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We give a comprehensive theoretical analysis and simulation of a FBG in single-multi-single mode fiber structure (FBG-in-SMS), based on the coupled mode analysis and the mode interference analysis. This enables us to explain the experimental observations, its asymmetric transmission and reflection spectra with the similar temperature responses near the spectral range of Bragg wavelengths. The transmission spectrum shift during FBG written-in process is observed and discussed. The analysis results are useful in the design of the SMS structure based sensors and filters.
© 2015 Optical Society of America
1. Introduction
Since the first demonstration by Meltz et al [1], fiber Bragg gratings (FBG) have been recognized as an important fiber-optic device used for spectral filtering, dispersion compensation, wavelength tuning, and sensing in optical communication and optoelectronics. Fiber gratings in single mode fibers (SMF) have been well studied and the spectral characteristics are well known [2–5]. Bragg gratings in multimode fiber (MMF FBG) have also been studied as the multimode fiber (MMF) has a merit of easy coupling with other light sources and a lot of work has been reported by using MMF FBGs as sensors [6–10].
The simple single-multi-single mode fiber (SMS) structure is also a type of multimode interference device [11–14], which could be designed as optical fiber filters [11] and sensors of temperature [15–20], strain [16–22], bending [23] and refractive index sensing [24–29]. SMS cascading a FBG in single mode fiber has been studied for multi-parameter sensing because of their different sensitivities to different parameters [18–20, 26]. However, the locations of the FBG and the sensing part of SMS are not the same, which may introduce measurement error, especially when the temperature varies over their separation distance. In order to avoid the problem, we wrote a FBG in the multimode section of a SMS structure and realized a FBG-in-SMS as a multi-parameter sensing module experimentally. To understand and utilize better the interesting characteristics of the FBG-in-SMS, we developed a comprehensive theoretical analysis here. The transmission and reflection spectra of a FBG-in-SMS are decided by both the mode coupling and the interference of the different excited modes in the multimode fiber section. Hence the theoretical analysis of FBG-in-SMS combines both the couple mode analysis and the interference analysis. This theoretical analysis method explained its asymmetric transmission and reflection spectra but with similar temperature response found in experiments. The SMS transmission spectrum shift during FBG written-in process is also discussed at discussion part.
2. Model and simulation of FBG-in-SMS
The structure of FBG-in-SMS is shown in Fig. 1. A ~5.0 cm section of multimode fiber is spliced in between two single mode fibers. Then a ~1.0 cm FBG is to be written into the multimode fiber through a phase mask to realize the FBG-in-SMS.
The output field of FBG-in-SMS results from several processes: 1. the mode coupling from the input SMF to the MMF at the left splice point; 2. the propagation of modes in MMF and the mode coupling caused by the FBG in MMF; 3. the mode re-coupling back to the SMF from the MMF and the interference happened between the re-coupled modes in SMF. The transmission output field is the mode re-coupling back to the output SMF and the reflection output field is the mode re-coupling back to the input SMF. The transmission spectrum is measured from the output SMF directly and the reflection spectrum is measured from input SMF by adding an optical fiber coupler.
The input SMF will excite multiple modes in the MMF. The modes excited in the MMF can be derived by the mode excitation efficiency, which determines the amount of the input field power that couples to each specific mode in the MMF, associated with the propagating modes within the MMF [30]. In our simulation, the refractive indices of the MMF and SMF are 1.4655 / 1.4462 and 1.4544 / 1.45, while the core and cladding diameters are 105 /125 μm and 9 / 125 μm, respectively. When light launched into MMF, the mode excitation efficiency ηm versus mode number m is shown in Table. 1. ηm is less than 6% when the excited mode is higher than LP06. Hence we will consider LP01-LP06 modes in the MMF approximately.
Table 1. Mode excitation efficiency, ηm, versus mode number for MMF using a SMF launch.
When a FBG is written into the MMF, the refractive index of the MMF will be changed and this will introduce mode coupling between the modes of MMF. The coupled mode theory is a commonly used tool for treating interactions between confined modes with refractive index perturbation, and the coupled mode equations can be expressed as [2]
where An and Bn are slowly varying amplitudes for the transverse mode fields traveling in the + z and –z directions, respectively. The initial values of An(0) and Bn(L) are related to the mode excitation efficiency, ηm of different excited modes. m and n stand for the mode index of LP modes and βm and βn are the corresponding propagation constants. In optical fiber, the coupling coefficients could be expressed by, here ε0 is the permittivity of vacuum, Em and En are the field pattern of the corresponding LP modes. δn is the perturbation to the refractive index. If δn is a constant of the fiber section, Kmn is nearly zero when m ≠ n because of the symmetry of the field pattern. The solutions of Eq. (1) are related to the mode excitation efficiency ηm and the coupling coefficients Kmn, simply for calculation, we consider the modes which has a large ηm or a small ηm but large Kmn*ηm.
