Study of the Verdet constant of the holmium-doped silica fiber

Zhenmin Liu; Zhenyi Chen; Na Chen; Yi Huang; Shupeng Liu; Yana Shang; Tingyun Wang

doi:10.1364/OSAC.390111

1. Introduction

The Faraday effect is one of the magneto-optical effects, that is, a phenomenon in which the polarization plane of linearly polarized light is rotated when passing through a magneto-optical material in the direction of an applied magnetic field or a direction of magnetization. The rotation angle of the polarization plane is proportional to the magnitude of the magnetic induction intensity and the effective length of light transmission. The proportionality factor describing this relationship is called the Verdet constant, which is an important parameter characterizing the magneto-optical properties of the material. Since the discovery of Faraday effect, magneto-optic materials with strong Faraday effect have been widely used in optical isolators, magneto-optic modulators, optical circulators, magnetic field sensors, and other fields [1–3].

To improve the material's magneto-optical properties, various rare earth elements were doped into the material. D.R. MacFarlane obtained the Verdet constants of Dy-doped glass, Pr-doped glass, Tb-doped glass and Ce-doped glass through experiments [4]. Large Verdet constants were found in the Fe₂O₃:Tb_(10%) and Fe₂O₃:Tb_(10%)/SiO₂ complex films by Peiweb Lv, the values were 17470.7 rad T^-1m^-1 and -6754.4 rad T^-1m^-1 respectively [5]. Yan Lin Aung et al. developed optical grade (Tb_xY_1-x)₃Al₅O₁₂ ceramics, which was 1.5 times the Verdet constant of ordinary TGG [6]. Zhe Chen et al. developed (Tb_(1-x)Er_x)₃Ga₅O₁₂ (TEGG) single crystal, and the Verdet constant of TEGG at 532 nm comes up to 224.5 rad T^-1m^-1, obviously larger than that of TGG [7]. In the same year, the team produced (Tb_(1-x)Pr_x)₃Ga₅O₁₂ (TPGG) single crystal. The Verdet constant of TPGG at 632.8 nm was 182.43 rad T^-1m^-1, which was 36.14% larger than that of TGG [8]. In the past, the research on magneto-optical materials mainly focused on glass, thin film, ceramics and crystals. However, the application range of these materials is limited by the difficulty of material integration and high loss. Therefore, how to improve the Verdet constant of optical fiber with easy integration and low loss has become a research topic. L. Sun et al. reported that the Verdet constant of Tb-doped silicate glass fiber with 56 wt.% doping solubility was -24.5 ± 1.0 rad T^-1m^-1 at 1053 nm [9]. After that, Yi Huang et al. developed the Eu-doped silica fiber whose Verdet constant is twice that of conventional single-mode fiber (SMF) at 660 nm, and its Verdet constant is -4.563 rad T^-1m^-1 [10]. The above series of studies have shown that doping rare element ions with magnetic moment into materials can improve the Verdet constant. Also, the researchers doped the Ho ions with 10.6µ_B emu (µ_B is Bohr magneton) magnetic moment [11], which has the largest magnetic moment of the lanthanide elements, into different materials to get the larger Verdet constant. In 2016, Zhe Chen produced THGG ((Tb_(1-x)Ho_x)₃Ga₅O₁₂) crystals, and the Verdet constant values measured at 632 nm and 1064 nm were 214.9 rad T^-1m^-1 and 77.8 rad T^-1m^-1, respectively. These values were twice that of TGG [12]. In 2017, Hiroaki Furuse and Ryo Yasuhara produced transparent Ho₂O₃ ceramics. The Verdet constant measured at 1064 nm was 46.3 rad T^-1m^-1, which was 1.3 times that of TGG [13]. In 2018, Sadia Sharif successfully prepared Bi_(1-x)Ho_xFeO₃ thin films with better magnetic properties, and their magnetism increased with the increase of Ho concentration [14]. In 2019, Abo-Naf et al. prepared silicate glass co-doped with Mn and Ho, and found that the co-doped glass had greater magnetism and showed linear paramagnetism [15]. In addition, numerous studies have shown that doping Ho into materials can improve the magneto-optical properties of different materials, such as films and ceramics [16–19].

