# Effective refractive index of three-layer modes

This tutorial demonstrates the use of the Optical Fibre Toolbox for calculation of the three-layer (core-cladding-surrounding) fibres. First, the specification of such structures is explained. Then the effective refractive index of the guided modes is calculated vs. fibre diameter (typical task for tapered microfibres).

(cc-by) K. Karapetyan et al., AG Meschede, Uni Bonn, 2011

http://agmeschede.iap.uni-bonn.de | kotya.karapetyan@gmail.com

## Contents

calculatemodes = true; % decide to enable or disable (for speed) calculation if ~calculatemodes % if calculation is disabled the data should be previously stored in a mat file load tutorial3ls.mat modesMonerieCore modesTsaoTClad modesTwoClad modesTwoCore modesErdoganClad end

## Three layer fibre

Usually, optical fibres are considered consisting of just two layers: the core and the cladding. Light field is well confined in the core and in the evanescent field around it in the cladding and does not reach the external surface of the cladding.

However, there are cases when this model is not valid. One of them is the tapered optical microfibre, consisting of the untapered ends, tapers, and micrometer-diameter waist. In such fibres, the light is first guided as usual, by the core in the untapered part. In the taper region, the light field expands and is no longer confined close to the core. At some point, the guidance is actually not by the core-cladding interface, but by the cladding-surrounding (usually --- air, liquids, or special coating having the refractive index lower than that of the cladding). In the waist the core is so small in diameter that it can be neglected in most cases and the two layer model can again be used, this time for cladding-surrounding. So, it is the taper region where the two-layer model is not enough to simulate light propagation in the fibre, and OFT provides the three-layer solutions for these cases.

## Known solutions

Simulation of the guided modes in the three-layer structure is an analytically challenging task. Several attempts have been made to publish the solutions in the literature. In 1976, BELANOV et al. have made the first known to me attempt to write the equations for the three-layer structure. However they have only solved them to the end for the weakly guiding fibre (linearly polarised modes approximation). It was also independently done by MONERIE in 1982. In 1989, TSAO has made an attempt to obtain the solution for the HE/EH, TE and TM modes, i.e. the full vector exact solution, in the three-layer structure. Unfortunately, while the published results are fine for TE and TM modes, the equations for the hybrid modes probably contains some typo and do not lead to the correct numerical solution. In 1997 ERDOGAN has approached the same problem once again and published the solution for the cladding modes (when light is guided by the cladding-surrounding interface) and suggested using the two-layer LP solution for the core-guided modes. His paper also contained a number of typos, and the errata was published in 2000. Finally, in 2005 ZHANG and SHI published the full solution using the same approach as TSAO (1989). Unfortunately, this solution again seems to contain a mistake.

The above relates to the eigen-value equation used to calculate the effective refractive index (or the propagation constant) of the guided mode. For full simulation, the light field (E and H vectors) should also be calculated. This is not implemented yet due to some problems with the references. BELANOV (1976) and MONERIE (1982) do not give the explicit solutions. ERDOGAN (1997, 2000) only gives it for the cladding-guided modes. ZHANG (2005) was not yet checked.

## Simulation of n_eff in the tapered microfibre

I suggest the following approach to simulation of the three-layer modes of the tapered microfibres. Initially, the fibre is weekly guiding and the Monerie solution is appropriate to calculate the effective refractive index in the core (will be proven below). As soon as the effective refractive index reaches the refractive index of the cladding, Erdogan solution is applied. In the region where the effective refractive index approaches that of the surrounding, this solution gets close to the two-layer solution --- classical HE/EH, TE and TM (proven below). modes), but there is no need to do so.

Let's first calculate the exact two-layer modes to use them as a reference.

We start by specifying the three materials in the order core, cladding, surrounging. As usual, the materials can be specified with refractive indices or names known to `refrIndex` function. In the latter case, material dispersion is automatically taken into account. In the former case it is ignored.

materials = {1.455, 1.45, 'air'}; fibre.materials = materials; fibre.coreCladdingRatio = 5/125; % Specify the simulation task, first for the core task = struct(... 'nu', [0 1],... % first modal index 'type', {{'hybrid', 'te', 'tm'}},... % mode types 'maxmode', Inf,... % how many modes of each type and NU to calculate 'lambda', 900,... 'region', 'core'); % In the microfibre taper, the diameter varies from the diameter of the % microwaist upto the diameter of the standard untapered fibre (125 um). % All fibre dimensions in OFT are specified in micrometers, all wavelength % values in nanometers. argument = struct(... 'type', 'dia',... % calculate mode curve n_eff vs. fibre diameter 'harmonic', 1,... % required 'min', 0.01,... % minimum diameter 'max', 125); % maximum diameter addpath('..'); % path to OFT functions infomode = false; % if true, OFT functions will output more information and % illustrate the simulation process with more pictures

Now calculate the mode curves:

if calculatemodes modesTwoCore = buildModes(argument, fibre, task, infomode); end

Which mode was found?

t = modeDescription(modesTwoCore, false, false); if iscell(t) for i = 1:numel(t)-1, fprintf(' %s,', t{i}), end, fprintf(' %s\n', t{end}); else fprintf(' %s\n', t); end

HE 1 1

This is a so-called single-mode fibre similar to the well-known Fibercore SM800: in the core it guides only the fundamental mode HE_11 for all wavelengths above approximately 800 nm.

