Expand this Topic clickable element to expand a topic
Skip to content
Optica Publishing Group
Journal Home About Issues in Progress Current Issue All Issues Feature Issues

Improved measurement method for the Faraday rotation distribution using beam splitting

Zhiyuan Jiang, Jian Wu, Ziwei Chen, Zhenyu Wang, Huantong Shi, and Xingwen Li

Open Access Open Access

Abstract

In this work, two Faraday channels and one shadow channel are constructed by two non-polarizing beam splitters and one reflector to measure the Faraday rotation distribution. The intensity of the Faraday and shadow images is related to the state of polarization (SOP) of the incident light, thus achieving two-dimensional accurate measurement. The measurement sensitivity is influenced by the settings of two polarization analyzers in the Faraday channels and the parameters of beam splitters, which are explored numerically and verified with experiments. The fluctuation of the probe light is eliminated by using three evaluation indexes. Also, the measurement range and error sources under different experimental settings are discussed.

© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement

1. Introduction

The Faraday effect is a phenomenon that the polarization plane of linearly polarized light rotates when the light passes through a magneto-optical medium along the direction of the magnetic field [1]. Based on this effect, magneto-optical properties of matter can be studied by measuring the rotation angle accurately, and the magnetic field around the medium can also be determined. It is widely used in industry, scientific research, aerospace, and other fields. Optical fiber sensor based on Faraday rotation of polarization is an important approach to measure distributed magnetic field [2–6]. In this case, polarization-sensitive reflectometry is used to measure the state of polarization (SOP) of Rayleigh backscattered light [7], from which the magnetic field component parallel to the fiber axis can be acquired with proper data analysis. The Verdet constant, describing the polarization rotation of light in the magneto-optical media caused by a magnetic field, can be studied with the Faraday effect [8,9]. Also, it is a reliable method to measure the vapor or electron density using the Faraday effect [10,11].

Several measurement methods are available for rotation angles, such as the extinction method [12,13], the Faraday modulation method [14], and the beam splitting method [15–17]. The extinction method is the simplest way to measure the Faraday rotation: the probe laser passes through the Faraday medium which is placed between two linear polarizers, and the difference is measured between the angular settings of the analyzer yielding minimum transmitted intensity [12]. As the change rate of light intensity near the extinction position is small, it is difficult to determine the extinction position precisely. Also, this method is commonly used for visual measurements with polarizing microscopes and is not suitable for the measurement of small-angle rotation. The Faraday modulation method can measure the rotation angle by the phase-locked amplification technique. By converting the optical signal into an electrical signal and amplifying it through an amplifier, the measurement of small-angle rotation can be realized [14]. But this method uses square wave excitation which causes the inductive effect in the solenoid and makes it difficult to achieve the ideal magneto-optical modulation [18]. This method is difficult to use for large amounts of data samples due to the complex calculation. Another important measurement method is the beam-splitting method. When a polarizing beam splitter is used, the light intensity of p and s components are divided and measured separately to obtain the SOP [15]. This method is confined by the original state of the probe laser, which will cause large errors when measuring small angles. On the contrary, when the non-polarizing beam splitters are used, a polarization analyzer at a specific angle to the incident light helps determine the distribution of the rotation angle and promotes the measurement sensitivity [16]. Upon this foundation, two Faraday channels with symmetrical polarization analyzers were set, which further improved the measurement sensitivity [17]. After this research, there are similar works in practical applications of this measurement method [19–21]. However, previous researchers didn’t consider the reflection and transmission coefficients of the beam splitter but only treated them as ideal elements. Therefore, the influence of the parameters of beam splitters needs urgent considerations.

This paper proposes an improved measurement method for Faraday rotation angle with high sensitivity. Two Faraday channels and one shadow channel were used. A proportionality factor of the intensity of measured pictures is defined as the evaluation index, which can eliminate the influence of the energy fluctuation of the laser probe in the experiment. The measurement sensitivity can be improved by modifying the settings of the analyzer and the parameters of the beam splitters. The change of sensitivity with specific parameters is calculated and demonstrated in the experiment by substituting the influence of Faraday rotation with a polarizer. The analysis of the measurement range and error sources is performed.

