Polarimetric current sensor based on polarization division multiplexing detection

Shaoyi Xu; Wei Li; Fangfang Xing; Yuqiao Wang

doi:10.1364/OE.22.011985

1. Introduction

There are several advantages of the optical current sensors (OCSs) over conventional current sensors, including fast response time, immunity to electromagnetic interference, high accuracy, wide dynamic range and wide bandwidth. Most of OCSs are based on the Faraday magneto-optic effect that the rotation of the plane of polarization is proportional to the intensity of the applied magnetic field in the direction of light propagation [1].

The OCSs are divided into two types: bulk-glass [2–4] and optical fiber [1, 5–12]. Bulk-glass current sensor employs the glass with high Verdet constant to achieve enough sensitivity, which is subject to alignment and temperature drifts. Optical fiber current sensor includes interferometer current sensor [5–9] and polarimetric current sensor [1, 10–12]. The interferometer current sensor (ICS) measures the non-reciprocal phase shift with high accuracy. However, the configuration of ICS is usually complex. Moreover, the cost of ICS may be expensive. The polarimetric current sensor (PCS) measures the rotation of a linear polarization, whose configuration is always simple.

The PCS with polarization diversity (PD) detection has been reported [1, 2, 4, 10, 12]. Compared with the single channel detection, the final output with PD detection is doubled, while the signal to noise ratio (SNR) is better. However, we believe that PD detection is not the best method for PCS. For example, we have found that PD detection is not suitable for PCS with the phase modulation error.

In this paper, we demonstrate a PCS based on the polarization division multiplexing (PDM) detection. PDM detection is applied to processing the orthogonal signal from a polarization beam splitter (PBS) and improving the system performance. Compared with PD detection, PDM detection mainly considered the effect of the static component on the final output of PCS. In addition, the sensor head with the heat insulation cavity is designed.

It should be noted that a PCS with PDM detection has been used to measure the stray current in railway transit systems. Stray current is known to induce the corrosion of any metallic elements that are in contact with the earth, such as reinforced concrete structures and armored cables, which is about a few amperes on a single metallic element in a real transit system [13, 14].

2. Configuration and principle

As shown in Fig. 1, the configuration of the proposed PCS mainly includes nine parts: (1) a broadband light source, whose center wavelength is 1550 nm; (2) a polarizer, whose extinction ratio isn’t lower than 40 dB; (3) a coupler; (4) a sensing fiber, which is the low-birefringence fiber, whose mode field diameter is 10 um and numerical aperture is 0.1; (5) a mirror, whose reflectivity isn’t lower than 99%; (6) a polarization controller, which is used to modulate the state of the polarization (SOP); (7) a polarization beam splitter; (8) a high-speed optical power meter, which has two input channel; (9) an industrial personal computer (IPC).

Fig. 1 Optical configuration of this sensor.

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Firstly, the light beam from the light source passes through the polarizer to form a linearly polarized light, which is parallel to the fast or slow axis of the polarizer (x axis or y axis). The linearly polarized light is coupled into the sensing fiber. In this fiber, the first Faraday effect is occurred due to the applied magnetic field. The SOP is rotated with the Faraday angle F. The mirror attached at the end of the sensing fiber reflects the linearly polarized light. When the reflected light propagates back to the sensing fiber, the second Faraday effect is occurred in the same magnetic field and the SOP is rotated with the same angle F again. The reflected linearly polarized light travels through the coupler, and then is coupled into the PBS. We use the polarization controller to rotate the SOP 45 degrees counterclockwise, which is set between the coupler and the PBS [1, 11]. The orthogonal optical signal from the PBS is detected by the two input channel of the high-speed optical power meter. The detected value of the power meter is sent to the IPC by RS232 serial interface.

