**S. K. Morshnev, N. I. Starostin, Y. V. Przhiyalkovskiy, A. I. Sazonov**

Nonlinear Dependence of the Verdet Constant on Concentrations of Paramagnetic Impurities Into the Optical Fiber Core

Paramagnetic contributions to the Verdet constants were measured in optical fibers whose cores are doped with paramagnetic impurities Gd+3 (0.4 at. %) and Tb+3 (1 at. %). It is established that the linear dependence of the Verdet constant on small concentrations of paramagnetic impurities in the optical fiber core requires nonlinear corrections when the concentrations significantly exceed 1–2 at. %. This phenomenon is explained by the formation of exchange pairs of ions of paramagnetic impurities, exchange triples, etc., the probability of which has the same dependence on the concentration of paramagnetic impurities. Exchange pairs of magnetic moments enhance the effect of the magnetic field on the sample and are the precursors of magnetic domains in ferromagnetic materials.

S. K. Morshnev, N. I. Starostin, Y. V. Przhiyalkovskiy, A. I. Sazonov

Kotelnikov Institute of Radio Engineering and Electronics

(Fryazino Branch) Russian Academy of Sciences,

Fryazino, Moscow region, Russia

Paramagnetic contributions to the Verdet constants were measured in optical fibers whose cores are doped with paramagnetic impurities Gd+3 (0.4 at. %) and Tb+3 (1 at. %). It is established that the linear dependence of the Verdet constant on small concentrations of paramagnetic impurities in the optical fiber core requires nonlinear corrections when the concentrations significantly exceed 1–2 at. %. This phenomenon is explained by the formation of exchange pairs of ions of paramagnetic impurities, exchange triples, etc., the probability of which has the same dependence on the concentration of paramagnetic impurities. Exchange pairs of magnetic moments enhance the effect of the magnetic field on the sample and are the precursors of magnetic domains in ferromagnetic materials.

Keywords: Faraday effect, Verdet constant, concentration of paramagnetic impurities, exchange pairs of ions

Article received: 23.08.2021

Article accepted: 30.09.2021

1. Introduction

The Faraday effect [1], which consists in the fact that when linearly polarized light propagates through an optically inactive substance in a magnetic field, the rotation of the plane of polarization of light is observed, is the easiest to imagine on the basis of the fundamental Zeeman effect [2]. Under the influence of a magnetic field, the spectral line from the region of the electronic absorption band is split into two lines separated by the Larmor frequency, and differing in the circular polarizations of the absorbed light. Their anomalous dispersions occur at a large distance along the spectrum, in particular at the operating frequency. Due to the Larmor shift, the anomalous dispersions are also frequency shifted, and the refractive indices of the right and left circular components of the radiation at the operating frequency are different, which leads to a phase delay between these orthogonal components (otherwise, to a rotation of the plane of polarization of linearly polarized radiation), i. e. to the Faraday effect.

Physically, this effect manifests itself due to the Larmor precession of electron orbitals with nonzero orbital moments and, accordingly, magnetic moments (diamagnetic Faraday effect) or due to a similar precession of uncompensated spin magnetic moments (paramagnetic Faraday effect). The precession of electron clouds (according to the Le Chatelier’s principle) decreases the external magnetic field, while the precession of paramagnetic spin moments increases it (the Boltzmann distribution). Hence the conclusion: the diamagnetic Faraday effect has a different sign compared to the paramagnetic effect. In addition, the strong temperature dependence of the paramagnetic Faraday effect, which depends on the difference in the populations of the magnetic levels, becomes clear.

Quartz, as well as many other transparent materials, does not contain paramagnetic impurities, but has diamagnetic properties, which allows them to be used in optical fibers for magnetic field and electric current sensors. This is also facilitated by the very weak temperature dependence of the diamagnetic parameters [3], in comparison with the paramagnetic parameters.

The magnetic properties of materials in the Faraday effect reflect their Verdet constants, V, proportionality coefficients between the angle of rotation of the plane of polarization at an angle dθ and the action of the induction of the magnetic field over the length of the segment :

.