At present, there are many magnetic materials, such as the yttrium iron garnet (YIG) crystal and its rare-earth substituted compositions [20–23], which possess a very low saturation magnetization and exhibit a few orders higher Faraday rotation than the presented Ho-doped silica fiber. However, easier fabrication process, lower loss, and the characteristic that the Faraday effect accumulates with the increase of fiber length of the Ho-doped silica fiber make it have great application prospects in the preparation of fiber based optical current transducer (OCT), high-resolution heat-resistant magnetic field sensors and probes for biomedicine [24]. Therefore, we propose doping Ho ions into the silica fiber in this paper, and study the magneto-optical properties of the Ho-doped silica fiber that is fabricated with the modified chemical vapor deposition (MCVD) method.

2. Theoretical analysis

Since light has wave-particle duality, when considering the interaction of a beam of linearly polarized light with a magneto-optical material, it should be analyzed from both the wave characteristics and particle characteristics of the light. Based on this, the contribution of Faraday magneto-optical effect on the Verdet constant can be divided into wave contribution and transition contribution [25,26].

The wave contribution of the Faraday magneto-optical effect to the Verdet constant means that when the left-handed circularly polarized lightwave and the right-handed circularly-polarized lightwave, which are decomposed by linearly polarized light, interact with the substance, the two beams interact with the ions of the magnetic moment present in the material, and they are coupled into two electromagnetic fields oscillating with opposite rotation directions [27]. When the two electromagnetic oscillations are subjected to the direct (or indirect) exchange effective field, the spin-orbit interaction effective field and the applied magnetic field, the Larmor procession is generated [28]. Therefore, the two electromagnetic-coupling oscillations propagate in different velocity in the material, which eventually leads to the rotation of the polarization plane.

At this point, the wave contribution of paramagnetic Verdet constant is given by Eq. (1) [25].

(1)$$V_w^{Par} = \frac{{e{\mu _0}\lambda }}{{2mc}}\frac{{dn}}{{d\lambda }} \cdot (1 + \nu \frac{{{T_c}}}{{T - {T_P}}}).$$

where e is the amount of electron charge, µ₀ is the magnetic permeability in vacuum, λ is the wavelength of the incident light, m is the electron mass, c is the velocity of light, dn/dλ is the dispersion of the magneto-optical material, ν is the coefficient related to the molecular field constant, T is the temperature at which the experiment was conducted, T_c is the Curie constant, and T_P is the Curie temperature.

The transition contribution of the Faraday magneto-optical effect to the Verdet constant means that the electronic ground-state energy levels of the atoms or ions in the material will split under the interaction of the exchanged effective field and the applied magnetic field. At the same time, due to the complex electronic probability distribution on the ground-state energy levels, the right-handed circularly polarized light and the left-handed circularly polarized light excite different electronic transitions, resulting in a Faraday magneto-optical effect.

Based on this theory, van Vleck-Hebb obtained the expression of paramagnetic Verdet constant [29]:

(2)$$V_t^{Par} = \frac{{4{\pi ^2}{\upsilon ^2}\hbar }}{{g{\mu _B}c}} \cdot \sum\limits_{ij} {[\frac{{{\varphi _{ij}}}}{{{\upsilon ^2} - \upsilon _{ij}^2}}]} .$$

In Eq. (2), g is Lande factor, ħ is Planck constant, υ is the frequency of incident light, and φ_ij represents the probability of υ_ij frequency transition.

Due to different transition probabilities and microscopic uncertainties, it is difficult to get the exact picture of the electronic transition. Therefore, the dominant wavelength transition model is introduced and only the transition (φ_t, υ_t) that plays a dominant role on Verdet constant is considered [30], where υ_t is the frequency at which the dominant wavelength excites the electron transition, and φ_t is the probability of υ_t frequency transition. Therefore, Eq. (2) can be simplified as follows:

(3)$$V_\textrm{t}^{Par} ={-} \frac{{\lambda _t^2}}{{{\lambda ^2} - \lambda _t^2}}\frac{E}{{T - {T_P}}}.$$

In Eq. (3), λ_t is the dominant transition wavelength, E=4π²φ_tT_c/(gµ_Bcħ) is a coefficient independent of the wavelength of the incident light, which is a negative value.

