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Abstract
When radiation propagates in an optical ring resonator, apart from the main peaks, additional peaks are observed. It is known that the strain of optical fiber induces birefringence. Assuming that an additional peak is associated with the main peak, we proposed a model describing the relative position of the main and additional resonance peaks. The model predicts induced birefringence based on the frequency difference between the main and additional peaks. For experimental verification, we fabricated an optical ring resonator, measured the difference between the main and additional peaks and obtained the birefringence value. This value was –(3.42 ± 0.51)·10−7, consistent with values obtained in other studies.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Ring resonators (RR) are used for stabilizing radiation frequency in laser technology [1,2], for ultra-precise measurement of displacements in metrology [3], for measuring the concentration of various substances in spectroscopy [4], and optoelectronic generators [5]. An optical ring resonator can also be used as a sensing element in a miniature resonance gyroscope [6–10].
Investigating the fiber optic ring resonator revealed the occurrence of additional resonance peaks due to the bending-induced birefringence in a single-mode fiber [11,12]. Some authors explain the occurrence of additional peaks by the existence of two eigenmodes in a bent fiber [13]. In order to develop a miniature resonance gyroscope, it is necessary to reduce the effect of additional resonance. Several attempts have been made to eliminate additional peaks, few studies, however, have investigated the causal relationship between additional peaks and birefringence in a systematic way. This phenomenon, however, can be useful in some applications such as high-precision temperature measurement. Therefore, the reason behind its occurrence must be studied. In this study, we propose a model for evaluating the birefringence Δn induced in a single-mode fiber. For this purpose, the difference between the main and additional resonance peaks positions obtained in the experiment is used.
2. Methodology
2.1 Parameters of the ring resonator
An RR comprises an interferometric scheme, two straight waveguide channels, coupled regions and a closed optical cavity. The light interference in an RR resembles multi-beam interference in a Fabry–Pérot interferometer [14].
In the coupling region, two types of interferences occur. At a specific wavelength, when the phase shift is a multiple of 2π, we observe constructive interference inside the resonator and destructive interference in the straight waveguide at the outlet. The phase shift can be used to determine the resonance condition:
where m is an integer. This formula can be represented in terms of wavelength: where L - geometrical length of the resonance cavity, neff - effective refractive index in the principal mode.Several operating parameters such as free spectral range (FSR) and Q-factor are identified for the RR. FSR constitutes the difference between resonance frequencies (or resonance wavelengths) [14]:
Q-factor can be defined as the total energy inside the resonator divided by energy lost in a single pass of optical radiation in a closed cavity. The Q-factor is an essential parameter since it can be used for evaluating the minimum angular velocity when using the RR as a sensor in a gyroscope [10,15].The effect of induced birefringence and the appearance of two resonance peaks are explained in the following works [13,16–24].
Hotate et al. [13] derived equations of radiation propagation in a fiber-optic ring resonator (FORR), that includes birefringence effect, and have also introduced the concept of polarization eigenstate. The RR comprises a single-mode fiber loop and a coupler. In this work, the authors established the relationship between the output power at the photodetector and the incoming electromagnetic wave. Furthermore, under fluctuating polarization of the incoming wave, the changes in power ratio of the peaks of two resonances corresponding to two orthogonal polarizations are observed. The authors note that the full width at half maximum (FWHM) of the two resonance peaks differ from each other, which affects the overall Q-factor of the resonator.
In the study conducted by Morichetti et al. [16], the authors consider an integrated optical RR and the optical pumping between polarizations that occurs due to a bent waveguide. It was established that the induced birefringence causes a phase shift between two orthogonal polarizations. The strength of polarization coupling increases linearly with an increase in the refractive index contrast of the waveguide. The authors used this effect to develop a module with a controllable phase shift.
The study conducted by Yu et al. [21] examines the polarization noise in an optical resonance gyroscope. The polarization noise is influenced mainly by three factors - temperature, the polarization state of the incoming wave and the quality of polarization in the waveguide resonator. Increasing the FWHM of the main resonance peak when an additional resonance peak emerges, causes noise leading to the degradation in the accuracy of gyroscope. Therefore, to improve the characteristics of the gyroscope, it is necessary to reduce the effect appearance of an additional peak.