In a diamagnetic medium, the Verdet constant is positive Vd > 0, and in a paramagnetic medium, it is negative VP < 0. As follows from the Van Vleck theory [4], the contributions to the Verdet constant from various spectral lines add up additively:

, (1)

where ηi is the concentration of centers of the corresponding line. Usually, λi is an experimental value that combines several lines of the same type: diamagnetic λi = λD or paramagnetic λi = λP; λ is the working wavelength. Thus, for example, fused quartz has a diamagnetic Verde constant Vq, described by formula (1) with a coefficient Ki = 1.419 rad ∙ μm2 / (T ∙ m) > 0 and λi = λD = 0.118 μm [3] that, e. g., at a wavelength λ = 1.55 µm gives: V (λ = 1.55 µm) = 0.594 rad / (T ∙ m).

Optical fibers are combined (layered) media, and both the core medium and the reflective cladding medium contribute to the Verdet constant for an optical fiber, since part of the radiation travels along it in exponentially decaying wings of the corresponding mode [5]. That is why the Verdet constant in optical fibers is often less than in a homogeneous medium. Some impurities, such as Ge, which is introduced into the fiber core, markedly increase the Verdet diamagnetic constant compared to pure quartz.

The paramagnetic impurity makes a negative contribution to the constant V [4]:

. (2)

It is clear that a paramagnetic impurity of low concentration placed in a diamagnetic quartz lattice can only decrease the value of the total positive Verdet constant. An increase in the concentration of the paramagnet leads first to the compensation of the positive diamagnetic component, and only after that to the appearance of the negative paramagnetic Verdet constant.

We observed a decrease in the diamagnetic component of the Verdet constant at relatively low (ηi ≤ 1 at.%) concentrations of both Tb3+ and Gd3+ ions. In combination with the results of other authors, this made it possible to predict the nonlinear dependence of the Verdet constant on the concentration of paramagnetic impurities.

2. Experiment

2.1. Methodology

The studies were carried out on the setup shown in Fig. 1 and Fig. 2. Fig. 1 shows a conventional ellipsometric unit (described in detail in [6]) for measuring the phase delay in optical fibers of type Lo-Bi (method I). This setup was used to study samples of fibers with a core doped with terbium Tb+3 with a concentration of ~1 at.%. Diode modules operating at wavelengths λ1 = 670 nm and λ2 = 820 nm were used as sources 1 of optical radiation. The light from the source 1 (Fig. 1), after passing through the linear polarizer 2, becomes linearly polarized, then modulated by a mechanical chopper 3 and enters the LoBi fiber 4 and becomes weakly elliptical due to a small phase delay. Suppose that the components of electric field of the light wave along the principal axes of the ellipse allocated by analyzer 5 are, respectively, a and b (а >> b). The modulated signal taken from the photodetector 6 is proportional to а2 (maximum) or b2 (minimum), depending on the orientation of the analyzer 5.

The magnetic field for the Faraday effect was created by a coreless coil 7 through which fiber was passed. Under these conditions, the rotation angle θ of polarization ellipse is equal: θ = A · V · I · N, where A = 2.19, I is current in ampers, V is Verdet constant in rad / (T · m), N ≈ 5 000 – the number of turns of the coil. The current through the coil was stabilized at a level of 1.07 ± 0.01 A. To increase the signal-to-noise ratio (S / N > 100), the modulated radiation at the fiber output was fed to a lock-in amplifier 8 together with a reference signal from modulator 3. Then the signal was fed to computer 9, where it was recorded. The signal began to be recorded before the switch 10 was turned on, as a result of which a current of ~1 A flowed through the coil 7. Then the direction of the current through the coil was reversed using the switch 11.

Signals with the direct passing of current Δ+ and with the reverse – Δ– were recorded by a computer. The difference between these values was used to determine the angle of rotation θ:

(Δ– – Δ+) = (a2 – b2) ∙ sin2ϕ ∙ sin2θ.