In summary, according to the theory of wave-transition contribution, the Verdet constant of paramagnetic materials can be expressed as follows [25]:

(4)$$V = V_w^{Par} + V_t^{Par} = \frac{{e{\mu _0}\lambda }}{{2mc}}\frac{{dn}}{{d\lambda }} \cdot (1 + \nu \frac{{{T_c}}}{{T - {T_P}}}) - \frac{{\lambda _t^2}}{{{\lambda ^2} - \lambda _t^2}}\frac{E}{{T - {T_P}}}.$$

Equation (4) indicates that the Verdet constant of paramagnetic materials is derived from the sum of the wave contribution of the light and the contribution of the electron transition. The form is the addition of the expressions of classical theory and quantum theory, but it is essentially the embodiment of the wave-particle duality of light.

3. Preparation and characterization of Ho-doped silica fiber

The process of preparing Ho-doped silica fiber by MCVD method is as follows: First, a high-purity silica-based tube is placed on a clamping device that can be synchronously rotated at both ends of a lathe for melting optical fiber preforms, and an oxyhydrogen flame blowtorch is used to perform high-temperature heating under the base tube. Then, SiCl₄ and GeCl₄ are transported into the silica-based tube through high-purity oxygen. At the same time, the Ho chloride material is also brought into the silica-based tube by high-purity oxygen after high temperature gasification. After high temperature oxidation reaction, it is deposited on the inner wall of the silica-based tube. A soot of silica doped with Ge dioxide and Ho ions is contracted into the Ho-doped silica optical fiber preform after high-temperature vitrification. Finally, the fiber preform was drawn into the Ho-doped silica fiber by a homemade specialty fiber drawing tower. In addition, the doping concentration of the fiber samples prepared by the MCVD process and the high temperature evaporation process will not be very uniform doping, which is determined by the MCVD process, so there will be a difference in the Verdet constant of the two fiber samples of the same length. However, the concentration difference is larger in the fiber drawn from both ends of the optical fiber preform, and the ion concentration distribution of the fiber drawn from the middle of the optical fiber preform is more uniform. In addition, the concentration difference between two consecutive optical fibers is very small. In order to reduce the error caused by the uneven concentration distribution, we choose the continuous optical fiber drawn from the middle of the preform. This allows the effect of the difference in fiber concentration to be negligible.

The cross section and refractive index profile of the Ho-doped silica fiber are shown in Fig. 1. Where, the core diameter and cladding diameter are 8.22 µm and 125.15 µm, respectively. The results obtained by the fiber refractive index analyzer show that the refractive index difference between the core and the cladding of the Ho-doped silica fiber is 0.014. Through calculation and simulation, the fiber is a few-mode fiber with a wavelength range of 1310 nm-1550 nm. The fiber loss measured by the cut-back method is 0.18 dB/m at 1550 nm. In the sample Ho-doped silica fiber, the concentration of Ho ions is less than 0.13 at %.

Fig. 1. Cross section and refractive index profile of Ho-doped silica fiber.

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4. Verdet constant measurement of Ho-doped silica fiber

The measuring setup for the Faraday rotation angle of the Ho-doped silica fiber in the magnetic field is shown in Fig. 2. The light generated by the laser is transmitted through a single-mode fiber, and becomes collimated light after passing through the collimator. Then, after it passes through two polarizing plates (P1 and P2) and a quarter-wave plate (Q1), the light is coupled into the Ho-doped fiber by an objective. In this part, the extinction ratios at P1, P2 and Q1 are 34.5 dB, 0.5 dB and 35.7 dB, respectively. The optical fiber to be tested is placed in a solenoid with a direct-current (DC) source supply, which provides an external magnetic field for the fiber. As the dc power supply increases from 0 A to 6 A, the magnetic induction intensity inside the solenoid will increase from 0 mT to 135 mT. The length of the optical fiber to be measured is 70 cm, and the effective length of the magnetic field generated by the device is 30.4 cm. Finally, the information of the light passing through the Ho-doped fiber is collected by the Stokes detector and sent to a computer for data processing. By analyzing the changes of the polarization angle of the linearly polarized light before and after applying a magnetic field, the Verdet constant of the Ho-doped fiber can be obtained with theoretical calculations.

Fig. 2. Schematic diagram of Verdet constant measuring setup.

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What’s more, the whole fiber should be kept straight to avoid high-order mode excitation due to physical bending of the fiber or any applied stress during the experiment [31]. In order to avoid such a situation, we placed the Ho-doped silica fibers on the smooth thin plastic plate, and fixed the two ends of the Ho-doped silica fibers with optical fixtures.

