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Abstract
This erratum corrects an error in the simulation of my published paper [Opt. Express 24(17), 19841 (2016). [CrossRef] ]. All figures and some related texts are revised. Other results and conclusions are not affected by this correction.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Correction
I realized in the published article in Ref. [1] that field intensity instead of field was mistakenly used in the integrations for calculating the mode field diameter (MFD). I corrected this and consequently revised all the figures. The corrected figures and related text are given below.
The relative mode size is plotted against the normalized thermal lensing parameter ξQ0w02 in Fig. 1. It can be seen that the rate of change in the relative mode size is higher at small ξQ0w02, but slows down at larger ξQ0w02. There is very little difference in the curves for different V values from the perturbation model. The relative mode size can collapse to small values at high heat. The most interesting observation is the fact that the effect of thermal lensing is fully characterized by the normalized thermal lensing parameter ξQ0w02, where ξ is fully determined by material properties and laser wavelength. This normalized thermal lensing parameter scales linearly with heat load Q0 and also scales quadratically with MFD. This MFD dependence of thermal lensing was also shown in [7]. Since the total effect of thermal lensing is determined by the integrated effect seen by the entire mode, it is reasonable to expect it to scale with the mode area, i.e. w02.
Fig. 1. Mode collapse under the influence of thermal lensing. Equation (3) is evaluated for normalized frequency V = 2-6. A numerical simulation is also performed for comparison for fibers with core diameters of 10µm, 15µm, 20µm, 25µm, and 30µm and NA of 0.06 at 1.06µm. The results are plotted as MFD for near-field MFD.
A comparison of the temperature profiles from the numerical model used in this study and the uniform heat load is shown in Fig. 2. The fiber has a NA of 0.06 and a core diameter of 20µm, operated at 1.06µm. We use a cladding diameter of 400µm throughout this study. The total heat load in the core is 100w/m in both cases and there is no heat load outside the core. The normalized heat load profile, i.e. the normalized mode intensity profile, is also shown. The numerical model used an iteration process described in the next paragraph to find the optical mode under heat load (see Fig. 3). It can be seen clearly that the solution in the cladding is the same for both cases and the uniform heat load model underestimates the temperature in the core. This is due to the fact that more heat is deposited near the core center than that accounted in the uniform heat load model.
Fig. 2. Comparison of the temperature profile from the numerical method used in this work and the parabolic solution for a uniform heat load. ΔT is set to be 0 K at the cladding boundary. Core diameter is 20µm; cladding diameter is 400µm and NA is 0.06. The total heat load is 100w/m in both cases. The laser wavelength is at 1.06µm. The normalized heat load profile, i.e. normalized mode intensity profile, is also shown.
Fig. 3. Typical run for the case in Fig. 2. Core diameter is 20µm and NA is 0.06. The total heat load is 100w/m. The laser wavelength is at 1.06µm. Relative changes are shown on the vertical axis on the right. The heat load is gradually increased over 30 steps.
The refractive index profiles obtained by the numerical model for the 20µm-core fiber are shown in Fig. 4 for various total heat loads. The refractive index profile is truncated at a radius 20µm, i.e. twice the core radius in this case. In our following analysis of optical mode, we are only concerned with the part of the waveguide seen by the optical mode of interest. We typically truncate the refractive index profile to 1.05 to 4.1 times of the core radius depending on guiding strength of the waveguide. Using an unnecessarily large cladding radius in the analysis can lead to poor numerical stability in the optical mode solver in addition to longer computation time. Care is taken in each case to ensure that the truncation does not compromise accuracy.
Fig. 4. Refractive index profiles under various heat loads. Core diameter is 20µm and NA is 0.06. The laser wavelength is at 1.06µm.