As an approximation to the UV laser launch focused to the fiber [31–34], δn could be expressed as
where δn0 is the maximum core refractive index change induced by UV in the fiber cross section at a given point z. Here δn0 depends on the illumination intensity and , s and z0 control the width and the central of Gaussian function, respectively; α is the asymmetry coefficient; ρ is the radius of the photosensitive area. When ρ = 52.5 μm, α = 0.1 μm−1, Λ = 530.345 nm, the refractive index change induced by UV laser is shown in the upper left corner of Fig. 2. The periodical change of refractive index will form the FBG in the MMF. Though we assumed only LP01 mode to LP06 mode excited by the SMF launched, numbers of high-order modes occurred in –z direction. According to the criterion mentioned above, several values of Kmn*ηm are significant, which are illustrated in Fig. 2. Its x-coordinate is marked as wavelength λ in order to show the phase match condition for different modes in MMF. Here the mode coupling between LP02 and LP12, LP03 and LP13, and LP04 and LP14 are all considered even though the small coupling coefficients. They could change the intensity and phase distribution of the light field in MMF.
As the Bragg wavelength of different modes showed in Fig. 2, the spectra of the neighboring modes may be overlapped at some wavelength, so we solve Eq. (1) with the coupling of LP06-LP06, LP05-LP05, LP04-LP04-LP14-LP14, LP03-LP03-LP13-LP13 and LP01-LP01-LP02-LP02-LP12-LP12, respectively. Substitute the coupling coefficients in Eq. (1) when δn0 = 0.0008, z0 = 0.5 cm and s = 6, the transmission and the reflection spectra of FBG for only coupled modes are simulated separately without considering the interference, shown in Figs. 3(a) and 3(b), respectively. The results, considered the coupling of LP01-LP01, LP02-LP02-LP12-LP12, LP03-LP03-LP13-LP13, LP04-LP04-LP14-LP14, LP05-LP05 and LP06-LP06, are given by the blue line, green line, red line, cyan line, pink line, and yellow line. The dips of the transmission spectrum of LP modes are symmetric to the reflection peaks of the reflection spectrum.
Fig. 3 (a) Transmission spectrum of FBG without interference. (b) Reflection spectrum of FBG without interference. (c) Phases of transmission waves in FBG-in-SMS. (d) Transmission spectrum of FBG-in-SMS. (e) Reflection spectrum of FBG-in-SMS (L0 = 0.2 cm). (f) Reflection spectrum of FBG-in-SMS (L0 = 3.8 cm).
The true transmission and reflection spectra of FBG-in-SMS should consider the interference between excited modes re-coupling to output SMF and input SMF. And only zero-order modes could be re-coupled back to SMF and the interference happened between them. So the spectra of those modes are phase sensitive. Their phases are related to all FBG-in-SMS device. The total MMF length LSMS and FBG are important for the transmission spectrum and the MMF length L0 and FBG are crucial for the reflection spectrum, shown in Fig. 1.
The transmission spectrum of FBG-in-SMS, shown in Fig. 3(d), is the interference spectrum between the excited modes (LP01-LP06 modes) of output fiber. Compared to Fig. 3(a), none of FBG dips of the excited modes in the transmission spectrum could be separated easily even though there are other dips in Fig. 3(d), such as a narrower dip around 1540.90 nm. They are the interference results. Here FBG is working more like a phase modulator for all excited modes and also as a filter at different wavelengths of different excited modes. The phase difference between any two excited modes LP0m-LP0n, could be expressed as
where Δφmn_FBG is the FBG induced phase difference that could be gotten from the coupled mode Eq. (1). The phase difference caused by SMS is Δφmn_SMS = (βm -βn)(LSMS - LFBG), determined by the mode propagation constants β and the lengths of SMS and FBG. They, Δφmn_FBG and Δφmn_SMS, are all wavelength dependent. The phases of transmission waves in FBG-in-SMS are shown in Fig. 3(c). The phase of LP01-LP06 modes at 1540.90 nm are 0.250π, 0.095π, 0.974π, −0.985π, −0.714π and 0.042π, respectively. For the larger mode excitation efficiency ηm, more weight of LP02, LP03 and LP04 modes affect the dip of the transmission spectrum. Subtract the phase of modes, then the phase different Δφ23 = 0.879π, Δφ24 = 1.080π, and Δφ34 = 1.959π, can be obtained from Fig. 3(c). As Δφ23 and Δφ24 are near π, though Δφ34 is near 2π (0), the transmission spectrum at 1540.90 nm is a dip. In the same way, we can explain the spectrum at other wavelength. For example, at 1540.82 nm, Δφ23 = 0.875π, Δφ24 = 0.719π, and Δφ34 = 0.156π, so the interference intensity is larger than 1540.90 nm.Considering the reflection spectrum, Δφmn_SMS = 2L0 (βm -βn) and Δφmn_FBG is also gotten from coupled Eq. (1). Here L0 is the distance between the input SMF and FBG, illustrated in Fig. 1. Figures 3(e) and 3(f) showed the reflection spectra when L0 = 0.2 cm and L0 = 3.8 cm, respectively. The reflection peaks of excited modes, marked in Figs. 3(e) and 3(f), are identified more easily in the reflection spectrum except LP01 and LP02 modes. The LP01 and LP02 modes are overlapped in their FBG wavelengths and they are coherent and phase sensitive, but other modes are separated well in their FBG wavelength and will not coherent strongly in the reflection spectra. On the other hand, the transmission spectrum in Fig. 3(d) is strongly affected by modal interference because of the excited modes in transmission are not spectrally well separated compared with those in reflection. The spectrum is related to both the power of each mode and their interference. So the transmission spectrum of FBG-in-SMS is dissimilar to that of FBGs because of the mixing effects of modal interference..