The technique uses the fact that for a fixed value of the current through the coil, the polarization ellipse rotates through the same angle θ, regardless of which angle ϕ the analyzer is oriented. The dependence (Δ– – Δ+) on sin(2ϕ) is plotted, the slope of which is proportional to sin(2θ). The relative error of the experiment is 2–5%.

A study of the single-mode fiber with a core doped with gadolinium Gd+3 with a concentration of 0.4 at. % was carried out using a standard fiber-optic current sensor (FOCS) [7] based on a reflective interferometer (method II). The setup diagram is shown in fig. 2. Formed in the electronic-optical unit of the FOCS 1 radiation at a wavelength of 1550 nm in the form of two linear orthogonally polarized waves is transmitted through fiber 2, which retains the polarization of radiation (PM line) to the quarter-wave plate 3. From the output of plate 3, radiation in the form of circularly polarized modes enters the sensing element of the FOCS, which consists of two series-connected straight fiber sections of the same length (40 cm): standard fiber 4 of the SMF‑28 type and a single-mode fiber under study 5, doped with gadolinium ions, that was spliced to it. Both sensitive fibers were alternately located inside the coil, with the remaining parameters of the circuit unchanged. The coil had N = 800 turns of copper wire, through which an electric current of I = 2А was passed. The relative error of the method is 0.1%.

The output signal of the FOCS was recorded both when the magnetic field of the coil was applied to the SMF‑28 sensor fiber and to the gadolinium-doped fiber. The signal ratio allows you to find the ratio of the Verdet constants of these fibers.

2.2. Results

The results obtained by method I (Fig. 1) are shown in Table 1 for two wavelengths. Fibers A, B, and C did not contain the paramagnetic impurity Tb+3, but they contained different concentrations of Ge, which made it possible to determine the angle of rotation θq for the diamagnetic Verdet constant of pure quartz under the conditions of our experiment on the intersection of the graph of the dependence of θ on the Ge concentration ηGe with the ordinate axis at ηGe = 0. θq (λ = 670 нм) = 1.03° and θq (λ = 820 nm) = 0,7°.

Table 1 shows that the introduction of 1 at .% Tb+3 led to a decrease in the angle of rotation of the plane of polarization of radiation in the Faraday effect without changing the direction of rotation. The latter means that the introduction of 1 at. % of the paramagnetic impurity Tb+3 did not allow us to overcome the contribution of the diamagnetic component of the Verdet constant.

Measurements at two wavelengths made it possible to obtain an important parameter λP for comparing the data of various authors (see (1)). From Table 1 it follows: λP = 475 nm, which contrasts with the result from [8]: λP = 215 nm and [9]: λP = 250 nm. In work [5], for a fiber with high concentrations, the result was obtained: λP = 385 nm, which also does not coincide with the data of works [8, 9]. We agree with the authors of [5] that this quantity depends on the Tb+3 concentration and the intensity distribution in the fiber. A decrease in its concentration leads to the expansion of the region occupied by the wave into the cladding region, which is reflected in an increase in the wavelength λP to λP = 475 nm at a low concentration ηTb ~ 1 at.%. The λP value is needed to recalculate the data (according to formula (1)) to a single operating wavelength (we chose λ = 820 nm).

Method II (Fig. 2) was used to obtain results for Gd+3 at a wavelength of λ = 1550 nm, namely: the ratio VGd / Vq = –0.1. Hence, VGd(λ = 1550 nm) = –0.059 rad / (T ∙ m).

Recalculation to the wavelength λ = 820 nm gives VGd (0.4 at.%, λ = 820 nm) = –0.3 rad / (T∙m), given in Table 2.

The last column of Table 2 shows the values of the Verdet constants in relation to the diamagnetic Verdet constant of quartz Vq (λ = 820 nm) = +2.155 rad / (T ∙ m). It can be seen that not only the introduction of 0.4 at. % of the paramagnetic impurity Gd+3, but also the introduction of 1 at. % of the Tb+3 impurity did not allow to exceed the diamagnetic component of pure quartz, despite the fact that the concentration of paramagnetic ions reached 0.26 ∙ 1021 ions / cm3 (1 at. %). This is in contradiction with the data of [10] and partially with [11]. The latter paper demonstrates the linear dependence of the Verdet constant on high concentration values, see below.