Display calculated curve

hTwoCore = showModes(modesTwoCore); set(hTwoCore, 'Color', 'black')

Now repeat the same for the cladding modes. There is a lot of modes that can be guided in the cladding with 125 um diameter. To calculate all of them would take quite a bit of time, so we will calculate only the first mode of each type (hybrid, TE and TM).

We can use the same task specified above, just change it a little.

task.region = 'cladding'; task.maxmode = 1; % Calculate only the first of all families of modes, i.e. % HE11, TE01 and TM01 if calculatemodes modesTwoClad = buildModes(argument, fibre, task, false); end hTwoClad = showModes(modesTwoClad); set(hTwoClad, 'Color', 'black') set(hTwoClad(2), 'LineStyle', '--') % Make TE curve dashed

Which modes have been calculated?

t = modeDescription(modesTwoClad, false, false); fprintf('Calculated cladding two-layer modes: '); if iscell(t) for i = 1:numel(t)-1, fprintf(' %s,', t{i}), end, fprintf(' %s\n', t{end}); else fprintf(' %s\n', t); end

Calculated cladding two-layer modes: HE 1 1, TE 0 1, TM 0 1

The core modes are basically invisible now because

We can resolve the curves by zooming vertically:

ylim([1.445 1.455])

The HE11 mode in the core does not reach the diameter of zero microns, due to numerical limitations. Theoretically, HE11 mode is guided at any fibre diameter, however small.

The TE_01 and TM_01 cladding modes are almost coinciding. We can zoom horizontally:

```
ylim auto
xlim([0 10])
```

## Monerie modes

Let's now calculate the Monerie modes in this fibre. In a separate demo on Monerie modes, I show that they should be traced from the core. We are interested in the LP01 mode corresponding to the HE11 mode as well as in LP11 corresponding to TE01 and TM01 modes. Our fibre core is single-mode, the LP11 mode is not supported. In order to trace it, we artificially increase the maximum considered diameter so that this mode is found and traced. Note: if argument.max is set to 180, there is a mode jump as explained in the Monerie tutorial

task = struct(... 'nu', [0 1 2],... % first modal index 'type', {{'monerie'}},... % mode types 'maxmode', Inf,... % how many modes of each type and NU to calculate 'lambda', 900,... 'region', 'core'); argument.max = 200; if calculatemodes modesMonerieCore = buildModes(argument, fibre, task, false); end hMonerieCore = showModes(modesMonerieCore); set(hMonerieCore, 'Color', [1 0.5 0.5], 'LineWidth', 5) set(hMonerieCore(2), 'LineStyle', '--') uistack(hTwoCore, 'top') uistack(hTwoClad, 'top') xlim([0 125]) % Which modes have been found? t = modeDescription(modesMonerieCore, false, false); fprintf('Monerie (nu=[0 1 2]) modes found in the cladding: '); if iscell(t) for i = 1:numel(t)-1, fprintf(' %s,', t{i}), end, fprintf(' %s\n', t{end}); else fprintf(' %s\n', t); end

Monerie (nu=[0 1 2]) modes found in the cladding: MONERIE 0 1, MONERIE 1 1

As mentioned before, the Monerie solution is derived in the scalar approximation valid for small refractive index steps. Therefore we can expect it to coincide with the exact solution (hybrid, TE and TM modes) in the regions where the low refractive index surrounding does not play much role. This is the case at the large diameter because most of the field is still inside glass. At the same time, unlike the two-layer HE, TE and TM solutions, the Monerie solution gives a smooth transition between the core- to the cladding-guidance regions:

ylim ([1.449 1.451]) xlim([0 125])

For thin fibres, in the waist, the high refractive index step at the outer surface (between cladding and surrounding) makes the scalar Monerie solution for LP01 mode fully invalid, it strongly deviates from the HE11 mode:

ylim([1 1.45]) xlim([0 3])

The Monerie LP11 curve coincides, even for small diameters, with the TE01 mode. This can be explained by the fact that a linear polarisation approximation is quite valid for the TE01 mode, which is transversal for electric field. For TM01 mode, this approximation is not valid.

