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Finite Element Beam Propagationo Method (FE-BPM)

A similar deviation of local maxima is also seen in Fig. According to Eq. The deviations at minimum positions are similar in magnitude. They occur from numerical errors. The source of errors lies in discretization, the paraxial approximation, and the reflection from the boundaries even though transparent boundary condition and absorption layers have been employed. Further tackling these problems seems unnecessary because the analytical perturbation solution holds at the limit of small modulations. However, both methods show similar trends in Fig. According to Fig. Any fields outside the left and right boundaries in Fig.

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By fitting the power flow within the study window, we estimate the attenuation constants using an exponential decay function with a single attenuation constant as shown in Fig. The exponential decay function with attenuation constant predicted by the perturbation solution is plotted for comparison. The difference in power flow between Fig. Normalized field distribution of a beam propagating in modulated core radius waveguides. In the design shown in Fig. Since one attenuation constant may correspond to several possible spatial periods of modulations, it is, in principal, possible to choose any period together with a certain amplitude of modulation.

However, notwithstanding technical issues in the manufacture of waveguides, if periods of modulation are chosen over a wide range, also the radiation angles deviate over a wider range, what compromises emission homogeneity. We therefore prefer to select periods of modulation in an as-narrow-as-possible range. As shown in Fig. This indicates that if we choose different periods in Fig. It provides a scattering loss of up to From the viewpoint of sensitivity, the former one produces a higher spatial resolution due to smaller dimension , but would also require higher precision due to smaller range in the fabrication of periodic structures.

In the previous sections, we demonstrated the ability of tailoring radiation loss by introducing core radius modulations at constant core refractive index. In an analogous approach, similar emission behavior can be generated by sinusoidal modulation of the core refractive index at constant core radius. This is of particular relevance when the modulations are not created through modifications of the fiber drawing process, but, e. Using the above-introduced BPM models, the radiation losses from fundamental modes which arise from such modulations of the core refractive index in slab waveguides and step index fibers are shown in Fig.

The curve shown in Fig.

This similarity implies that a master curve as in Fig. When the radiation losses are extremely low Fig. Simulated attenuation constants for radiation losses from fundamental modes due to axial modulation of the core refractive index. For the modulation of core radius, the radiation loss is proportional to the square of b Eq. The lowest positions are corresponding to the maximum radiation losses, indicating the highest total overestimation. This was already discussed in the previous sections. The main reason for the deviation from 2 arises from the fact that the modulation amplitudes are large.

If we follow the same process for the modulation in core refractive index, we get the results as shown in Fig. This result implies that the amplitude of modulation in core refractive index is still within the limit of validity of analytical solutions. For the slab waveguides, similar observations were made not shown.

Simulated attenuation constants for radiation losses from fundamental mode due to different modulations on core radius a and on refractive index axially b. If it is smaller than 2 the attenuation constants develop due to increasing amplitude of modulation slower than the perturbation solution predicts and vice versa. As seen from Fig.

Light extraction from fundamental modes in modulated waveguides for homogeneous side-emission

The advantage is that tailoring the core refractive index in a sinusoidal fashion is already well developed in the fabrication of fiber Bragg gratings using interferometric laser writing The period of the modulation in refractive index can be varied through the phase mask or the distance between fiber and mask.

We presented the design of homogeneous side-emitting waveguides by introducing axial modulations in core radius or core refractive index. Available algorithms for calculating the quantitative relationship between radiation losses and modulation parameters were extended to large amplitudes of modulation via verified FDBPM for symmetric slab waveguides and HTBPM for step index fibers. For modulation in core refractive index, we find that it is sufficient to use 0. Finally, we deduce exemplar conditions for homogeneously side-emitting one-meter-long slab waveguides and step index fibers using modulation in core radius as well as modulation in core refractive index.

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The presented model also applies without the assumptions of single-mode guidance, zero absorption, zero Rayleigh scattering and no backward propagation. Initial use of such assumptions only simplifies the case result validation. Beams propagation in the present case occurs mainly in axial direction. Such directional and not fast-spreading beams allow for a plane wave paraxial wave approximation and therefore can be modelled with an envelope function multiplied by a plane wave function:. With the weakly guiding approximation, substituting Eq. If n 0 is suitably chosen, e.

We now express the slowly varying envelope approximation in the second derivative,.

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Following the Crank-Nicolson scheme 13 , Eq. To solve Eq.

N equations and two boundary conditions all together form a tridiagonal system which can be solved iteratively using the Thomas method Appendix B in ref. For step index fibers, it is suitable to expand Eq. The expansion coefficient a n is given by. To obtain Eq.


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According to the BPM technique, field propagation as described in Eq. The first step can be realized in the expansion coefficient a n by. It is then possible to reconstruct the field using Eq.


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