3. Experimental results of the FBG-in-SMS spectrum at the Bragg wavelengths of the FBG
We also do some experiment to prove our model. Hydrogenated MMF was used to increase its UV sensitivity. A UV beam is projected on the MMF section of the SMS structure through a phase mask. A fiber amplified spontaneous emission (ASE) source is launched into the SMS structure and the transmission spectrum and the reflection spectrum are monitored by two optical spectrum analysers (OSA). The UV beam at 248 nm is from a KrF laser (Coherent Bragg Star Industrial 1000) with a pulse energy E = 10 mJ and a repetition rate 50 Hz.
In experiment, we were writing a ~1.0 cm FBG on a ~5.0 cm MMF with ~0.2 cm gap to the splice point between the SMF and MMF, and the parameters of the fibers are mentioned previous. The spectra of FBG-in-SMS in experiment are shown in Fig. 4. The transmission spectrum of FBG-in-SMS in experiment, shown in Fig. 4(a), have a few dips and the strongest one is at 1541.15 nm, which is marked with the circle. The reflection spectra, measured from two sides of the FBG-in-SMS device, are shown in Figs. 4(b) and 4(c), corresponding to the cases of L0 = 0.2 cm and L0 = 3.8 cm, respectively. The reflection spectrum and the transmission spectrum are asymmetric obviously. The FBG reflection peaks in the reflection spectrum are easier to be identified than the dips in the transmission spectrum because they are separated well in wavelength and not affected strongly by modal interference. However, the Bragg reflection bands of LP01 and LP02 modes are overlapped partly, which made the interference possible. The reflection spectrum at that band are phase sensitive and relates to the MMF length L0, which are consistent with our analysis and simulation results in Figs. 3(e) and 3(f).
Fig. 4 (a) Transmission spectrum of FBG-in-SMS in experiment. (b) Reflection spectrum of FBG-in-SMS in experiment when L0 = 0.2 cm. (c) Reflection spectrum of FBG-in-SMS in experiment when L0 = 3.8 cm.
We can use the FBG-in-SMS as a temperature sensor. The shapes of the transmission and reflection spectra near Bragg wavelengths are kept well and the red-shift is happened when the temperature is increased. The dips in the asymmetric transmission and or the peaks in reflection spectra of the structure could be used as characteristic wavelengths. Figures 5(a) and 5(b) show the good linear relationships between the temperature and the characteristic wavelengths in reflection and transmission spectra, respectively. The temperature response of the reflection peaks of LP01 to LP06 modes are 13.57 pm/°C, 13.13 pm/°C, 13.57 pm/°C, 13.47 pm/°C, 13.24 pm/°C and 13.43 pm/°C and their norm of residuals are smaller than 0.0144. The temperature response of the transmission dip is 13.77 pm/°C and its norm of residuals is 0.00860. The temperature responses are similar, even though their spectra are asymmetric. The temperature response ratios of the reflection peak wavelengths and the transmission dip wavelength of FBG-in-SMS are shown in Fig. 5(c) and they are all in the range of [0.96, 1] for simulations and experiments, which also shows that the reflection spectrum and transmission spectrum have the almost same sensitivity.
Fig. 5 (a) The measured temperature sensitivity of the reflection peaks when L0 = 0.2 cm. (b) The measured temperature sensitivity of the transmission dip. (c) The temperature response ratio of the reflection peak wavelength and the transmission dip wavelength of FBG-in-SMS.
