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We demonstrate an experimental study of the chromatic dispersion properties for a series of microstructured fibers (MSFs) dedicated for a supercontinuum generation. With white-light interferometry application we analyze experimentally how the small variations of structural parameters, i.e. an air-hole diameter and a lattice constant, influence dispersion characteristics in different groups of MSFs. Our study provides useful information on how to design the fiber which is less sensitive to the fabrication imperfections. Moreover those investigations are the initial step to the development of the customized or tunable supercontinuum light sources based on MSFs with slightly changed structural parameters which can generate light with a different spectrum range, adapted to a proper application.
©2013 Optical Society of America
1. Introduction
Nonlinear effects, including Raman scattering, four-wave mixing, self-phase modulation [1], etc., are of special interest in optical fibers due to the very large power intensity accumulated in a fiber core and the large interaction length between light and the fiber material. On the one side, those effects are mitigated by researchers, for example in high power lasers and amplifiers, in which these effects are detrimental [2]. On the other side however, they try to improve the efficiency of nonlinear effects in applications such as a generation of supercontinuum (SC) [3].
In this paper we focus on the latter case and more specifically on nonlinear fibers which can be used for SC generation. The nonlinear effects which are responsible for a broadening of the spectrum of light are strongly dependent on dispersion characteristics of a fiber. Thus to generate the SC with a customized spectrum, the precise tailoring of dispersion characteristics is necessary. The accurate control of the dispersion characteristics is also obligatory in another kind of applications, such as pulse recompression [4, 5], mode locked fiber lasers [6] or soliton delivery [7]. With respect to the managing a chromatic dispersion (CD) the microstructured optical fibers (MSFs) have shown the superiority over conventional optical fibers [1]. In MSFs the dispersion can be controlled and tailored with a great freedom and accuracy. Some of the remarkable dispersion properties, unattainable in a case of conventional optical fibers have already been reported [8–13].Then numerical simulations allow for the calculation of the CD and anticipation of various nonlinear effects including SC generation [3]. However, one has to be aware of the fact that fabrication inaccuracies, which are inevitable, may lead to changes of the dispersion characteristics compared to the expected ones, and thus may originate completely altered nonlinear effects. Therefore these characteristics have to be verified by experimental characterization of actual MSFs.
Most of the published studies, which compare results of the CD measurements with numerical simulations [14–21], concentrates on one or two MSF samples and discusses their particular properties [15, 18, 21]. Although the influence of fabrication errors on dispersion properties of specific fibers has been investigated in [11, 22], these errors are commonly underestimated and their influence on dispersion characteristics of MSF's has not been studied experimentally in details yet. Additionally, these papers do not clearly reveal the very divergent influence of geometrical parameter variations on the CD for dissimilar fiber structures. The fabrication inaccuracies are inevitable and are caused for example by the limited ability of fast and accurate measurement of the structural parameters during the fiber drawing process, the difference between diameters of capillaries, which construct the fiber preform, temperature stability in the furnace, stability of applied pressure, preform and fiber drawing speed, etc.. Therefore, in this paper we investigate dispersion characteristics of a series of MSFs with slightly modified parameter values to identify what kind of fibers is sensitive and to what extent to the small deviations of geometrical parameter values, without looking for the exact reason of these deviations, since this reason can be different in each fiber drawing process. In addition, our studies open up the possibility of developing MSFs with features dedicated for customized SC sources.
Our paper is structured as follows. In section 2 we provide the basic information about CD directed to discuss the concept of CD measurement with the use of white-light interferometry, which is commonly used to measure the CD and which permits characterization of short fiber lengths [19, 20, 23], similar to those used for the SC generation. In section 3 we validate our measurement setup by comparing measurement results of the CD for a standard SMF-28 with catalog data. By doing so we also demonstrate step by step our measurement methodology. Section 4 shows the results of the CD measurements carried out for the series of MSFs. We analyze this results and discuss the influence of MSFs fabrication inaccuracies on the CD characteristics. Section 5 delivers information on the accuracy of the measurements which is estimated by means of Monte-Carlo experiments [24] and we close our paper with remarks and conclusions in section 6.
2. Measurement of the chromatic dispersion by white-light free-space interferometry
The CD measurement setup employing white-light free-space interferometer is schematically presented in Fig. 1 . It consists of a beam splitter (BS), one fixed and one moving mirror and a variable optical attenuator (VOA), which is used in the reference arm to compensate the loss of a fiber under test (FUT) and achieves high interference contrast. The Fianium SC-450 supercontinuum system is used as a broad spectrum (400 nm – 1800 nm) light source whereas interferograms are detected with an optical spectrum analyzer (OSA) Yokogawa AQ-6319.
