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

Dedicated control of axial light emission from light-guides enables a new generation of functional light sources for volumetric illumination. A primary challenge is to ensure homogeneous emission intensity across the full length of the device. Here, we introduce an approach towards homogeneously side-emitting waveguides which do not rely on imposing local scattering centers such as bubbles, micro-/nanoparticles, and rough or abrupt interfaces, but on modulated core radius. Previous quantitative studies of the relationship between structural parameters and radiation losses provide initial conditions for tailoring side-emission through core-diameter modulations, however, with strongly limited amplitude of modulation. We now employ and verify numerical simulation to overcome this limitation towards meter-long homogeneously side-emitting waveguides in which the amplitude of core-diameter modulation is of the same order of magnitude as the core diameter itself. Similar emission properties can be obtained through modulation of the core refractive index instead of the core diameter, or through a combination of both approaches. Using the present model, we deduce exemplary conditions for homogeneous side-emission in which the power flow within the waveguides decays linearly, what may present another interesting feature for applications beyond illumination.


Model Construction and Assumptions
In our approach, we initially limit our discussion to weakly-guiding single mode fiber, similar to Marcuse's perturbation solutions 11,15 . This enables to neglect the effects of polarization, and also to employ scalar homogeneous Helmholtz equations for the description of beam propagation. Then, the Helmholtz equations lead to paraxial wave equations by introducing the slowly varying envelope approximation in FDBPM. However, such treatment is not necessary for the HTBPM, in which an average reference refractive index is used instead of the effective refractive index of the propagation mode. Both numerical models and the analytical reference are restricted to real propagation constants, meaning that evanescence or leaky modes are not included in our discussion. Furthermore, the core medium has no absorption and both core and cladding medium are free from Rayleigh scattering, therefore, our result underestimates experimental expectations. As assumed in Marcuse's reference solutions, both the forward and backward radiation losses 11 from a forward-propagating mode are considered, while the radiation losses from a backward-propagating mode are weak and therefore neglected. The present numerical reference is consequently built on a one-way beam propagation algorithm. Both models evaluate waveguides supporting only fundamental modes (TE 1 even mode for a 2D slab waveguide and LP 01 mode for step index fiber). Besides the simplicity in calculation, the absence of other higher order guiding modes prevents complicated energy exchange. It therefore becomes easier to tailor the radiation losses. Furthermore, it seems safe to assume in this case that no energy transfer occurs from a radiation mode back into the fundamental mode (line [11][12] in page 114 in ref. 11 ). Table 1 summarizes all assumptions which were used in the following.
It should be noted that the radiation loss discussed in this paper accounts only for the light escaping from the core, which is equivalent to the presumption in the reference solutions that the cladding is infinitely large. Such an assumption is still valid when we consider the following practical situations shown in Fig. 2. If the waveguide is working in a medium with distinct refractive index (e.g., in air or water), a scattering layer (e.g., polymer coating, slightly phase separated/crystallized glass) with a somewhat higher refractive index can be attached to the exterior Figure 1. Schematic of a side emitting fiber with sinusoidal modulation of the core diameter, obtained from a separate finite element simulation on the starting section with Gaussian incident beam profile (approximate LP 01 mode). The radiation loss from the core is illustrated through the normalized axial power flow in the fiber. The rings surrounding the cladding layer show the magnitude of axial Poynting vector arriving on the coatingcladding interface. of the cladding ( Fig. 2(a)). The radius of the cladding is chosen so that it is big enough for the fundamental mode to vanish on the boundary. The radiation arriving in the interior boundary of the coating scatters forward to the outside. The backscattered radiation can be safely ignored if we keep the mechanism of scattering within the Mie regime; i.e., the dimension in effective radius of the scattering centers is between half to about 300 times of the incident wavelengths. The scattering layer is only required to provide a constant scattering ability. Figure 2(b) demonstrates another working situation in which the refractive index of the working medium, n 2 , is just slightly lower than the core refractive index n 1 . In this case, neither a cladding nor an auxiliary scattering layer would be necessary in theory. Even though the radiation fields coming out of the periodic core interfere in the far field region, there is no obvious interference (fringes) in the near field, that is, in the vicinity of the scattering layer, to be confirmed in the following sections. The set-up shown in Fig. 2(b) corresponds to applications such as light sources for aquaponics and agriculture or water disinfection, in which the surrounding lossy medium absorbs the radiation in the near field so that no far field interference occurs. Under these considerations, the interference effect can be ignored in our discussion.