The Jones calculus can be exploited for theoretical analysis of this PCS in Fig. 1. The main contents are as follow:

2.1 Jones matrix of optical elements

The Jones matrix of the polarizer is L_p. In the first rotation, the Jones matrix of the sensing fiber is L_f1. And in the second rotation, the Jones matrix of the sensing fiber is L_f2 [15]. The Jones matrix of the mirror is L_m. In addition, the Jones matrix of the polarization controller is L_R.

\begin{array}{l} L_{p} = [\begin{matrix} 1 & 0 \\ 0 & 0 \end{matrix}] . L_{m} = [\begin{matrix} - 1 & 0 \\ 0 & 1 \end{matrix}] . L_{R} = [\begin{matrix} \cos \frac{π}{4} & \sin \frac{π}{4} \\ - \sin \frac{π}{4} & \cos \frac{π}{4} \end{matrix}] . \\ L_{f 1} = [\begin{matrix} A + i B & - C \\ C & A - i B \end{matrix}] . L_{f 2} = [\begin{matrix} A^{'} + i B^{'} & - C^{'} \\ C^{'} & A^{'} - i B^{'} \end{matrix}] . \\ A = \cos p . A^{'} = \cos q . \\ B = \frac{δ}{2} (\frac{\sin p}{p}) . B^{'} = \frac{δ}{2} (\frac{\sin q}{q}) . \\ C = (T + F) (\frac{\sin p}{p}) . C^{'} = (T - F) (\frac{\sin q}{q}) . \\ p = \sqrt{{(T + F)}^{2} + {(δ / 2)}^{2}} . q = \sqrt{{(T - F)}^{2} + {(δ / 2)}^{2}} . \end{array}

Where T is the circular birefringence and $δ$ is the linear birefringence of the sensing fiber.

2.2 Power signal

The input light from the light source is represented by the Jones vector E_i = [E_xi E_yi] ^T. The output light from the PBS is represented by the Jones vector E_o = [E_xo E_yo] ^T, which can be derived as:

\begin{matrix} E_{o} = L_{R} \cdot L_{f 2} \cdot L_{m} \cdot L_{f 1} \cdot L_{p} \cdot E_{i} \\ = [\begin{matrix} - (A A^{'} - B B^{'} + C C^{'} + A C^{'} - A^{'} C) - i (A^{'} B + A B^{'} + B C^{'} + B^{'} C) \\ (A A^{'} - B B^{'} + C C^{'} - A C^{'} + A^{'} C) + i (A^{'} B + A B^{'} - B C^{'} - B^{'} C) \end{matrix}] \frac{\sqrt{2} E_{x i}}{2} . \end{matrix}

Thus, according to Eqs. (1), when the magnetic field is induced by the applied current, the power signal detected by the power meter is given by

{\begin{cases} J_{x 1} = \frac{E_{x i}^{2}}{2} [{(A A^{'} - B B^{'} + C C^{'} + A C^{'} - A^{'} C)}^{2} + {(A^{'} B + A B^{'} + B C^{'} + B^{'} C)}^{2}] \\ J_{y 1} = \frac{E_{x i}^{2}}{2} [{(A A^{'} - B B^{'} + C C^{'} - A C^{'} + A^{'} C)}^{2} + {(A^{'} B + A B^{'} - B C^{'} - B^{'} C)}^{2}] \end{cases} .

Without the magnetic field, the detected power signal is given by

{\begin{cases} J_{x 0} = \frac{E_{x i}^{2}}{2} [{(A_{0}^{2} - B_{0}^{2} + C_{0}^{2})}^{2} + {(2 A_{0} B_{0} + 2 B_{0} C_{0})}^{2}] \\ J_{y 0} = \frac{E_{x i}^{2}}{2} [{(A_{0}^{2} - B_{0}^{2} + C_{0}^{2})}^{2} + {(2 A_{0} B_{0} - 2 B_{0} C_{0})}^{2}] \end{cases} .

Where

A_{0} = \cos k . B_{0} = \frac{δ}{2} (\frac{\sin k}{k}) . C_{0} = T (\frac{\sin k}{k}) . k = \sqrt{{(T)}^{2} + {(δ / 2)}^{2}} .