2. Theoretical background

2.1 Faraday rotation

The Faraday effect is a magneto-optical phenomenon, describing the rotation of the polarization plane as linearly polarized light traveling through a Faraday medium in a magnetic field. When plasma is applied to a Faraday medium, the Faraday rotation can be calculated by [17]

$$\alpha = \frac{{{e^3}{\lambda ^2}}}{{8{\pi ^2}{\varepsilon _0}m_e^2{c^3}}}\int\limits_0^L {{n_e}{\boldsymbol B}d{\boldsymbol l}}$$

Where λ is the wavelength of the probing laser beam, L is the distance of light propagation in plasma, ne is the electron density, B is a projection of the magnetic field onto the direction of the probing laser beam. From Eq. (1), the magnetic field and the parameters of the Faraday medium can be determined by measuring the distribution of the rotation angle.

2.2 Measurement of the Faraday angle

The probe laser is divided into three channels, including two Faraday channels (CCD1, CCD2) and a shadowgraph (CCD3), as shown in Fig. 1. Two Faraday channels were analyzed at angles ±β on either side of the polarization plane of incident light θ0. The rotation angle can be obtained by analyzing the normalized spatial intensity distribution of three channels during an experiment [17].

 figure: Fig. 1.

Fig. 1. A 3D Schematic of the Faraday rotation measurement. BS: beam splitter, RM: reflector mirror, PA: polarization analyzer, CCD: charge-coupled device. P and S represent the p- and s-polarization. The blue rectangle represents the optical plane. θ0 is the polarization angle of incident light. θ1 and θ2 are the polarization angle of the light reflected by BS1 and BS2, respectively. β and -β are the bias angle of PA1 and PA2 concerning θ0, respectively.

Download Full Size | PDF

However, while the transmission and reflectance efficiency of the p and the s components are different, this diagnosis method would be more complex. In this research, the polarization direction of all polarized light is defined as the angle between the polarized light and p-polarization. The shadow image is reflected by a protected silver mirror, which has the same transmission and reflectance efficiency of the p and s components.

When the polarization analyzers are not considered, the intensity of incident light captured by three CCDs depends only on the laser intensity and θ0. Therefore, the ratio of captured light intensity to incident light intensity, named Di for the i-th CCD, is a function of θ0. When the parameters of the beam splitters are determined, especially the reflectance and transmission efficiency of the p and s components, Di for the i-th CCD is calculated by the interpolation algorithm before considering the effect of the polarization analyzers. Then, Considering the polarization analyzers, the intensity distributions captured by the i-th CCD is Ii(x,y) when the Faraday rotation does not exist:

$$\begin{aligned} {{I_1}(x,y)} &= {D_1}({\theta _0})I(x,y){{\cos }^2}({\theta _1} - {\theta _0} - \beta ))\\ {{I_2}(x,y)} &= {D_2}({\theta _0})I(x,y){{\cos }^2}({\theta _2} - {\theta _0}\textrm{ + }\beta ))\\ &\quad{{I_3}(x,y) = {D_3}({\theta _0})I(x,y)} \end{aligned}$$
where I(x,y) is the intensity distribution of initial light, θ1 and θ2 are the polarization angle of the reflected light from BS1 and BS2 respectively. In this equation, θ1, θ2, and Di are the functions of the polarization angle of the incident light θ0. When the Faraday rotation exists and is distributed in α(x,y), the polarization angle of the incident light is θ0+α(x,y). The intensity distribution captured by the i-th CCD is Ii’(x,y):
$$\begin{aligned} {{I_1}^{\prime}(x,y)} &= {D_1}({\theta _0} + \alpha (x,y)){I^{\prime}}(x,y){{\cos }^2}({\theta _1}^{\prime} - {\theta _0} - \beta ))\\ {{I_2}^{\prime}(x,y)} &= {D_2}({\theta _0} + \alpha (x,y)){I^{\prime}}(x,y){{\cos }^2}({\theta _2}^{\prime} - {\theta _0} + \beta ))\\ &\quad{{I_3}^{\prime}(x,y) = {D_3}({\theta _0} + \alpha (x,y)){I^{\prime}}(x,y)} \end{aligned}$$
where I’(x,y) is the intensity distribution of incident light considering the fluctuations of laser energy. At this time, θ1’, θ2’, and Di depend on θ0+α(x,y). Therefore, six images were captured by three CCDs for each experiment. Defining three evaluation indexes to be:
$$\begin{aligned} &{{I_F}(x,y) = (\frac{{{I_1}^{\prime}(x,y)}}{{{I_1}(x,y)}} - \frac{{{I_2}^{\prime}(x,y)}}{{{I_2}(x,y)}})/\frac{{{I_3}^{\prime}(x,y)}}{{{I_3}(x,y)}}}\\ &\quad{{I_{F1}}(x,y) = \frac{{{I_1}^{\prime}(x,y)}}{{{I_1}(x,y)}}/\frac{{{I_3}^{\prime}(x,y)}}{{{I_3}(x,y)}}}\\ &\quad{{I_{F2}}(x,y) = \frac{{{I_2}^{\prime}(x,y)}}{{{I_2}(x,y)}}/\frac{{{I_3}^{\prime}(x,y)}}{{{I_3}(x,y)}}} \end{aligned}$$