3. Discussions

Our results and those of other authors were obtained at different optical wavelengths and for two different paramagnetic impurities: Tb+3 and Gd+3, which required a recalculation of the results. Concentrations are calculated for the number of ions per unit volume (ions / cm3). Let the total mass of the components be m0, and the fraction of the substance of interest (containing paramagnetic ions) is γ wt.%. Let the molar mass of this substance, say, Tb2O3, be μ = 366 g / mol), then the number of Tb+3 ions in a cubic centimeter will be:

, (3)

where NA is Avogadro’s number, ρ is the density of the resulting glass. The calculation in terms of atomic % is simpler: the ratio of embedded ions to the number of atoms in the glass network of a unit volume, however, the calculation becomes more complicated at high concentrations and a large number of constituent components. In work [5] γTb = 54 wt.%, and the density value ρ = 3,3 g / cm3 is given, which makes it possible to accurately calculate the concentration (shown in Table 2).

Unfortunately, in [12] the density of the obtained glass (for the fiber core) is not given, therefore the exact concentration cannot be calculated, and the concentration shown in Fig. 2 in [12] η = 10.8 ∙ 1021gives when calculating by the formula (3) unrealistic density ρ = 5.0 g / cm3. Indeed, γTb = 54 wt.% [5] gives an increase in density compared to quartz Δρ = 0.65 g / cm3, and an additional increase in the value of γTb to γTb = 65 wt. % [12], i. e. by ΔγTb = 0 wt.%, cannot give an increase in density by Δρ = 1.70 g / cm3. A proportional increase in density gives Δρ = 0.13 g / cm3, and the density of glass in [12] ρ = 3.43 g / cm3. According to formula (3), this value leads to the concentration ηTbindicated in Table 2. In this case, the linear dependence of the Verdet constant shown in Fig. 2 in [12] is not obtained.

Figure 3 shows the ratios of the Verdet constant of silica fibers to the diamagnetic Verdet constant of quartz Vf / Vq as a function of the concentration of paramagnetic impurities of gadolinium and terbium. It can be seen that in the absence of paramagnetic impurities ηGd, Tb = 0, the Verdet constant of the fiber Vf is positive and equal to the diamagnetic valueVf / Vq = 1. The introduction of an impurity of paramagnetic Tb+3 ions of low concentrations decreases the positive Verde constant and only at a concentration of the order of ηTb ~0.5 ∙ 1021 it changes sign. The linear dependence of Vf on low concentrations is well represented by graph I described by formula (4):

(Vf / Vq)I = 1 – 1.9∙10–21 ∙ η, (4)

passing through two experimental points and zero in the region of low concentrations. However, in the region of high concentrations ηTb = 5.8 ∙ 1021 graph I gives the value Vf / Vq = –10.7, while experimentally, for a fiber optic, Vf / Vq = –18.6 [5]. On the contrary, if through the point Vf / Vq = –27.28 and the reliable point Vf / Vq = 1 at ηTb = 0 draw a straight line (graph II):

(Vf / Vq)II = 1 – 3.9 ∙ 10–21 ∙ η, (5)

then in the region of low concentrations there will be an equally dramatic discrepancy between theory and experiment. In particular, a negative value of Vf / Vqwill be achieved already at a concentration of terbium equal to ηTb = 0.25 ∙ 1021, which was not observed by us at ηTb = 0.26 ∙ 1021 ions / cm3. It can be argued that a linear dependence on the concentration of a paramagnetic impurity can adequately describe the behavior of the Verdet constant only at low concentrations of this impurity no more than 1–2 at.%.

Correction to formula (5) in the form of a quadratic dependence:

(Vf / Vq)IIΙ = 1 – 1.9 ∙ 10–21 ∙ η – 0.23 ∙ (10–21 ∙ η)2

leads to graph III in Fig. 3. The correction is also for the cubic dependence (6):

(Vf / Vq)IV = 1 – 1.85 ∙ 10–21 ∙ η – 0.24 ∙ (10–21 ∙ η)2 –

– 0.004 ∙ (10–21 ∙ η)3 (6))

allows plot IV in Figure 3 to go through all experimental points.