Modelling homogeneous side emission. Since the power flow in waveguides consists of guiding and radiating contributions, constant transversal radiation loss (homogeneous side-emission) translates into a linear decay of the guided power P 0 over the length L. Then, the propagating power P(z) along position z has the form  (1) and (2), the attenuation constant α m for each group to approximate linear decay is where z m , z m+1 , P(z m ) and P(z m+1 ) are known parameters. Figure 3(a) shows an example of a set of scattering losses to approximate a linear decay function using 10 sections of exponential decay by varying from 4.6 dB/m to about 30.1 dB/m according to Eq. (3). It is shown that using ten sections would be sufficient (with R 2 = 0.999971 for the linear fit over position 0 to 0.9 m in Fig. 3(a)) to approximate linear decay (whereby the last section is always ignored because the exponential decay never reaches zero, while the linear decay does). The result can be easily extended to any other length L if the same amount of sections (e.g. 10 sections) are desired, although Eq. (3) seems to depend on L. If we set z m to cL, where c is the fraction of length (0 < c < 1), the L in Eq. (3) cancels each other and the variation of α m depends only on M (the number of sections).
As pointed out by Marcuse 5 and Snyder 6 , any imperfection in core radius or refractive indices initiates energy exchange between any of the guided and radiation modes. If the relationship between attenuation constant α and modulation parameters is quantitatively known, our main objective is reached. Here, we take step index fibers as example (for slab waveguides, one may start from Eq. (79) in ref. 15 ). The radiation loss which occurs in a single mode fiber due to a sinusoidal modulation of the core radius has been studied by Marcuse. The perturbation solution of the attenuation coefficient is given as Eq. (4.4-13) in ref. 11 : Figure 2. Schematic of possible working situations, corresponding to waveguides with infinitely large cladding as illumination sources. Environmental illumination is achieved via radiation losses due to sinusoidal modulation of the core radius. (a) Application in air or a medium with significantly lower refractive index than the core. (b) Application in a working medium whose refractive index is a bit lower than that of the core. In the latter case, the medium itself acts as an effective cladding, forming a step-index waveguide. where α is the attenuation constant for the field amplitude (therefore, 2α for power attenuation), b is the amplitude of the sinusoidal modulation, and β φ is the propagation constant of the radiation mode at angle φ to the fiber axis. The ρ parameter is defined in Eq. (2.4-14) in Ref. 11 . K denotes coupling coefficients from the LP 01 mode to forward (p = 1) and backward (p = 0) radiation modes, and μ is the azimuthal symmetry of the radiation mode.
The subscript x → x indicates coupling from x to x polarized radiation modes while x → y means coupling from x to y polarized radiation modes. It is found that only the first term in the sum of mode square of K (representing coupling between identical polarization without azimuthal variation) is significant. The other three terms are neglectable. Eq. (4) is a perturbation solution at the limit of small amplitude of modulations as compared to the average core radius. The attenuation constants are evaluated at axial position far from a single period of modulation. Figure 3(b) shows one exemplar relation between normalized attenuation constants (valid for small modulation limit) and radiation angles Θ, which are correlated to periods of modulation Λ via Eq. (4.4-11) in ref. 11 . The reproduced curve in Fig. 3(b) is oscillating and symmetric, which differs from practical observation of random perturbation in the core radius as shown in Fig. 2(a) of Mazumder et al. 18 . The reason for this is that in CMT, the fundamental mode couples only to one single radiation mode with specific radiation angle due to the fact that only one spatial frequency exists after the Fourier transformation of the profile with fixed Λ (Eq. (4.4-13) in ref. 11 .). In practice, for a core radius profile with random perturbation, the Fourier transform of the core radius profile contains a range of spatial frequencies. This results in coupling to a broad range of radiation angles. Notwithstanding Rayleigh scattering, Fig. 3(b) shows the scattering loss at different specific radiation angles due to different structures with single spatial frequency. Figure 2(a) in ref. 18 , on the other hand, shows angle-dependent scattering losses due to a specific core radius profile with broad spatial frequencies after Fourier transform and, therefore, exhibits a broadened peak for the scattering losses. In practice, the modulation function will result from superposition of the desired sine modulation and a random modulation due to surface roughness. Therefore, practical measurement may hardly observe the oscillating results as the broadening effect will smoothen the curve as in Mazumder's work. The radius modulation can be expressed as where r(z) is the core radius at axial position z, b and Λ are the amplitude and period of the sinusoidal modulation, respectively. The thus reproduced curve in Fig. 3(b) is exactly corresponding to Fig. 4.4.2 on page 159 of ref. 11 , where the ratio of b:a is kept at ~1%. For more pronounced modulation amplitude, this means that the single-mode condition can hardly be maintained. For example, a relevant practical scenario for side-emission would be to have a and b in the micrometers range, e.g., b = 1 μm, a = 10 μm, resulting in b:a = 10%. In this case, Eq. (4) breaks down. To extend the approach to large modulation amplitude, we now utilize beam propagation methods to solve the problem numerically. It will be shown that Eq. (4) is acceptable for amplitudes of modulation which do not exceed 5% of the average core radius. Otherwise (for example, at 10%), the reference solutions overestimate radiation losses especially at periods of modulation where maximum losses are predicted.