When θ0 is fixed, the intensity distribution of captured images is relevant to the Faraday rotation angle α and the polarizer bias angle β. It can be seen that all of Ii’(x,y)/ Ii(x,y) include this item: I’(x,y)/ I(x,y), thus eliminating fluctuations of the laser energy in the calculation. The changing trends of three indexes with different Faraday rotation angle α(x,y) is shown in Fig. 2. The reflectance and transmission efficiency of the p and the s components are assumed to be 0.5 and the initial incident angle θ0 is 0. There is a nonlinear mapping between the Faraday rotation angle α and the evaluation indexes when θ0 and β are determined. Therefore, for each position of the measured images, there is a one-to-one correspondence between Faraday rotation α(x,y) and calculated evaluation indexes. When α is in the range of -45° to 45°, IF is monotonic whereas IF1 and IF2 are not, which is the primary benefit of using two faraday channels simultaneously. Also, IF is more sensitive than IF1 and IF2. Therefore, the two-dimensional distribution of Faraday rotation α(x,y) can be obtained by calculating the corresponding IF. When β increases, dIF/dα can be affected positively, which means the improvement of measurement sensitivity.

 figure: Fig. 2.

Fig. 2. Calculated evaluation indexes with different rotation angle α, assuming the splitting ratio of two beam splitters is 50:50. Dashed dot red lines represent β=10°, dashed green lines represent β=30° and solid blue lines represent β=50°. The solid gray lines represent α=±45°.

Download Full Size | PDF

2.3 Influence of beam splitter parameters

When the reflectance and transmission efficiency of the p and s components are not equal, the measurement sensitivity will be likewise influenced. Assuming θ0 = 0, the polarization angle of two Faraday channels depends on the Faraday rotation, which is calculated by consulting [1]:

$$\begin{aligned} {\theta _1} &= {\cos ^{ - 1}}(\sqrt {\frac{1}{{1 + {{\tan }^2}(\alpha )/({P_r}/{S_r})}}} )\\ {\theta _2} &= {\cos ^{ - 1}}(\sqrt {\frac{1}{{1 + {{\tan }^2}(\alpha )/({P_t}{P_r}/{S_t}{S_r})}}} ) \end{aligned}$$
where θ1 and θ2 are the polarization angle of the first and second Faraday channel separately, Pr and Sr are reflectance efficiency of the p and the s components while Pt and St are the transmission efficiency of the p and the s components. Particularly, θ1 and θ2 are decided by Pr/Sr and PrPt/SrSt respectively, and have similar changing trends. The changing trends of θ1 and dθ1/dα with Pr/Sr are shown as an example in Fig. 3. When Pr/Sr is not 1, θ1 and θ2 is not equal to α and change nonlinearly with α. If dθ1/dα>1, it means that the non-polarizing beam splitter has an extra rotation effect on the polarized light during the reflection process. dθ1/dα<1 means a depolarization effect conversely. Also, smaller Pr/Sr and PrPt/SrSt can achieve a stronger extra rotation effect on the polarized light. The extra rotation would improve measurement sensitivity, which would facilitate choosing the polarizer model, as shown in Fig. 4. This figure represents calculated evaluation indexes with different rotation angles α with different practical commercial beam splitters and ideal elements simultaneously, detail parameters are shown in [22]. dIF /dα with the splitting ratio of 8:92 is 2.5 times higher than that with the splitting ratio of 45:55. Larger measurement sensitivity will decrease the minimal measurable threshold of the small-angle Faraday rotation. What’s more, if we treat available non-polarizing beam splitters as perfect ones, there will be significant effects to the evaluation index IF. The error is related to the parameters of the beam splitter.

 figure: Fig. 3.

Fig. 3. The changing trends of θ1 and dθ1/dα with different rotation angles, assuming the polarization angle of incident light is 0. Dashed red lines represent Pr/Sr =0.5, solid black lines represent Pr/Sr =1 and dashed dot blue lines represent Pr/Sr =1.5.