4. Properties of the FBG-in-SMS spectrum far from the Bragg wavelength of the FBG
We focused on the spectra of FBG-in-SMS around FBG wavelength region in previous part. However, the SMS spectrum has some interesting properties in the regions far from FBG wavelength. The whole SMS transmission spectra would be changed as the FBG written-in process. The SMS dip, caused by the interference in the SMS structure, can be shifted, shown in our previous work [35]. Here we explain that shift of the SMS spectrum by solve Eq. (1) between 1515 nm and 1535 nm with different δn0, then plot the interference spectra in Fig. 6.
Fig. 6 (a) The initial transmission spectrum of FBG-in-SMS. (b) Spectra of FBG-in-SMS during FBG writing process. (c) The shift of SMS dip versus δn0.
Figure 6(a) shows the initial transmission spectrum of SMS without writing FBG (black line) and a transmission spectrum of SMS with FBG written in (red line). The spectra properties in FBG region are discussed in detail in previous sections. The SMS dip around 1534 nm has a blue shift during the FBG writing process, shown in Fig. 6(b) obviously. When the UV beams irradiation, the UV induced refractive index δn0 is increased, and the corresponding phase change Δφmn in Eq. (3). So the dip of the interference spectrum between the excited modes (LP01-LP06 modes) shift and the transmission level is varied. The shift values of the dip versus δn0 are shown in Fig. 6(c). With the enlargement of δn0, the wavelength shift linearly. The simulation explains the previous experimental results qualitatively. It proved that UV irradiation is a method to realize the tuning of SMS spectrum, which may be interested in the fabrication of the SMS based filter.
5. Discussions
In previous sections, we show the asymmetric transmission and reflection spectra of FBG-in-SMS structure. Compared with the simulation results, the experiment spectra have more ripples, which are showed in Figs. 4. One possible reason is that there could be more LP modes (such as LP11 and LP15) excited in MMF in our experiments than the LP modes used in simulation. The LP11 and LP15 modes, with their Bragg wavelengths are 1541.77 nm and 1540.75 nm, have smaller coupling coefficients under ideal condition, which were neglected in our simulation. Moreover, splicing the SMF and MMF under the experimental conditions may be excited other modes in MMF. Hence the experimental exciting condition may not be exactly the same as the exciting condition in simulation and this could also result in the ripples as seen in the experimental spectra. Another possible reason is the deviation of the Gaussian apodization function during the grating inscription. It could introduce the side lobes near Bragg wavelengths. The interference between the side lobes and the Bragg band of different modes may be happened and could introduce some ripples as well. Though more ripples in the experiment spectra, the simulation provided a simple and effective explanation for FBG-in-SMS.
It is also to be noted that the dips of the SMS spectrum are caused by the destructive inference between the excited modes when the transmission spectrum of interest is far from the Bragg wavelength of the FBG. The dip near 1530 nm is mainly formed by the interference between LP02 mode and LP03 mode, while the dip around 1560 nm is the interference between LP04 mode and LP06 mode. Since the two dips are related to two different sets of modes, they vary differently during FBG writing.
6. Conclusion
In this paper, a comprehensive theoretical analysis and simulation of a FBG-in-SMS, based on the coupled mode analysis and the mode interference analysis, are given to analyze the experimental reflection and transmission spectrum properties of the FBG-in-SMS. The transmission and reflection spectra of FBG-in-SMS are asymmetry because of the multimode interference, which is related to the excited modes in MMF and the MMF length and the location of written-in FBG. In the transmission spectrum, FBG dips of any excited mode become less obvious and difficult to be recognized even though some new dips appear strongly. Those new dips are introduced by the interference of the excited modes in MMF and the written-in FBG is a phase modulator for these modes. The FBG dips are mixed in the interference spectrum. In the reflection spectrum, FBG reflection peaks for those excited modes in MMF are easier identified because they are separated well in wavelength and not coherent strongly except the first two low-order modes, LP01 and LP02. Their Bragg reflection bands are overlapped partly, which make the interference happened possibly and the reflection spectrum at that band are phase sensitive and relates to the MMF length. Though the transmission spectrum and the reflection spectrum are asymmetric, their temperature responses are similar. Based on our analysis, the SMS transmission spectrum shift during FBG written-in process is also explained and consistent with experimental results. These results are useful in the design of the SMS based sensors and filters.
Acknowledgements
Authors thank for the support by National Science foundation projects (11178010 61377096, 61227013, and 61422505), National Key Scientific Instrument and Equipment Development Project (No. 2013YQ040815),Heilongjiang young researcher support project (1253G018), the Program for New Century Excellent Talents in University (NCET-12-0623), Harbin Science foundation (2011RFLXG004), the Fundamental Research Funds of the Central University, China, and the 111 project (B13015), to the Harbin Engineering University.
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