We have found that the Monerie solution is well suitable for the large diameter regions. We need a solution, which would work in the small diameter region and at the same time take into account the three-layer structure (so smoothly connect with the Monerie solution).

## Tsao modes

As I mentioned before, the first known to me publication to treat this task was TSAO 1989. The eigen-value equation for the HE/EH modes is erroneous. But the TE and TM equations are fine.

Let's calculate the TE01 and TM01 Tsao modes and see if they coincide with the two-layer T*01 solutions at small diameters and with the Monerie LP11 solution at the large diameter.

task = struct(... 'nu', [0],... % first modal index 'type', {{'tsaote', 'tsaotm'}},... % mode types 'maxmode', 1,... 'lambda', 900,... 'region', 'cladding'); argument.max = 125; if calculatemodes modesTsaoTClad = buildModes(argument, fibre, task, false); end % for i = 1:numel(modesTsaoTClad) % modesTsaoTClad(i) = addPointsToMode(modesTsaoTClad(i), [0 3]); % modesTsaoTClad(i) = addPointsToMode(modesTsaoTClad(i), [0 3]); % end hTsaoTClad = showModes(modesTsaoTClad); set(hTsaoTClad, 'Color', 'green', 'LineWidth',3) set(hTsaoTClad(1), 'LineStyle', '--') uistack(hTwoCore, 'top') uistack(hTwoClad, 'top') ylim([1 1.45]) xlim([0 3])

As we see, at small diameters the Tsao TE and TM modes coincide well with the two-layer TE01 and TM01 modes. At larger diameter, the Tsao modes deviate from the two-layer solution and follow the Monerie LP11 mode, as expected.

ylim([1.44997 1.449975]) xlim([123 125])

## Erdogan mode

The three-layer fundamental mode (HE11) can be calculated using the solultion of ERDOGAN (1997, 2000).

task = struct(... 'nu', 1,... % first modal index 'type', {{'erdogan'}},... % mode types 'maxmode', 1,... 'lambda', 900,... 'region', 'cladding'); if calculatemodes modesErdoganClad = buildModes(argument, fibre, task, false); end % modesErdoganClad = addPointsToMode(modesErdoganClad, [0 3]); % modesErdoganClad = addPointsToMode(modesErdoganClad, [0 3]); hErdoganClad = showModes(modesErdoganClad); set(hErdoganClad, 'Color', 'cyan', 'LineWidth',3) uistack(hTwoClad, 'top')

The found Erdogan mode nicely follows the two-layer HE11 mode in the small diameter region...

```
ylim auto
xlim([0 3])
```

...and then deviates from it and "goes into the core", following the Monerie LP01 mode.

ylim ([1.4485 1.4505]) xlim([0 60])

## Conclusions

In this tutorial I have shown how to calculate the modal curves (n_eff vs. d) for three-layer fibres using OFT. As the example system, I've used a tapered optical microfibre. Calculation can be done using both two-layer and three-layer solutions.

The two-layer solutions are exact in the regions where the third layer can be neglected and cannot simulate the transition region.

The scalar approximation-based Monerie solution is valid for three-layer fibres when light is confined inside glass so that it "does not see" the high refractive index step at the outer surface. This solution is therefore not valid for small diameters, at which a significant portion of light propagates in the evanescent field, but can be used to consider the transition region, where the two-layer solution does not provide enough information.

For small diameter three-layer fibres, the Tsao and Erdogan solutions can be used. The Erdogan solution is only available for the cladding-guided modes (when n_eff < n_cladding). Therefore it seems reasonable to use the simpler Monerie solutions as long as the mode is core-guided (n_eff > n_cladding) and switch to the exact Erdogan and Tsao modes for n_eff < n_cladding.

## Acknowledgements

I am thankful to Timothy Lee and Peter Horak from ORC in Southampton, Ariane Stiebeiner from AG Rauschenbeutel in Vienna, and Fabian Bruse, our former diploma student, for their help during investigation of the available three-layer solutions.

## References

- Belanov et al., 1976: http://dx.doi.org/10.1070/QE1976v006n01ABEH010808
- Erdogan, 1997: http://dx.doi.org/10.1364/JOSAA.14.001760br/
- Erdogan, 2000: http://dx.doi.org/10.1364/JOSAA.17.002113
- Monerie, 1982: http://dx.doi.org/10.1109/JQE.1982.1071586
- Tsao et al., 1989: http://dx.doi.org/10.1364/JOSAA.6.000555
- Zhang, Shi, 2005: http://dx.doi.org/10.1364/JOSAA.22.002516

save tutorial3ls.mat modesMonerieCore modesTsaoTClad modesTwoClad modesTwoCore modesErdoganClad