Download Full Size | PDF

 figure: Fig. 4.

Fig. 4. The changing trends of three indexes with different rotation angle α, assuming β=20°. Dashed dot green lines represent ideal beam splitters, dashed red lines represent the splitting ratio of 45:55, and solid blue lines represent the splitting ratio of 8:92.

Download Full Size | PDF

3. Experimental setup and results

In this experiment, the Nd: YAG (Beamtech: SGR750) laser generates 8 ns laser pulses at a 1 Hz repetition rate and 1064 nm wavelength, which is used for a Faraday probe. The laser beam is expanded through a beam expander. The initial SOP of the pulsed laser is p-polarized. Then, a polarizer with a high extinction ratio (>107:1) and high acceptance angle (±20°) is used to change the polarization direction of the incident laser and simulate the Faraday rotation (Thorlabs: LPVIS100-MP2). The polarized light is divided into three channels by two splitters and one mirror, as shown in Fig. 5. The reflected light of two splitters is analyzed at angles ±β on either side of the polarization plane of incident light. The reflected light of the mirror is a shadow image used as a control. The images of all three channels are captured by CCDs with high linearity of 0.99 to ensure linearity between the picture luminance and the light intensity of each pixel(Atik 383L+). Also, each CCD is coupled with a bandpass filter (λ = 1064 nm, δλ=10 nm) for blocking stray light. To verify the theoretically calculated results, two different models of non-polarizing beam splitters, Thorlabs-BP208 (8:92) and Thorlabs-BP245B3(45:55) were used in the experiment. Here the splitting ratio is the reflectance efficiency and transmission efficiency of unpolarized light. Relevant detailed parameters of the beam splitters at the wavelength of 1064 nm can be obtained from the official website of Thorlabs [22], which is shown in Table 1.

 figure: Fig. 5.

Fig. 5. Diagram of the experimental arrangement. BE: beam expander, RM: reflect mirror, BS: beam splitter, PA: polarization analyzer, BF: bandpass filter, CCD: Charge-coupled device.

Download Full Size | PDF

Tables Icon

Table 1. detailed parameters of used beam splitters

View Table

The laser images were measured for different faraday rotation α varying from -20° to 20°, and the evaluation indexes were measured by setting the value of β to 10° and 22° respectively. For each faraday rotation α, three repeated experiments were undertaken to eliminate the accidental error. The measured and calculated evaluation indexes using Thorlabs-BP208 are shown in Fig. 6. The experimental results agree well with the theoretical results. The fluctuation of the evaluation index generated by repeated experiments is very small, and the maximum error is less than 5%.

 figure: Fig. 6.

Fig. 6. (a)Results of calculated and measured evaluation index with the splitting ratio of 8:92 when β is 10°. (b)Results of calculated and measured evaluation index with the splitting ratio of 8:92 when β is 22°. The solid lines (blue) represent the calculated IF, dashed lines (blue) the calculated IF1, and dash-dot lines the calculated IF2, respectively. Square dots on graphs represent the measured IF, circular dots on graphs the measured IF1, and triangle dots on graphs the measured IF2.

Download Full Size | PDF

When the experiment is performed using another kind of beam splitter, BP245B3, the experimental results obtained are shown in Fig. 7 by setting the value of β to 18° and 35°. From Fig. 6(a) and 7(a), dIF /dα with the splitting ratio of 8:92 when β is 10° is bigger than that with the splitting ratio of 45:55 when β is 18°. When β is 10°, dIF /dα with the splitting ratio of 8:92 is much bigger than the one with the splitting ratio of 45:55 accordingly. Therefore, the beam splitter with a large transmission or reflection ratio of the p component to s component help lift the measurement sensitivity. At the same time, when the ratio of the transmission coefficient to the reflection coefficient of the beam splitter is small, to achieve the same sensitivity as the one with a higher ratio, the angle of the analyzer needs to be increased.

 figure: Fig. 7.

Fig. 7. (a)Results of calculated and measured evaluation index with the splitting ratio of 45:55 when β is 18°. (b)Results of calculated and measured evaluation index with the splitting ratio of 45:55 when β is 35°. The solid lines (blue) represent the calculated IF, dashed lines (blue) the calculated IF1, and dash-dot dot lines the calculated IF2, respectively. Square dots on graphs represent the measured IF, circular dots on graphs the measured IF1, and triangle dots on graphs the measured IF2.