2.3 Sensor output

This sensor applies PDM detection to process the detected power signals in the IPC with PD detection as a reference. Based on PD detection and PDM detection, the output of this sensor is given as D₁ and D₂, respectively.

D_{1} = \frac{J_{y 1} - J_{x 1}}{J_{y 1} + J_{x 1}}; D_{2} = \frac{J_{y 1} - J_{y 0}}{J_{y 0}} - \frac{J_{x 1} - J_{x 0}}{J_{x 0}} .

2.4 Experimental setup

The sensor head mainly includes a multilayer solenoid and the sensing fiber, which is visualized in Fig. 2. The multilayer solenoid is used to generate a magnetic field in the direction of the light propagation. The sensing fiber passes through the solenoid along the axial magnetic field direction induced by the applied current. The distribution curve of the axial magnetic field intensity H is illustrated in Fig. 2, which is not uniform. Thus, the non-uniform part is isolated by the metal casings, which are made of Ferro-manganese. The intensity of the uniform part can be calculated based on the empirical formula (Eqs. (5)) [16].

Fig. 2 Configuration of the sensor head: 1. Sensing fiber; 2.Solenoid; 3.Framework; 4. Metal casings; 5. Heat insulation cavity; 6. L-form holder.

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H (t) = n_{1} \cdot n_{2} \cdot (L / 2) \cdot \ln \frac{b + \sqrt{b^{2} + {(L / 2)}^{2}}}{a + \sqrt{a^{2} + {(L / 2)}^{2}}} \cdot i (t) .

Where, n₁ is the axial number of turns per unit length; n₂ is the vertical number of turns per unit length; L is length of the solenoid; a and b are the internal and external diameter of the solenoid, respectively; i(t) is the applied current.

At a specific time (for example t₀), the Faraday angle F (t₀) can be derived as

F (t_{0}) = V \cdot N \cdot l \cdot H (t_{0}) = (V \cdot N \cdot l \cdot n_{1} \cdot n_{2} \cdot (L / 2) \cdot \ln \frac{b + \sqrt{b^{2} + {(L / 2)}^{2}}}{a + \sqrt{a^{2} + {(L / 2)}^{2}}}) \cdot i (t_{0}) = m \cdot i (t_{0}) .

Where, V is Verdet constant of the sensing fiber; N is the number over which the magnetic field and light interact; l is the length over which the magnetic field and light interact. In our experiment, the Verdet constant is 0.804 urad/A at 1550 nm; N is equal to 3; l is equal to 0.17 m; n₁ is equal to 639.7 /m; n₂ is equal to 545.2 /m; L is equal to 0.39 m; b is equal to 0.054 m; a is equal to 0.032 m. Thus, the parameter m in Eqs. (6) is about 0.0031 rad/A.

In Eqs. (2) and (3), the parameters $δ$ , T and F need to be determined. The linear birefringence $δ$ is mainly induced by temperature, bend or the inherent linear birefringence, etc. The circular birefringence T is mainly induced by the torsional method or level of the sensing fiber. The Faraday angle F is changed with the applied current.

A heat insulation cavity is set between the solenoid and sensing fiber, which fills the glass cotton to cut off the heat-transfer path. The digital temperature indicator is used to measure the temperature variation of the sensing fiber. According to our experimental results, the temperature variation is less than 0.03 ${}^{\circ}C$ when a 20 A current is applied on the solenoid and lasted for five hours. Thus, the effect of the ohmic heat generated by the solenoid on the sensor head can be ignored in our experiment.

The low-birefringence sensing fiber is produced by Oxford Electronics Ltd. (type LB 1550-125), whose inherent linear birefringence δ_i is about $4^{\circ} / 4 m$ . As shown in Fig. 2, the bend-induced linear birefringence δ_b is calculated by Eqs. (7) [17, 18], which is equal to 0.1811 rad/m. And the total bending length is about 0.184 m. Therefore, the total bend-induced linear birefringence is about ${1.91}^{\circ}$ . Thus, the total linear birefringence $δ$ is about ${5.91}^{\circ}$ . In addition, the sensing fiber is not twisted and the circular birefringence T is approximated at zero.