The observed nonlinearity can be explained by some centers formed from paramagnetic impurity ions at their concentrations above 1 at%. The so-called “exchange pairs of paramagnetic ions” [12, 13] may well fit the role of such centers. Beginning with a concentration of ~1 at. % with a random distribution, there is a noticeable probability for two ions to be at such a small distance that an exchange interaction of the form I∙(S1 S2) occurs between them, where I is the exchange integral. The quantity I exponentially increases as the ions approach each other. Since the exchange interaction depends on both spin S1 and spin S2, the strength of this interaction depends on the square of the concentration of paramagnetic ions η2 [12, 13].

It is known, that the exchange interaction produces a magnetic ordering of spins in space, which leads to cooperative effects that are realized, for example, in the formation of domains in ferromagnets, and to a multiple increase in the effect of external magnetic fields on a material with a paramagnetic impurity. Apparently, something similar it happens with the Faraday effect in fibers with a high concentration of paramagnetic impurity, which leads to the addition of a correction quadratically dependent on the concentration to the linear dependence of the Verde constant. It is clear that a combination of three spins is possible with a lower probability (proportional to the third power of the concentration η3), etc. In our case, the Verdet constant grows nonlinearly with increasing concentration due to the alternate inclusion of the contributions of “exchange unions of magnetic moments”.

The work was carried out within the framework of the state assignment of the Kotelnikov Institute of Radio Engineering and Electronics, Russian Academy of Sciences.

The authors are grateful to K. M. Goland, Yu. K. Chamorovskiy, V. A. Isaev, V. A. Aksyonov, V. V. Voloshin, I. L. Vorobyev for the fibers provided. The authors are also grateful to V. A. Atsarkin for fruitful discussions.

AUTHORSHIP CONTRIBUTION STATEMENT

Morshnev S. K.: concept, concept, planning and conducting experiment, t, processing and analysis of the results, writing an article. Starostin N. I.: organization and conducting of the experiment, analysis of the results, discussion, suggestions, writing an article. Przhiyalkovskiy Y. V.: conducting an experiment, analysis of the results, discussing, editing, design. Sazonov A. I.: preparation and conducting of the experiment, processing of the results, discussion, design.

ABOUT AUTHORS

Morshnev S. K., Dr. of Science (Phys.&Math), Senior Researcher,

e-mail: morshnev@profotech.com, Kotelnikov Institute of Radio Engineering and Electronics (Fryazino Branch) Russian Academy of Sciences, Fryazino, Moscow region, Russia. Research interests: physical principles of fiber optics.

ORCID: 0000-0001-5095-2745

Starostin N. I., Cand. of Science (Phys.&Math), Leading Researcher,

e-mail: nistar53@mail.ru, Kotelnikov Institute of Radio Engineering and Electronics (Fryazino Branch) Russian Academy of Sciences, Fryazino, Moscow region, Russia. Research interests: fiber optics, fiber optic sensors.

ORCID: 0000-0001-9013-8588

Przhiyalkovskiy Y. V., Cand. of Science (Phys.&Math), Senior Researcher,

e-mail: yankus.p@gmail.com, Kotelnikov Institute of Radio Engineering and Electronics (Fryazino Branch) Russian Academy of Sciences, Fryazino, Moscow region, Russia. Research interests: fiber optics, fiber optic sensors.

ORCID: 0000-0003-0591-8323

Sazonov A. I., Cand. of Science (Eng.), Senior Researcher, e-mail: sazonov_alexandr_48@mail.ru, Kotelnikov Institute of Radio Engineering and Electronics (Fryazino Branch) Russian Academy of Sciences, Fryazino, Moscow region, Russia. Research interests: special fibers and fiber sensors.

CONFLICT OF INTERESTS

All members of the authoring team agree with the text of the submitted manuscript and with the specified distribution of the contribution of each of them, the authors guarantee the originality of the results and that the manuscript is not being considered for publication in another journals.