Download Full Size | PDF

4. Discussion

The initial image intensity of each CCD is known when the light source and polarization of two analyzers are set up. When the angle of the faraday rotation changes, the image intensity captured by each CCD varies simultaneously. The light intensity of the captured image in one Faraday channel becomes smaller while that of another Faraday channel will be bigger. Also, high measurement sensitivity enlarges the change ratio of light intensity compared with initial light in the same α angle.

A particular value of α would result in brightening by a multiplicative factor named Fb on one CCD, and darkening by a multiplicative named Fd on another CCD. For the 16-bit CCD used in this experiment, the identifiable image brightness is in the range of 0-65535. In the experiment, the dark current noise of the camera causes the picture to have a basal brightness of 300, which can be regarded as the minimum resolvable light intensity. Therefore, there is a range of 300-65535 for image brightness captured by a CCD during the experiment, which limits the measurement range of α and depends on the choice of β. When the Faraday rotation exists, the image brightness of CCD1 changes from I1(x,y) to I1’(x,y), the image brightness of CCD2 changes from I2(x,y) to I2’(x,y). Ii(x,y) and Ii’(x,y) should be both in the range of 300-65535. If the direction of Faraday rotation is unknown, nor do we know the image brightness of which CCD will increase or decrease. A suitable method is to adjust the initial brightness of both CCDs in Faraday channels to the same value by using neutral density filters. Defining R to be the ratio of the brighter image intensity to the darker image intensity of CCD1 and CCD2:

$$\begin{aligned} &R < {I_{\max }}/{I_{\min }}\\ &{I_N} = {I_{\max }}/\sqrt R \end{aligned}$$
where Imax and Imin are the maximum and minimum identifiable light intensity, respectively. IN is the corresponding initial luminance. As stated above, Imax and Imin are 65535 and 300, respectively. It can be calculated that the R should be smaller than 218.5. The ratio of captured image intensity when α≠0 to that when α=0 by CCD1 and CCD2 is drawn in Fig. 8(a). The proportion of CCD1 and CCD2 are mirror-symmetric for two analyzers at opposite angle orientations.

 figure: Fig. 8.

Fig. 8. (a) The ratio of captured light intensity when α ≠ 0 to that when α = 0 by CCD1 and CCD2. Different colored lines represent the dynamic changes in light intensity. (b) The ratio of the brighter image intensity to the darker image intensity of CCD1 and CCD2. Different colored lines represent the dynamic changes in R. α varies from -45° to 45° and β varies from 0° to 90°. The beam splitters parameters of BP208 are used.

Download Full Size | PDF

When β is determined, there will be a situation where the polarization plane of the reflected light and the analyzer are approximately perpendicular on the darker Faraday channel, thus resulting in very weak image intensity. This phenomenon causes R to be greater than 218.5. These pixels of Faraday images with high R cannot be processed precisely in the way mentioned in Section 2.2. Considering the intensity change of the laser beam measured with CCD1 and CCD2, the measurement range with different β can be determined and shown as the red areas in Fig. 8(b). The blue areas mean unresolvable situations of α and β. When β is smaller than 75°, a higher measurement sensitivity will result in a different measurement range. The measurement range of α is sectioned by weak image intensity caused by approximately perpendicular angles.

In addition, for the settings of the analyzer and the polarizer, the adjustment precision of the rotation mount is up to 1°. When in specific experiments, the precision error causes the setting of the polarizer to be difficult to completely coincident with p-polarization when Faraday rotation does not exist, and there may be an error of 1°. The measurement error caused by precision error will be impacted in different values of β. Similarly, the bias angle β of both polarization analyzers may produce an adjustment error of 1°, which will also lead to a certain deviation of the calculated α value from the actual value. The specific errors of IF caused by initial polarization angle and β are shown in Fig. 9. From this image, larger β will cause higher measurement sensitivity and greater errors simoutanouly. When the Faraday rotation is close to 45°, the evaluation index has the largest measurement error. Also, for a given initial polarization angle and β, the uncertainty of the initial polarization angle would cause a greater error than that of β, which means it is crucial to regulate the polarization direction of initial light precisely.

 figure: Fig. 9.

Fig. 9. (a) The error bands of IF with different incident angles arise from a 1° adjustment error of β. The blue-tinted error band represents the total uncertainty of IF when β=20°, the red-tinted error band represents the total uncertainty of IF when β=40°. (b) The error bands of IF with different rotation angles arise from a 1° adjustment error of the initial polarization angle from 0. The blue-tinted error band represents the total uncertainty of IF when β=20°, the red-tinted error band represents the total uncertainty of IF when β=40°.