δ_{b} = \frac{π}{λ} E C \frac{r^{2}}{R^{2}} .

Where $E = 7.5 \times 10^{8} Pa$ is Young’s modulus, $C = 3.05 \times 10^{- 12} {Pa}^{- 1}$ is the stress-optic coefficient at 1550 nm [19], r = 62.5 um is the radius of the fiber, R = 10 mm is the radius of the four bends and the length of each bend is about 15.3 mm, $λ = 15 50 nm$ is the operating wavelength.

3. Experimental results and discussion

In our experiment, PDM detection is carried out as follows: Firstly, the power signal without the magnetic field is detected and recorded, which is called static component in this paper, i.e., J_x0 and J_y0 in Eqs. (3). Then, the power signal with the magnetic field is detected and recorded, which includes the SOP rotation induced by the magnetic field, i.e., J_x1 and J_y1 in Eqs. (2). Finally, the detected data is processed to get the final output D₂ according to Eqs. (4). In addition, the sensitivity of the sensor is determined by the experiments. Thus, the applied current will be measured.

3.1 Sensitivity

According to Eqs. (6), when the applied current is from 0 A to 20 A, which represents the real distribution of the stray current on a single metallic element, the total Faraday angle is about 0 rad to 0.062 rad. Referring to Eqs. (2)-(4), the output of the sensor can be simulated based on PD detection and PDM detection, as shown in Fig. 3. The simulation includes two cases, one from the linear birefringence is ${5.91}^{\circ}$ and one from the linear birefringence is $0^{\circ}$ , which represents $D_{1} = \sin (4 F)$ and $D_{2} = 2 \sin (4 F)$ in Eqs. (4).

Fig. 3 Simulation result of the sensor sensitivity.

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Referring to Fig. 3, we can find that the linear birefringence has little effect on the output of this sensor, which due to the linear birefringence of the sensor head is small. We approximate the output D₁ at 4F and the output D₂ at 8F. It can be seen that sensitivity of PDM detection and PD detection is about 0.0248 /A and 0.0124 /A respectively, which indicates that the sensitivity of PDM detection is the double of PD detection.

For the simulation results validation, the sensitivity experiment was carried out and the result is shown in Fig. 4. The fitted curves are based on the least square method. The actual sensitivity of PDM detection and PD detection is about 0.0261 /A and 0.0131 /A, respectively. The former sensitivity is about the double of the latter, which is consistent with simulation result. However, we can find that the output with PD detection at 0 A is about 0.0182 while the output with PDM detection at 0A is about 0. We believe that the phase modulation error from the polarization controller is the main factor. In the simulation, the phase modulation angle is accurate 45 degrees, which is not achieved in the experiment (In this experiment, the power J_x0 and J_y0 is 876.9 uW and 909.3 uW, respectively, i.e., the modulation error is about 0.52 degrees). Thus, PDM detection will be proved to reduce the influence of the modulation error, which is illustrated in next section.

Fig. 4 Experimental result of the sensor sensitivity.

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3.2 Modulation error

The polarization controller is expected to rotate the SOP 45 degrees counterclockwise. However, it is difficult to accurately modulate the angle to 45 degrees in the experiment. $θ$ is expressed as the modulation error. And the Jones matrix L_R is redefined as follows:

L_{R} = [\begin{matrix} \cos (\frac{π}{4} + θ) & \sin (\frac{π}{4} + θ) \\ - \sin (\frac{π}{4} + θ) & \cos (\frac{π}{4} + θ) \end{matrix}] .