5. Experimental results and analysis

When the incident light wavelength is 1310 nm and 1550 nm, the Faraday rotation angles of the Ho-doped silica fiber and the SMF at different magnetic induction intensity are θ₁ and θ₂, respectively. Their measured values are shown in Table 1.

Table 1. Faraday rotation angle of Ho-doped silica fiber and SMF at different magnetic induction intensity.

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According to the rotation angle of the two optical fibers collected in the above table at two wavelengths, the Verdet Constant can be calculated by V=θ/BL [32], where B is the external magnetic induction intensity, L is the effective length of the fiber, and V is the Verdet constant to be solved. The results are shown in the Table 2, and the confidence level of the results in the table is 95%. It can be obtained through calculation that the Verdet constants of Ho-doped silica fiber at wavelengths of 1310 nm and 1550 nm are 3.915 rad T^-1m^-1 and 1.287 rad T^-1m^-1, respectively, which are 4.6 times and 1.6 times the Verdet constants of single-mode fiber under the same conditions.

Table 2. The Verdet constant for Ho-doped silica fiber and SMF at different wavelengths.

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The paramagnetic Curie temperature of Ho ions is -7.5 K [33]. When the temperature is greater than the Curie temperature, the thermal motion energy is greater than the exchange energy, which leads to the chaotic arrangement of the ion magnetic moment and eventually becomes paramagnetic. This means that the Ho-doped silica fiber can be considered a paramagnetic substance in the case where the experimental temperature is 293 K, and the Curie-Weiss law is satisfied.

Since the outer layer of the Ho ion in the Ho-doped silica fiber is 4f¹⁰, under the action of the magnetic field, the electron is highly prone to 4fⁿ → 4f^n-15d¹ migration. The electronic structure is susceptible to the influence of temperature, resulting in electron transition, so that the Verdet constant of the material has a temperature-dependent characteristic. Based on the dielectric dispersion equation, the wave contribution of the Verdet constant of Ho-doped fiber was calculated. Because the concentration of Ho³⁺ ions is very low, the influence of Ho³⁺ ions on fiber dispersion can be ignored. Therefore, the refractive index of Ho-doped silica fiber is considered still meeting the modified Sellmeier equation [34,35]:

(5)$${n^2}(\lambda ) = A + \frac{B}{{{\lambda ^2} - C}} - D{\lambda ^2}.$$

In Eq. (5), A=2.3694, B=0.009 µm², C=0.0181 µm², D=0.0383 µm^-2.

According to Eq. (1), Eq. (5) and the coefficients A, B, C and D, the wave contribution V_w of the Verdet constant of the Ho-doped silica fiber is obtained:

(6)$${V_w} = \frac{{{q_e}}}{{2mc}} \times \frac{{ - B{\lambda ^2} - D{\lambda ^2} \times {{({\lambda ^2} - C)}^2}}}{{{{({\lambda ^2} - C)}^2} \times \sqrt {A + \frac{B}{{{\lambda ^2} - C}} - D{\lambda ^2}} }} \times \frac{T}{{T - {T_P}}}.$$

According to Eq. (4) and Eq. (6), the transition contribution V_t of the Verdet constant of Ho-doped silica fiber can be calculated, and the calculation results are shown in the Table 3.

Table 3. Contribution analysis of the Verdet constant of Ho-doped silica fiber.

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In the Table 3, V is measured by the experiment. According to Eq. (3) and the data in the fourth column in the Table 3, the coefficients of transition contribution of Ho-doped silica fiber can be obtained: E = -264.627 rad·K·T^-1·m^-1, λ_t = 1182.402 nm, and the value of λ_t is very close to the fluorescence wavelength of Ho in silica [36].

Substituting Eq. (6), E and λ_t into Eq. (4), the Verdet constant of Ho-doped silica fiber is obtained:

(7)$$V = \frac{{{q_e}}}{{2mc}} \times \frac{{ - B{\lambda ^2} - D{\lambda ^2} \times {{({\lambda ^2} - C)}^2}}}{{{{({\lambda ^2} - C)}^2} \times \sqrt {A + \frac{B}{{{\lambda ^2} - C}} - D{\lambda ^2}} }} \times \frac{T}{{T - {T_C}}} - \frac{{E \times {\lambda _t}^2}}{{({\lambda ^2} - {\lambda _t}^2)(T - {T_P})}}.$$

The theoretical curve of the contribution of each part of the Verdet constant of the Ho-doped silica fiber obtained by the calculation of the Table 3 is shown in Fig. 3, and the parameter values in Eq. (6) and Eq. (7) are shown in the Table 4.