Download Full Size | PDF

5. Summary

In this paper, an improved method for Faraday rotation measurement with high sensitivity is presented. The Faraday rotation is measured using two Faraday channels and a shadow channel with CCDs capturing the light intensity. Three evaluation indexes about rotation light intensity are defined to obtain the angle of the rotation. The ratio of light intensity of Faraday rotation to initial light eliminates the fluctuations in the incident light energy. The sensitivity of the measurement can be improved by changing the splitting ratio of the beam splitters in the optical path. At the same time, the angle setting of the analyzer used for the two Faraday channels is mirror-symmetric, which can further improve and adjust the magnitude of the measurement sensitivity. The influence of the parameters of two beam splitters on the rotation measurement is investigated. When the incident light is under the p polarization, smaller Pr/Sr and PrPt/SrSt can achieve higher measurement sensitivity. The theoretical calculations were verified by simulating the Faraday effect on the probe laser using a high extinction polarizer. Considering the CCD has a range-limited intensity of the received light, there is a different range of rotation angle measurements for different analyzers settings. The sources and specific effects of measurement errors are also discussed.

Funding

National Natural Science Foundation of China (51790523, 51807155, 51922087).

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.

References

1. I. H. Hutchinson, Principles of Plasma Diagnostics, 2 ed. (Cambridge University Press, 2002).

2. L. Palmieri, D. Sarchi, and A. Galtarossa, “Distributed measurement of high electric current by means of polarimetric optical fiber sensor,” Opt. Express 23(9), 11073–11079 (2015). [CrossRef]  

3. A. J. Rogers, “Polarization-optical time domain reflectometry: a technique for the measurement of field distributions,” Appl. Opt. 20(6), 1060–1074 (1981). [CrossRef]  

4. J. Ross, “Measurement of magnetic field by polarisation optical time-domain reflectometry,” Electron. Lett. 17(17), 596–597 (1981). [CrossRef]  

5. L. Palmieri and A. Galtarossa, “Distributed polarization-sensitive reflectometry in nonreciprocal single-mode optical fibers,” J. Lightwave Technol. 29(21), 3178–3184 (2011). [CrossRef]  

6. M. Aerssens, A. Gusarov, P. Moreau, P. Malard, V. Massaut, P. Mégret, and M. Wuilpart, “Development of a Jones vector based model for the measurement of a plasma current in a thermonuclear fusion reactor with a POTDR setup,” in Optical Sensing and Detection II, (International Society for Optics and Photonics, 2012), 84390D.

7. L. Palmieri, “Distributed polarimetric measurements for optical fiber sensing,” Opt. Fiber Technol. 19(6), 720–728 (2013). [CrossRef]  

8. J. Cruz, M. Andres, and M. Hernandez, “Faraday effect in standard optical fibers: dispersion of the effective Verdet constant,” Appl. Opt. 35(6), 922–927 (1996). [CrossRef]  

9. P. Yao, X. Chen, P. Hao, H. Xiao, Z. Ding, T. Liu, and X. S. Yao, “Introduction and measurement of the effective Verdet constant of spun optical fibers,” Opt. Express 29(15), 23315–23330 (2021). [CrossRef]  

10. E. Vliegen, S. Kadlecek, L. Anderson, T. Walker, C. Erickson, and W. Happer, “Faraday rotation density measurements of optically thick alkali metal vapors,” Nucl. Instrum. Methods Phys. Res., Sect. A 460(2-3), 444–450 (2001). [CrossRef]  

11. Y. Kawano, S.-i. Chiba, and A. Inoue, “Infrared laser polarimetry for electron density measurement in tokamak plasmas,” Rev. Sci. Instrum. 72(1), 1068–1072 (2001). [CrossRef]  

12. J. Liu, F. Guo, B. Zhao, N. Zhuang, Y. Chen, Z. Gao, and J. Chen, “Growth and magneto-optical properties of NaTb(WO4)2,” J. Cryst. Growth 310(10), 2613–2616 (2008). [CrossRef]  

13. F. Y. Guo, Q. P. Wan, Y. Hou, L. Z. Zhang, H. Fu, and J. Z. Chen, “Growth and Faraday rotation characteristics of Tb2Sn2O7 crystal,” J. Cryst. Growth 397, 19–23 (2014). [CrossRef]  

14. H. Ganser, W. Urban, and J. M. Brown, “The sensitive detection of NO by Faraday modulation spectroscopy with a quantum cascade laser,” Mol. Phys. 101(4-5), 545–550 (2003). [CrossRef]  