Verification of the BPM models. Figure 4 displays scattering losses evaluated by BPM models and by
Marcuse' perturbation solution. All methods evaluate radiation losses from the fundamental mode due to sinusoidal modulation of the core radius in slab waveguides (TE 1 incident mode) and optical fibers (LP 01 incident mode). The amplitudes of modulations are 1.0 µm, 0.5 µm and 0.1 µm over an average core radius of 10 µm. This corresponds to 10%, 5% and 1% modulations, respectively. Figure 4(a,b) show the best agreement between BPM simulations and the analytical results, with only minor deviations at the positions local maxima and minima. The deviation at maximum positions arises from the weak coupling assumption in the derivation of Marcuse's perturbation method (Eq. 3.4-1 in ref. 11 ), which assumes that the energy coupled to the radiation mode is so weak that the incident field does not change its shape and amplitude. This assumption is not valid when the coupling becomes stronger, resulting in an overestimation of radiation losses (the energy of the incident beam is still considered constant even though it decreases significantly). A similar deviation of local maxima is also seen in Fig. 4(c,d) where the amplitude b is twice in magnitude as compared to Fig. 4(a,b). According to Eq. (4), the radiation loss is proportional to the square of b, therefore, stronger overestimations arise in the perturbation solution shown in Fig. 4(c,d).
The deviations at minimum positions are similar in magnitude. They occur from numerical errors. Figure 4(e,f) show the radiation losses at 1% modulation of the core diameter. In this case, taking the analytical result as reference, the two BPM models exhibit different behavior in their numerical errors: random fluctuations and a bias for FDBPM and HTBPM, respectively. 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.
Practically, FDBPM differs significantly from HTBPM (2D FDBPM and 3D HTBPM). However, both methods show similar trends in Fig. 4(a-d), confirming that the numerical results are reliable and the overestimation from the analytical perturbation solution of the mode-coupling theory is not occasional. Figure 5 shows the normalized field distribution and normalized power propagation of a beam with the fundamental mode incident in a slab waveguide and a step index fiber, respectively. According to Fig. 5(a,b), the We chose a study window from −35 µm to 35 µm in radius for the integration of power flow to account for the confined power flow in the waveguide. Any fields outside the left and right boundaries in Fig. 5(a,b) are considered as radiation losses. 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. 5(c,d). The exponential decay function with attenuation constant predicted by the perturbation solution is plotted for comparison. For a period of Λ = 75 µm and a modulation amplitude of 1 µm, the perturbation solution overestimates the analytical result (Fig. 4(d)). The difference in power flow between Fig. 5(c,d) arises from the slowly varying envelope assumption in FDBPM, which is not employed in HTBPM for optical fiber, therefore, fast oscillations appear in Fig. 5(d). Fig. 3(a), ten discrete values of attenuation constants ranging from 1 to 7 m −1 (4.6 to 30.1 dB/m) are required to generate approximately linear decay in guided power and, correspondingly, laterally homogeneous side-emission. 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. 3(b), when it is far from 90°, there are eight peaks occurring between 0° to 40°, averaging in a spacing of 2.5° among each half peaks. This indicates that if we choose different periods (in Fig. 4) of modulation distributed within half a peak, the biggest deviation in radiation angle (radiation angle in Fig. 3(b) is equivalent to period in Fig. 4) is just 2.5°. Here, we take a step index fiber with an amplitude of modulation of b = 1 µm as an example. It provides a scattering loss of up to 39.1 dB/m, covering the required range for a one-meter-long homogeneously side-emitting fiber. As shown in  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 example shown in Table 2, we therefore use the latter window of modulations.