Fig. 1 CD measurement setup employing white-light Michelson interferometer (FUT – fiber under test, VOA – variable optical attenuator, BS – beam splitter).
The spectrum of two beams interfering in such interferometer can be expressed as:
where Es(ω) and Er(ω) are the frequency dependent magnitudes of the signal and reference beam, respectively. The phase function ϕ(ω)is defined as:where L is the length of a tested fiber, L0 is the length of the reference arm and βs = ωneff/c, βr = ω/c represent propagation constants in the fiber and in the air. For a given position of a moving mirror the interferometer is balanced for one wavelength (λ0), which means that for this wavelength the optical paths in both arms of the interferometer have the same length. The Taylor expansion of a phase function around the balanced wavelength expressed in terms of wavelength is presented as:A registration of the positions of the interference fringes in a wide spectrum range allows creating a phase diagram for a balanced wavelength. Therefore by comparing the polynomial, which is fitted to the phase diagram, to the Eq. (3) we can obtain the second order propagation constant β2. Thus the CD of a fiber can be calculated with the use of the formula:
The precise shift of the moving mirror, the localization of interference fringes and the analysis of the obtained results are performed automatically with the use of in-house developed software.
In addition to our measurements, we performed numerical simulations of the CD for each tested fiber with the use of a commercially available MODE Solutions software from Lumerical [25]. This software employs a fully vectorial finite difference method for solving wave equation, which is based on the analysis described in [26]. In a case of MSFs we imported the scanning electron microscope (SEM) images of the cross sections of MSFs to the MODE Solutions software and the simulations were performed for these images.
3. Validation of the CD measurement setup
To validate our measurement setup we performed the measurement of the CD for the standard telecom SMF-28 (Corning) and we compared results of these measurements with the catalog data [27]. The interference pattern for a given position of a moving mirror, balancing the wavelength of 1550 nm, is shown in Fig. 2 . The range of the spectrum which is used to calculate the dispersion for λ0 = 1550 nm is marked with a dashed rectangle. For short wavelengths the SMF-28 is multimode and the higher order modes (HOMs) participate in the interference. Thus, to mitigate the influence of HOMs while calculating the CD for a short wavelengths, we used the numerical filtering of the interference pattern.
Fig. 2 Fringe pattern obtained forSMF-28. Dashed region shows the region of the interferogram which is used for the analysis of the CD.
The neighboring minima and maxima of the interference pattern are separated with the phase difference of π. Therefore a registration of fringe positions in a wide spectrum range permits for the construction of a phase diagram (see Fig. 3 ) for a balanced wavelength. Such diagram should respect Eq. (3) in which the 4th-order and higher terms are neglected. Thus we can fit a 3rd-order polynomial:
to the phase diagram and in this way we can obtain the values of coefficients A1. A2 and A3.Fig. 3 Phase diagram for λ0 = 1550nmfitted by the third order polynomial (solid line). nπ indicates the phase difference between reference and signal path in a function of ω-ω0.
The coefficient A2 of Eq. (5) can therefore be used to calculate the second order propagation constant for a balanced wavelength:
Repetition of this measurement for different balanced wavelengths (different positions of the moving mirror) allows to calculate the CD with the use of Eq. (4). The CD obtained in this way for SMF-28 is presented in Fig. 4 . The results are in a very good agreement (R2 = 0.99975) with the catalogue data (see the solid red line in Fig. 4), so we can comment that our measurement setup delivers reliable and accurate results.
4. Measurements of a chromatic dispersion for a series of microstructured fibers
To demonstrate the influences of fabrication errors on the CD of MSFs we have fabricated series of ten MSFs at Maria Curie-Sklodowska University in Lublin (Poland). The investigated fibers were drawn from four different performs, each creating a fiber group named A, B, C and D. The arrangement of air holes in these fibers is demonstrated in Fig. 5 . In each fiber the structural parameters such as the air hole diameter d and the lattice constant Λ were varied, which enabled the investigation of the influence of fabrication inaccuracies on CD characteristics. To draw fibers with different geometrical parameters from one preform, we changed the drawing parameters (e.g. pressure, drawing speed, etc.) for each fiber. Thus, we obtained different fibers with longitudinally stable structures. The length of each measured fiber was in the range of 0.7-1.0 m. We measured also the CD of approximately 10 m long fibers. The results were in a very good agreement with the results of measurements conducted for short fibers. Therefore, for the sake of clarity, we present only the results of measurements of short fibers.
Fig. 5 Scanning electron microscope image of one of the fabricated MSFs (the fiber C3 specifically) (a) and its image imported to the Lumerical MODE Solution software (b).