15. J. Stamper and B. Ripin, “Faraday-rotation measurements of megagauss magnetic fields in laser-produced plasmas,” Phys. Rev. Lett. 34(3), 138–141 (1975). [CrossRef]  

16. M. Tatarakis, R. Aliaga-Rossel, A. E. Dangor, and M. G. Haines, “Optical probing of fiber z-pinch plasmas,” Phys. Plasmas 5(3), 682–691 (1998). [CrossRef]  

17. G. Swadling, S. Lebedev, G. Hall, S. Patankar, N. Stewart, R. Smith, A. Harvey-Thompson, G. Burdiak, P. de Grouchy, and J. Skidmore, “Diagnosing collisions of magnetized, high energy density plasma flows using a combination of collective Thomson scattering, Faraday rotation, and interferometry,” Rev. Sci. Instrum. 85(11), 11E502 (2014). [CrossRef]  

18. K. Jashnani, J. Shetty, and H. Desai, “Vibrant colors of uric acid crystals on polarization,” Indian J. Pathol. Bacteriol. 58(3), 404 (2015).

19. L. G. Suttle, J. D. Hare, S. V. Lebedev, G. F. Swadling, G. C. Burdiak, A. Ciardi, J. P. Chittenden, N. F. Loureiro, N. Niasse, F. Suzuki-Vidal, J. Wu, Q. Yang, T. Clayson, A. Frank, T. S. Robinson, R. A. Smith, and N. Stuart, “Structure of a magnetic flux annihilation layer formed by the collision of supersonic, magnetized plasma flows,” Phys. Rev. Lett. 116(22), 225001 (2016). [CrossRef]  

20. J. D. Hare, L. G. Suttle, S. V. Lebedev, N. F. Loureiro, A. Ciardi, J. P. Chittenden, T. Clayson, S. J. Eardley, C. Garcia, J. W. D. Halliday, T. Robinson, R. A. Smith, N. Stuart, F. Suzuki-Vidal, and E. R. Tubman, “An experimental platform for pulsed-power driven magnetic reconnection,” Phys. Plasmas 25(5), 055703 (2018). [CrossRef]  

21. G. V. Dowhan, A. P. Shah, N. M. Jordan, N. M. Jordan, S. N. Bland, S. V. Lebedev, R. A. Smith, L. Suttle, and S. A. Pikuz, “Updates on the X-Pinch Platform and Faraday Rotation Imaging Diagnostic on the Maize Pulsed Power Facility,” 2021 IEEE International Conference on Plasma Science (ICOPS), pp. 1-1 (2021).

22. Thorlabs, https://www.thorlabs.de/newgrouppage9.cfm?objectgroup_id=898s

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.

Cited By

Optica participates in Crossref's Cited-By Linking service. Citing articles from Optica Publishing Group journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (9)
Fig. 1.
Fig. 1. A 3D Schematic of the Faraday rotation measurement. BS: beam splitter, RM: reflector mirror, PA: polarization analyzer, CCD: charge-coupled device. P and S represent the p- and s-polarization. The blue rectangle represents the optical plane. θ0 is the polarization angle of incident light. θ1 and θ2 are the polarization angle of the light reflected by BS1 and BS2, respectively. β and -β are the bias angle of PA1 and PA2 concerning θ0, respectively.

View in Article | Download Full Size | PDF

Fig. 2.
Fig. 2. Calculated evaluation indexes with different rotation angle α, assuming the splitting ratio of two beam splitters is 50:50. Dashed dot red lines represent β=10°, dashed green lines represent β=30° and solid blue lines represent β=50°. The solid gray lines represent α=±45°.

View in Article | Download Full Size | PDF

Fig. 3.
Fig. 3. The changing trends of θ1 and dθ1/dα with different rotation angles, assuming the polarization angle of incident light is 0. Dashed red lines represent Pr/Sr =0.5, solid black lines represent Pr/Sr =1 and dashed dot blue lines represent Pr/Sr =1.5.

View in Article | Download Full Size | PDF

Fig. 4.
Fig. 4. The changing trends of three indexes with different rotation angle α, assuming β=20°. Dashed dot green lines represent ideal beam splitters, dashed red lines represent the splitting ratio of 45:55, and solid blue lines represent the splitting ratio of 8:92.