Exemplar conditions for homogeneously side-emitting fiber. In the design shown in
Index versus radius modulation. 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.g., through laser-assisted writing.
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. 6. The requirement for a one-meter-long homogeneously side-emitting waveguide can be met by introducing an amplitude of modulation in refractive index of Δn = 5 × 10 −4 (Fig. 6(c,d)). Detailed design parameters for such a fiber are provided in Table 2. The radiation losses due to a modulation amplitude of Δn = 1 × 10 −4 are too low for a one-meter-long side-emitting waveguide. The curve shown in Fig. 6(a) is similar to 6(c) while the Fig. 6(b) is similar to (d). This similarity implies that a master curve as in Fig. 3(b) exists by dividing a same function of b, e.g. b 2 . When the radiation losses are extremely low (Fig. 6(e,f)), numerical errors occur which prevent the further applicability of the model.
For the modulation of core radius, the radiation loss is proportional to the square of b (Eq. (4)) in the limit of validity of the perturbation solution. Figure 7(a) shows the radiation losses using the present numerical models with large modulation amplitudes in core radius, i.e., from 0.5 µm to 2.25 µm over an average radius of 10 µm. If we try to fit the radiation losses for different b in a power function, b γ , we find that γ is not a constant over varying periods, and that it deviates from 2, as shown in the red curve in Fig. 7(a). If γ < 2, there is a lower growth rate with increasing amplitude of modulation then predicted by the perturbation solutions. 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. 7(b). The γ in this case is approximately 2 over most of the considered range of periods, except for the minimum positions, where numerical errors occur. 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).
As seen from Fig. 7, the modulation in core refractive index is more sensitive than the modulation of the core radius. The maximum radiation loss reaches 166 dB/m with an amplitude of modulation in refractive index of Δn = 20 × 10 −4 . Even when using small periods at about 50 µm, homogeneous side-emission can readily be achieved with an amplitude of Δn = 20 × 10 −4 . 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 16 . The period of the modulation in refractive index can be varied through the phase mask or the distance between fiber and mask. Then, the remaining technical requirement is to provide a low loss photosensitive material that is able to generate refractive index changes in the magnitude of 5 × 10 −4 , and its combination with a non-photosensitive cladding [19][20][21][22] .

Conclusions
We presented the design of homogeneous side-emitting waveguides by introducing axial modulations in core radius or core refractive index. According to simulation data, these waveguides provide up to 33 dB/m for radius Step index fiber Slab waveguide Step index fiber    modulation and 166 dB/m for refractive index modulation in our calculation, compared to up to 10 dB/m in commercially available devices 2,6 . 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. We find good agreement between numerical models and the analytical perturbation solution from CMT when the amplitude of modulation in core radius is about 5% (relative to the core radius). Above 5% the radiation losses grow with increasing amplitude of modulation at different rates (predicted by the analytical perturbation solution), depending on the period of modulation. For modulation in core refractive index, we find that it is sufficient to use 0.03% modulation (relative to the core refractive index) to achieve a one-meter-long homogeneous side-emitting waveguide. 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. 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.
The FDBPM discretizes the envelope function ϕ n m at positions (n∆x, m∆z) with n = 1, 2, …, N, m = 1, 2, …, M. Following the Crank-Nicolson scheme 13 , Eq. (10) can be discretized into  ϕ + . To solve Eq. (11), two more boundary conditions are therefore required. For these, we take the transparent boundary condition 13 in the 2D FDBPM. 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. 13 ). If the initial envelope function n 1 ϕ is known as incident, the envelope function n m ϕ at positions m∆z can be evaluated by solving Eq. (11) iteratively. The incident field is TE 1 even mode using core thickness 2r 0 = 20 µm, wavelength 1.55 µm, core refractive index 1.460 and cladding refractive index 1.459.