The geometrical dimensions of fabricated MSFs are presented in Table 1 and were controlled to obtain the cut-off wavelength below 1060 nm. In addition, we compare our fibers in terms of the core diameter, dcore = 2Λ-d, and the filling factor, d/Λ. The values of d and Λ are measured for the two innermost air hole rings from the SEM images and then averaged for each fiber. Such approach is justified since these air holes have the largest influence on dispersion properties and the d and Λ are uniform for these air holes - the standard deviation is less than 1% of average values in each fiber.
Table1. Structural Parameters of Tested MSFs
For the fabricated fibers we performed measurements of the CD as described in sections 2 and 3. The measurement results for four fiber groups are presented in Fig. 6 . In addition we demonstrate that these results are in a good agreement with the simulation data represented by the solid lines in Fig. 6.
Fig. 6 Chromatic dispersion measurements (markers) and simulations (lines) of fiber groups: A, B, C and D.
Despite relatively small variations of structural parameters within each fiber group, an impact of these variations on dispersion characteristics can be substantial, either causing shifting or bending (as in the group C) of CD curve. Figure 6 reveals that in some fiber groups the CD is more sensitive to changes of d and Λ than in the others. However to allow fair comparison one has to compare fiber pairs, in which these changes are at the similar level. Therefore to assess in which fiber groups the fabrication inaccuracies may have detrimental effect on a fiber performance we introduce the following approach. Firstly, we evaluate how strongly the geometrical parameters of fibers vary within each fiber group. To do so, we calculate for each two fibers from the same fiber group (e.g. fibers C1 and C3) the relative changes of d and Λ (Δd/d and ΔΛ/Λ). The results are presented in Fig. 7 , in which the changes calculated for example for a fiber pair C1 and C3are represented by a data point C1-3.
Fig. 7 Relative changes of d and Λ for two fibers from one of the fiber group: A, B, C and D. The pairs of fibers with similar changes are grouped into classes I, II and III.
Secondly, basing on Fig. 7 we distinguish three classes of fiber pairs, in which the relative changes of d and Λ are at the similar levels. The smallest changes occur in class I (fiber pairs:B1-2, D1-2), whereas the largest changes are in class III (fiber pairs:C1-3 and D1-3). Class II represents the fiber pairs with intermediate changes of geometrical parameters. For these three classes we compare the CD changes (ΔD) caused by the variation of d and Λ(see Fig. 8 ).
Fig. 8 The calculated change of CD (ΔD) caused by the structural parameter variations for the class (a) I, (b) II and (c) III, as defined in Fig. 7.
Figure 8 shows that in the group of fibers A, B and D the change of the CD is weakly dependent on the wavelength even if the variation of geometrical parameters is large (see ΔD for fibers D1-3in Fig. 8(c)). On the other hand in the fiber group C the dispersion characteristics change significantly with the change of structural parameters and, in contrast to other groups, this change strongly depends on the wavelength. To explain this exceptional properties of the fiber group C we compare in Fig. 9 the spectral changes of normalized mode field diameter (MFD) in fibers:C1, C3, D1, D3.These fibers were chosen for a comparison, because they represent the same fiber class and the above mentioned two types of behaviors are well observed in fiber pairs C1-3 and D1-3.
Fig. 9 Normalized MFD of fibers C1, C3, D1, D3. MFD1060 indicates the MFD of a given fiber at the wavelength of 1060 nm.
Due to the smaller core diameter and the smaller filling factor (see Table 1) the fundamental mode in fibers from group C expands quicker with the wavelength than in the other groups as demonstrated on the example of group D. Consequently, when the MFD is larger, the mode stronger interacts with the structure of the photonic crystal cladding and thus the sensitivity of dispersion properties to the change of d and Λ is larger. In fibers with large core diameter and large filling factor the MFD is large however most of the optical power is localized inside the core. The mode does not spread out strongly to the photonic crystal cladding and does not increase largely with the wavelength. As a result, the impact of the fabrication imperfections on CD is smaller in this kind of fibers.
Therefore the conclusion is that fibers with small core diameters have to be fabricated with a high precision to preserve their dispersion characteristics. However they give also more freedom with respect to managing the dispersion features. On the one hand, it is possible to obtain the second zero dispersion wavelength (ZDW) in a desired wavelength range due to the bending of the dispersion curve [1,28]. Additionally, the ZDW can be shifted to the short wavelengths, as it is presented in Fig. 10 . In each fiber group, the smaller core diameter results in a blue shift of the ZDW. In a case of fiber pairs from class III, i.e. C1-3 and D1-3, the ZDW is shifted by approximately 100 nm. This is of particular importance in a case of fibers which are designed and fabricated for a SC generation. To efficiently generate SC the fiber has to be pumped close to the ZDW. Therefore the shift of the ZDW caused by the variation of structural parameters may have detrimental effect on the performance of the device.