View in Article | Download Full Size | PDF

Fig. 5.
Fig. 5. Diagram of the experimental arrangement. BE: beam expander, RM: reflect mirror, BS: beam splitter, PA: polarization analyzer, BF: bandpass filter, CCD: Charge-coupled device.

View in Article | Download Full Size | PDF

Fig. 6.
Fig. 6. (a)Results of calculated and measured evaluation index with the splitting ratio of 8:92 when β is 10°. (b)Results of calculated and measured evaluation index with the splitting ratio of 8:92 when β is 22°. The solid lines (blue) represent the calculated IF, dashed lines (blue) the calculated IF1, and dash-dot lines the calculated IF2, respectively. Square dots on graphs represent the measured IF, circular dots on graphs the measured IF1, and triangle dots on graphs the measured IF2.

View in Article | Download Full Size | PDF

Fig. 7.
Fig. 7. (a)Results of calculated and measured evaluation index with the splitting ratio of 45:55 when β is 18°. (b)Results of calculated and measured evaluation index with the splitting ratio of 45:55 when β is 35°. The solid lines (blue) represent the calculated IF, dashed lines (blue) the calculated IF1, and dash-dot dot lines the calculated IF2, respectively. Square dots on graphs represent the measured IF, circular dots on graphs the measured IF1, and triangle dots on graphs the measured IF2.

View in Article | Download Full Size | PDF

Fig. 8.
Fig. 8. (a) The ratio of captured light intensity when α ≠ 0 to that when α = 0 by CCD1 and CCD2. Different colored lines represent the dynamic changes in light intensity. (b) The ratio of the brighter image intensity to the darker image intensity of CCD1 and CCD2. Different colored lines represent the dynamic changes in R. α varies from -45° to 45° and β varies from 0° to 90°. The beam splitters parameters of BP208 are used.

View in Article | Download Full Size | PDF

Fig. 9.
Fig. 9. (a) The error bands of IF with different incident angles arise from a 1° adjustment error of β. The blue-tinted error band represents the total uncertainty of IF when β=20°, the red-tinted error band represents the total uncertainty of IF when β=40°. (b) The error bands of IF with different rotation angles arise from a 1° adjustment error of the initial polarization angle from 0. The blue-tinted error band represents the total uncertainty of IF when β=20°, the red-tinted error band represents the total uncertainty of IF when β=40°.

View in Article | Download Full Size | PDF

Tables (1)

Tables Icon

Table 1. detailed parameters of used beam splitters

View Table

Equations (6)

Equations on this page are rendered with MathJax. Learn more.

α = e 3 λ 2 8 π 2 ε 0 m e 2 c 3 ∫ 0 L n e B d l
I 1 ( x , y ) = D 1 ( θ 0 ) I ( x , y ) cos 2 ( θ 1 − θ 0 − β ) ) I 2 ( x , y ) = D 2 ( θ 0 ) I ( x , y ) cos 2 ( θ 2 − θ 0  +  β ) ) I 3 ( x , y ) = D 3 ( θ 0 ) I ( x , y )
I 1 ′ ( x , y ) = D 1 ( θ 0 + α ( x , y ) ) I ′ ( x , y ) cos 2 ( θ 1 ′ − θ 0 − β ) ) I 2 ′ ( x , y ) = D 2 ( θ 0 + α ( x , y ) ) I ′ ( x , y ) cos 2 ( θ 2 ′ − θ 0 + β ) ) I 3 ′ ( x , y ) = D 3 ( θ 0 + α ( x , y ) ) I ′ ( x , y )
I F ( x , y ) = ( I 1 ′ ( x , y ) I 1 ( x , y ) − I 2 ′ ( x , y ) I 2 ( x , y ) ) / I 3 ′ ( x , y ) I 3 ( x , y ) I F 1 ( x , y ) = I 1 ′ ( x , y ) I 1 ( x , y ) / I 3 ′ ( x , y ) I 3 ( x , y ) I F 2 ( x , y ) = I 2 ′ ( x , y ) I 2 ( x , y ) / I 3 ′ ( x , y ) I 3 ( x , y )
θ 1 = cos − 1 ( 1 1 + tan 2 ( α ) / ( P r / S r ) ) θ 2 = cos − 1 ( 1 1 + tan 2 ( α ) / ( P t P r / S t S r ) )
R < I max / I min I N = I max / R
Select as filters
Select Topics Cancel
Top
       
© Copyright 2024 | Optica Publishing Group. All rights reserved, including rights for text and data mining and training of artificial technologies or similar technologies.
Privacy | Terms of Use