Fig. 3. Wavelength dependence of Verdet constant of Ho-doped silica fiber (T=293 K).

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Table 4. The symbol of each parameter and its corresponding value.

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In Fig. 3, V_t represents the transitional contribution, V_w represents the wave contribution, and V is the theoretical curve of the wave transitional contribution model. As shown in Fig. 3, the transition contribution in the Verdet constant of Ho-doped silica fiber is greater than the wave contribution, making the overall performance positive. It can be seen from the figure that the Verdet constant of the fiber is inversely proportional to the wavelength, and its value decreases as the wavelength increases. Furthermore, the trend of the paramagnetic contribution is decreasing with the increasing wavelength because there is no transition. From the Fig. 3, the Verdet constant value of Ho-doped silica fiber can be predicted when the incident light is of different wavelength, which is help to the application of Ho-doped silica fiber.

6. Conclusion

In this paper, starting from the nature of the interaction between light and fiber material in the magnetic field, Ho with the maximum magnetic moment in rare earth elements was doped into silica fiber by the MCVD method to enhance the Faraday rotation effect of the optical fiber in the magnetic field. The obtained Ho-doped silica fiber has high Verdet constant that is 4.5 times and 1.6 times that of the single mode fiber at 1310 nm and 1550 nm, respectively.

In addition, the enhancement of Faraday effect of Ho-doped silica fiber is analyzed by the theory of wave-transition contribution, and the Verdet constant’s dependency on the wavelength is obtained. Ho-doped silica fiber can be used for optical fiber current transducers, high-resolution heat-resistant magnetic field sensors and probes that detect magnetic field in living organisms, etc.

Funding

State Key Laboratory of Advanced Optical Communication Systems and Networks. (SKLSFO2017-02, SKLSFO2018-05); National Natural Science Foundation of China (61475095, 61575120, 61635006, 61875118).

Disclosures

The authors declare no conflicts of interest.

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	1310 nm		1550 nm
B (mT)	Ho-doped fiber θ₁ (˚)	SMF θ₂ (˚)	Ho-doped fiber θ₁ (˚)	SMF θ₂ (˚)
22.5	1.728	0.365	0.494	0.322
45.0	3.209	0.690	1.035	0.621
67.5	4.928	1.024	1.514	0.921
90.0	6.294	1.362	2.027	1.229
112.5	6.762	1.689	2.546	1.516
135.0	7.833	1.988	2.963	1.793

	1310 nm		1550 nm
	V_Ho rad·T^-1·m^-1	V_SMF rad·T^-1·m^-1	V_Ho rad·T^-1·m^-1	V_SMF rad·T^-1·m^-1
V	3.915 ± 0.202	0.876 ± 0.014	1.287 ± 0.027	0.786 ± 0.01

λ (nm)	V (rad·T^-1·m^-1)	V_w (rad·T^-1·m^-1)	V_t (rad·T^-1·m^-1)
1310	3.915	0.046	3.869
1550	1.287	0.062	1.225

Item	A	B (µm²)	C (µm²)	D (µm^-2)	c (m·s^-1)
Value	2.3694	0.009	0.0181	0.0383	2.998×10⁸
Item	λ_t (nm)	T_C (K)	q_e (C)	m (kg)	E (rad·K·T^-1·m^-1)
Value	1182.402	-7.5	-1.6×10⁻¹⁹	9.1×10⁻³¹	264.627

	1310 nm		1550 nm
B (mT)	Ho-doped fiber θ₁ (˚)	SMF θ₂ (˚)	Ho-doped fiber θ₁ (˚)	SMF θ₂ (˚)
22.5	1.728	0.365	0.494	0.322
45.0	3.209	0.690	1.035	0.621
67.5	4.928	1.024	1.514	0.921
90.0	6.294	1.362	2.027	1.229
112.5	6.762	1.689	2.546	1.516
135.0	7.833	1.988	2.963	1.793

Study of the Verdet constant of the holmium-doped silica fiber

Abstract

1. Introduction

2. Theoretical analysis

3. Preparation and characterization of Ho-doped silica fiber

4. Verdet constant measurement of Ho-doped silica fiber

5. Experimental results and analysis

6. Conclusion

Funding

Disclosures

References

Cited By

Figures (3)

Tables (4)

Equations (7)

OSA Continuum