Fig. 10 ZDW for the investigated MSFs, extracted from the measurement results. The horizontal line represents the wavelength of a pump source, which is meant to be used to generate the SC in these MSFs.
The above conclusions can be used to mitigate the impact of the fabrication inaccuracies by a proper fiber design but also to properties and meant for different applications.
5. Analysis of the measurement errors
Although measurement results are in a good agreement with simulations, the former are affected by errors which is in particular evident in a case of fibers from group C (see Fig. 6). The analysis of the interference pattern and hence measurement accuracy is determined by the resolution of a detection device (OSA), a thermal stability, the measurement of the fiber length and the loss of the dispersion measurement setup. To calculate the tolerance of the measurement methodology on the accuracy of the peak localization caused by the limited resolution of OSA we performed the Monte-Carlo simulation with 500 000 runs per balanced wavelength. The spectral resolution of OSA was set to ± 0.25 nm due to the need of fast registering of the interference pattern. Therefore in each run of the Monte-Carlo analysis, the positions of peaks were drawn from a normal distribution with a standard deviation of 0.25 nm and the mean values equal to the measured values. To demonstrate what is the expected error of a CD measurement caused by the uncertainty of a peak localization related with the spectral resolution of the OSA we present in Fig. 11 the difference of a measured value and the mean value of the CD achieved in Monte-Carlo experiment as a function of wavelength. In addition, error bars represent the standard deviation of simulation results.
Fig. 11 The difference of a measured value of the CD (Dm) and the mean value of all runs of Monte-Carlo simulation (Ds) as a function of wavelength. Error bars represent the standard deviation of Monte-Carlo results.
The error does not exceed ± 1ps/(nm*km) for all wavelengths and all fibers. Thus, the limitation of the spectral resolution to the level of ± 0.25nm cannot be the reason of a spread of measurement results in the fiber group C. Our analysis confirmed also that the error of the fiber length measurement has not affected substantially the CD measurement accuracy. In addition, the influence of the temperature variation can be neglected due to the automated and fast process of registering the spectral response of the interferometer. The more meaningful in the case of the measurement accuracy is the power level of light, which is introduced to the tested fiber. In fibers with a small core diameter (as for example in the fiber C3), the portion of light launched to the fiber is smaller than in fibers with large cores. It results in a worse fringe contrast and a larger signal to noise ratio.
6. Summary and conclusion
In this paper we reported on the series of ten MSFs fabricated with slightly changed structural parameters. All the fibers were designed in order to obtain the ZDW around the wavelength of 1060 nm and to generate the SC while pumped at this wavelength. With the use of white-light interferometry we have measured the CD for each fiber. The measurement results were in a good agreement with simulation data and they indicated that fibers with a small core diameter are more sensitive to the inevitable fabrication inaccuracies than the fibers with a large core diameter and thus they have to be fabricated with a better accuracy. However the large flexibility of these fibers with respect to the managing of the dispersion characteristics allows to develop fibers with customized dispersion features. The change of geometrical parameters by approximately 30% in a case of fibers from group C shifted the ZDW by 100 nm and significantly reduced the CD for long wavelengths. It shows that fibers with a slightly changed geometry may be used to generate SC light with different spectral characteristics, adapted for a different applications. Therefore this work is the first step to the development of a customized SC light source with an exchangeable nonlinear fiber and tunable SC source, in which the tuning can be achieved by e.g. stretching the fiber. On the other hand we also demonstrated experimentally that it is possible to obtain the ZDW close to the 1060 nm wavelength in fibers which exhibit low sensitivity to the structural parameter deviations. This in turn is profitable because the dispersion characteristics of such fibers will not be affected by the fabrication inaccuracies. Therefore the designing process of a MSF should be a tradeoff between the insensitivity to the inevitable fabrication errors and the possibility of tuning the CD of the fiber and thus employing such fiber in a tunable SC light source. Our initial experiments reveal that with the use of our fibers (both sensitive and not sensitive to the variations of geometrical parameters) we can generate the broad spectrum, which covers wavelength range from approx. 400 nm to 1800 nm (with a deep at 500 nm). However the optimization of the fiber length and the comparison of our fibers with respect to the SC generation need some more in-depth analysis and will be the subject of our next paper.
Acknowledgments
The work described in this paper was partially supported by the EU FP7 as the COST action TD1001, by the Polish Ministry of Science and Higher Education within the Innovative Economy Programme as the key project POIG.01.03.01-14-016/08-07 and by the National Centre for Research and Development within the research project no. NR02 0074 10 and by the Polish Agency for Enterprise Development within the Innovative Economy Programme as the projects POIG.01.04.00-06-017/11 and POIG.01.04.00 18-008/10.
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