2.2. Theoretical approach
To the best of our knowledge, the dependence of the frequency (wavelength) difference between two polarization peaks on birefringence Δn induced in the FORR is missing in the literature. This characteristic can be important when using FORR as a sensing element for a physical quantity that can alter the birefringence Δn of a resonator. The signal of an optical resonant gyroscope is a set of peaks (main and additional) in the time domain. For more correct signal processing, additional peaks must be filtered out, since the calculation of the output signal should be conducted only on the main peaks. By calculating the fiber bend radius, we can predict the position of the additional peaks and consider them during processing without additional measurements.
The main and additional resonance peaks were assumed to relay two modes with mutually orthogonal polarization (Fig. 1). The refractive indices for the two orthogonal modes differ owing to the induced birefringence; as a result, the resonance for these modes is observed at different frequencies (wavelengths). Due to the tuning rate of the laser used in the experiment being calibrated in units of frequency, further reasoning will be conducted for the frequencies of the resonant peaks. The difference between the main and additional peaks is denoted as Δfp. This indicates that for the specific frequency f0 of optical radiation with some polarization and the corresponding frequency (f0 + Δfp) of the radiation with orthogonal polarization, there is a given identical integer number of waves m staying within the resonator, according to Eq. (2). Thus, the equation may be written as:
where f0 – center frequency of the radiation (corresponds to the wavelength λ0 = 1550 nm); neff = 1.46 – refractive index of silica fiber; neff 2 = neff + Δn – refractive index for wave with orthogonal polarization; Δn – induced birefringence.
Fig. 1. Schematic imaging of two mutually orthogonal polarized modes with the same mode number m in the optical resonant cavity. The blue line is the mode at frequency f0, it corresponds to the main resonant peak, the red line is the mode at frequency f0 + Δfp, it corresponds to the additional resonant peak
From Eq. (4) the value of birefringence is determined as follows:
2.3 Measurement procedure for resonance curves and resonator performance
The measurement of the spectral characteristics of a resonator is performed using the technique used by Valyushina et al. [25] and is shown in Fig. 2(a). The technique consists of a simple and relatively accurate method for calculating the characteristics of the resonator using a tunable laser and an oscilloscope. The advantage is that it requires neither additional synchronizations nor frequency calibrations as are required for other techniques [26,27].
Fig. 2. a - experimental scheme of measuring spectral characteristics; b - spectrum obtained from photodiode 2, as observed on oscilloscope screen; с – experimental setup (1 – fiber ring resonator, 2 – port 1, 3 – laser, 4 – port 2, 5 – port 3, 6 – photodiodes, 7 – oscilloscope)
To record the spectrum and measure the parameters of FORR, a narrow linewidth tunable laser (Pure Photonics PPCL300) was used. The basic requirement for a laser is as follows: the width of a laser peak should be narrower than that of the resonance peak.
Laser radiation enters port 1 of the resonator, and then passes through ports 2 and 3 to photodiodes 1 and 2 (FPD610-FC-NIR Menlo Systems), converting the optical signal into an electrical one. The laser exerts a central frequency sweep of the radiation with a given tuning rate v. Thus, the signal power taken from the photodiodes changes according to the frequency response of the resonator. To register a signal in the time domain, where a resonant spectrum is observed, photodiodes are connected to a Tektronix DPO7254 oscilloscope (Fig. 2(c)). The signal was recorded from both photodiodes. The resonance peak position obtained from photodiode 2 coincides with that of resonance dips from photodiode 1; therefore, it is enough to show only one spectrum from photodiode 2 (Fig. 2(b)).