We then proceed to study the impact of thermal lensing using our numerical model for five fibers with core diameters of 10µm, 15µm, 20µm, 25µm, and 30µm. All fibers have a NA of 0.06 and a cladding diameter of 400µm. This study is conducted for a wavelength of 1.06µm. The respective V values are respectively 1.778, 2.667, 3.557, 4.446, and 5.33. The 10µm-core fiber is in the single-mode regime and the other fibers are in the multimode regime. The results are shown in Fig. 1 for MFD (eMFD is similar, omitted for clarity). It can be seen that the results from the perturbation method are only reasonable for very small normalized thermal lensing parameter but overestimate the effect of thermal lensing when ξQ0w02 > 0.1. The results for the five fibers at ξQ0w02 > 15 are very close. In this regime, wave guidance is almost entirely from the effect of thermal lensing and the original waveguide plays a very small part. At smaller ξQ0w02, the single-mode fiber suffers slightly more mode size reduction. The mode is much larger than the core in the single-mode regime and this enhances the impact of thermal lensing on mode size. MFD and eMFD follows each other closely (eMFD is omitted for clarity). A convenient place to set the thermal lensing threshold is ξQ0w02 = 2, where the mode size is reduced by ∼10% and the effective mode area by ∼20% for larger V values. The thermal lensing limit is put at ξQ0w02 = 0.18 in [7].
Using the numerical study shown in Fig. 1 in the asymptotic case when V is large, ω/ω0 =−4.10470E−09×6 + 4.80781E−07×5 − 2.09893E−05×4 + 3.91479E−04×3 − 1.50598E−03×2 − 5.30000E−02x + 1.00093, where x =ξQ0w02, and we can also provide a theoretical mode size change for the experimental work in [9]. Quantum defect heating is used to convert heat load from the extracted power. Peak heat load at the output is used for the normalized thermal lensing parameter, which is estimated by multiplying a factor to the average heat load (see Fig. 5). The measured data for those fibers with large MFDs are highly scattered, indicating much larger measurement errors in this regime. This model does not consider effects such as anti-crossings with modes originated in the cladding and photo-darkening, which are known to take place in some of these fibers. It nevertheless provides a reasonable agreement with the measurements.
Fig. 5. Simulated MFD for the fibers in [9]. Quantum defect heating is used to convert heat load to extracted power. Peak heat load at the output is used and is estimated by multiplying a factor to the average heat load. This factor is respectively 6, 7, 6, 9, 5, 3, and 9 for LPF30, LPF35, LPF45, LPF60, LPF 75, 200/40, and 285/100 to get the best fit to the measured data.
We studied 5 fibers for TMI nonlinear coupling coefficient with core diameters of 10µm, 15µm, 20µm, 25µm and 30µm respectively. The fiber NA is 0.06 and cladding diameter is 400µm. This study is conducted at a wavelength 1.06µm. The V values are respectively 1.778, 2.667, 3.557, 4.446, and 5.335. The results are shown in Fig. 6. The 10µm-core fiber is single mode at low heat load, but the LP11 mode is guided when ξQ0w02 > 1. Its TMI nonlinear coupling coefficient χ increases until ξQ0w02=∼10 as the LP11 mode is increasingly guided. When ξQ0w02 > 10, thermal lensing becomes significant and χ starts to decrease. Thermal lensing pulls all modes to the core center, but this is more significant for the fundamental mode (see Fig. 7). This lowers the overlap between the LP01 and LP11 modes, consequently leading to a reduction in χ. A similar trend can be seen for the 15µm-core fiber. Since the LP11 is already well guided in this fiber with heat load, the initial increase in χ at low heat load is less pronounced. χ starts to decrease when ξQ0w02 > 4. The remaining fibers have similar χ at low heat load; this is due to that fact that χ changes very slowly at large V [6]. For these fibers, χ starts to decrease significantly when ξQ0w02>∼4. Similar studies were conducted for LP02 mode and similar trends were obtained. This reduction of TMI at high thermal load is, however, of limited practical use as the effect of thermal lensing is significant at this point. For most practical fibers, the TMI threshold is also well below this thermal load.
Fig. 6. Simulated TMI nonlinear coupling coefficient for fibers with core diameters of 10µm, 15µm, 20µm, 25µm and 30µm respectively. The fiber NA is 0.06 and cladding diameter is 400µm. This study is conducted at a wavelength of 1.06µm. Contour lines indicate constant thermal load.
Fig. 7. The LP01 and LP11 modes at 1.06µm in a fiber with a NA of 0.06 and a core diameter of 20µm without heat load and with a heat load of 1000w/m, i.e. ξQ0w02 = 3.81. The cladding diameter is 400µm.
Funding
Army Research Office (911NF-12-1-0332).
Disclosures
The author declares no conflicts of interests.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the author upon reasonable request.
References
1. L. Dong, “Thermal Lensing in Optical Fibers,” Opt. Express 24(17), 19841–19852 (2016). [CrossRef]
