1.3 | Fundamental Mode Values | Birefringence (TR) | Birefringence (TR) |

V | |||

1.5 | 1.495847758922450 | 0.000344065006270 | 0.000325722889410 |

1.7 | 1.501459058153440 | 0.000451639480470 | 0.000520210503450 |

1.9 | 1.506179273220210 | 0.000480411129310 | 0.000601842058430 |

2.1 | 1.510134432440230 | 0.000468166987430 | 0.000615857816190 |

2.3 | 1.513456882268610 | 0.000436878544030 | 0.000593498171210 |

2.5 | 1.516262818410440 | 0.000398422704380 | 0.000553682765940 |

2.7 | 1.518648025192190 | 0.000358904227450 | 0.000507155757960 |

2.9 | 1.520689537973830 | 0.000321272665480 | 0.000459732120240 |

3.1 | 1.522448794667530 | 0.000286796641510 | 0.000414381305640 |

3.3 | 1.523974761485380 | 0.000255871394240 | 0.000372472456950 |

Rectangular Core | Elliptic Core | ||
---|---|---|---|

a/b = 1.5 | Fundamental Mode Values | Birefringence | Birefringence (TR) |

V | |||

1.5 | 1.498120450723510 | 0.000278361237480 | 0.000295449461120 |

1.7 | 1.503463400911670 | 0.000357573590440 | 0.000442306154260 |

1.9 | 1.507908503417930 | 0.000376605061620 | 0.000497613358440 |

2.1 | 1.511610102671980 | 0.000365648210230 | 0.000501375052490 |

2.3 | 1.514709614765410 | 0.000341283334480 | 0.000478771792790 |

2.5 | 1.517324041962060 | 0.000312143747180 | 0.000444266137270 |

2.7 | 1.519546684126380 | 0.000282536889750 | 0.000405759261530 |

2.9 | 1.521451022056960 | 0.000254483058070 | 0.000367378187930 |

3.1 | 1.523094834995030 | 0.000228815613970 | 0.000331142666700 |

3.3 | 1.524523718254910 | 0.000205765806060 | 0.000297919827130 |

Rectangular Core | Elliptic Core | ||
---|---|---|---|

a/b = 2 | Fundamental Mode Values | Birefringence (TR) | Birefringence (TR) |

V | |||

1.5 | 1.501154370588750 | 0.000115100455570 | 0.00017851280204 |

1.7 | 1.505905455821760 | 0.000185956233950 | 0.00028097261437 |

1.9 | 1.509839664490200 | 0.000217496051150 | 0.00032331073887 |

2.1 | 1.513123501262850 | 0.000227441962700 | 0.00033221109868 |

2.3 | 1.515890438928040 | 0.000225620967080 | 0.00033221109868 |

2.5 | 1.518243818452060 | 0.000217498908240 | 0.00030589412246 |

2.7 | 1.520263082984900 | 0.000206163759050 | 0.00028459934024 |

2.9 | 1.522009559228890 | 0.000193389032200 | 0.00026227629912 |

3.1 | 1.523531014266760 | 0.000180203858540 | 0.00024038749997 |

3.3 | 1.524865056067340 | 0.000167204354840 | 0.00021967555642 |

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**MDPI and ACS Style**

Boucouvalas, A.C.; Papageorgiou, C.D.; Georgantzos, E.; Raptis, T.E. Resonant Transmission Line Method for Unconventional Fibers. *Appl. Sci.* **2019**, *9*, 270.
https://doi.org/10.3390/app9020270

**AMA Style**

Boucouvalas AC, Papageorgiou CD, Georgantzos E, Raptis TE. Resonant Transmission Line Method for Unconventional Fibers. *Applied Sciences*. 2019; 9(2):270.
https://doi.org/10.3390/app9020270

**Chicago/Turabian Style**

Boucouvalas, Anthony C., Christos D. Papageorgiou, Eurypides Georgantzos, and Theophanes E. Raptis. 2019. "Resonant Transmission Line Method for Unconventional Fibers" *Applied Sciences* 9, no. 2: 270.
https://doi.org/10.3390/app9020270