Further, we establish the difference between the peaks (maximum values on the spectrum) within the time domain ΔtFSR. This value is used to calculate FSR in the frequency domain:
where v is the given tuning rate. Figure 2(b) shows the spectral distribution of radiation transmitted through a FORR from a single-mode fiber twisted into a loop with a diameter of 9 cm. Additional resonant peaks are observed near the main peaks.3. Results
For the experiment, we manufactured a fiber-optic resonator from two fiber-optic splitters with a division ratio of 99:1 and a 1.13 m long fiber (SMF-28 Ultra, Corning) with a cavity diameter of 9 cm (4 loops). The v = 5 GHz/s tuning rate was chosen for the laser. As a result, a resonance spectrum is observed on the oscilloscope; the oscilloscope sweep is selected in such a way that at least 5 peaks are observed on the oscilloscope screen. To determine the maximum power of the resonance peak and its corresponding point on the time scale accurately, the resonance peak is approximated by the Lorentzian function [25] (Fig. 3(a)).
Fig. 3. a - Approximation of the resonant peak by the Lorentz function; b - spectrum from photodiode 2, observed on the oscilloscope screen
For adjacent resonant peaks with the same maximum power, the time difference is calculated and averaged over 5 peaks. The resulting value was ΔtFSR = 41.48 ± 2.88 ms. The difference between the main and additional peaks was also calculated from the values for 5 pairs of peaks; the value was Δtp = 9.08 ± 1.35 ms (Fig. 3(b)).
Substituting the values obtained from the experiment into Eq. (7), we obtain that in the frequency domain, the distance between the main peaks (FSR) is FSR = 207.38 ± 14.40 MHz, and the additional peaks are spaced from the main ones by Δfp = 45.38 ± 6.74 MHz. Further, we obtain the value of birefringence Δn from Eq. (8) Δn = −(3.42 ± 0.51)·10−7.
Note, that the amplitude of the additional and main peaks varies depending on the polarization of incoming radiation; however, the frequency difference between the additional and main peaks remains constant.
4. Discussion
In some of the works [14,23], only the intrinsic polarizations of radiation in the fiber cavity are considered; however, the relative position of resonance peaks arising from induced birefringence due to fiber bending remains unexplored. Valyushina et al. [24] built a simple mathematical model, where the dependence of the power of resonance peaks on the radiation wavelength was established, taking into account the induced birefringence Δn. Several cases of mutual arrangement of the main and side peaks at different values of birefringence are considered in this work, and the value of induced birefringence is quantitatively calculated from the dependence of birefringence Δn on the bending radius R [28]:
where Rf is the cladding radius of the fiber.Feng et al. [11] also obtained the dependence of induced birefringence on the bending radius of the fiber:
where k = 5.334·10−10 m2 for single-mode fiber.The values of birefringence Δn, calculated by Eqs. (8) and (9), for the bend radius R = 4.5 cm are Δn = − 2.51·10−7 and Δn = − 2.63·10−7, respectively, which is consistent with our result Δn = −(3.42 ± 0.51)·10−7.
The probable reason for the deviation of the obtained result from those available in the literature is the design features of the resonator (strong bends of the fiber in the region where the splitter is connected to the fiber cavity, additional birefringence in the optical divider, etc.).
5. Conclusions
In this work, we have established an analytical relationship between the resonance peaks (main and additional) and the induced birefringence in the optical fiber of a ring resonator. The values obtained using the proposed model agree with the literature data on birefringence induced during fiber bending. In addition, the work confirms the results obtained by other authors, who investigated the occurrence of additional resonance peaks. These results include the nature of the peaks, namely, their formation due to different resonance conditions for waves with mutually orthogonal polarization. Note that the proposed model is applicable in the case when the FSR is greater than the distance between the peaks Δfp, that is, in the case of a fiber resonator that is not very long length and does not have a large bending radius. Otherwise, there might be ambiguity as to which main peak the additional one belongs to.
In the future, the influence of temperature and other external factors on the relative position of peaks should be explored in more detail. In addition, it would be beneficial to verify the spacing of resonance peaks at critical fiber bending for additional information on the maximum value of induced anisotropy and limits to the applicability of our approach.
Funding
Ministry of Science and Higher Education of the Russian Federation.
Acknowledgments
The authors would like to thank Associate Professor M. Suvorova for helpful discussion of the study results. The work was carried out with the financial support of the Ministry of Education and Science of the Russian Federation within the framework of the program of activities of the world-class scientific and educational center "Rational Subsoil Use."
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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