To simplify the calculation, since the linear birefringence is small, the analysis about the effect of modulation error is carried out without thinking about the linear birefringence. In presence of the modulation error, the power J_x0, J_y0, J_x1 and J_y1 are given by

{\begin{cases} J_{x 0} = \frac{E_{x i}^{2}}{2} [1 - \sin (2 θ)] \\ J_{y 0} = \frac{E_{x i}^{2}}{2} [1 + \sin (2 θ)] \\ J_{x 1} = \frac{E_{x i}^{2}}{2} [1 - \sin (4 F + 2 θ)] \\ J_{y 1} = \frac{E_{x i}^{2}}{2} [1 + \sin (4 F + 2 θ)] \end{cases} .

Based on PD detection, the output D₁ is given by

D_{1} = \frac{J_{y 1} - J_{x 1}}{J_{y 1} + J_{x 1}} = \sin (4 F + 2 θ) \approx 4 F + 2 θ .

Based on PDM detection, the output D₂ is given by

\begin{matrix} D_{2} = \frac{J_{y 1} - J_{y 0}}{J_{y 0}} - \frac{J_{x 1} - J_{x 0}}{J_{x 0}} \\ = \frac{\sin (4 F + 2 θ) - \sin (2 θ)}{1 + \sin (2 θ)} - \frac{- \sin (4 F + 2 θ) + \sin (2 θ)}{1 - \sin (2 θ)} \\ \approx \frac{4 F}{1 + 2 θ} + \frac{4 F}{1 - 2 θ} = \frac{8 F}{1 - 4 θ^{2}} . \end{matrix}

In Eqs. (10) and (11), if $θ$ is equal to ${0.52}^{\circ}$ , i.e., 0.0091 rad, the output D₁ includes 4F and 0.0182 while the output D₂ is about equal to 8F. In addition, the actual sensitivity of PD detection is about 0.0131 /A by the experiment. Thus, the current error will be about 1.39 A. Comparing Eqs. (10) and (11), the output based on PD detection includes the error, but the output with PDM detection does not. It means that the PDM detection is more suitable in the presence of the modulation error.

For the above theoretical result validation, the modulation error experiment was carried out. The experimental result of PDM and PD detection is listed in Table 1 and shown in Fig. 5. The applied current is sinusoidal current, which is produced by a variable frequency drive. Figure 5(a) shows the result of 15 Hz current. And in Table 1, the difference of the positive amplitudes between PDM and PD detection is 1.38 A, and the difference of the negative amplitudes is 1.37 A. In addition, Fig. 5(b) shows the results of 30 Hz current. According to Table 1, the difference of the positive amplitudes is 1.41 A while that of the negative amplitudes is 1.38 A. Therefore, the theoretical result is proved by the experimental result.

Table 1. Result with PDM and PD detection

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Fig. 5 Experimental output in presence of the modulation error: (a) 15 Hz; (b) 30 Hz.

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3.3 Output power of light source

Effect of the output power of light source on this sensor is illustrated in Fig. 6. In the experiment, we test the source power of 6.01 mW, 3.99mW and 0.96 mW, respectively. And the frequency of the applied current is about 50 Hz. With the results of 6.01 mW as a reference, we find that the current waveform is gradually distorted with the reducing of the source power. Two main reasons about the distortion will be expressed: one is the noise, especially the effect of the noise becomes more remarkable with the reducing of the source power; the other one is the shifting of center wavelength as the power decline, which induces the shifting of the Verdet constant of the sensing fiber.

Fig. 6 The original signals.

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The band-pass filter is designed using Matlab software to evaluate whether the noise is the one of the main reasons about the distortion. And the sampling frequency is 1024 Hz. The center frequency of the filter is 50 Hz. The higher and lower cut-off frequency is 30 and 70 Hz, respectively. Figure 6 shows the original signal at the different source power while Fig. 7 shows the signal processed by the filter, which is called de-noised signal. Compared Fig. 7 with Fig. 6, the current waveform has been improved, especially at 0.96 mW. Then, the improvement is quantified by the similarity coefficient r² (Eqs. (12)), which is shown in Table 2. We can find that the similarity coefficient between 6.01 mW and 3.99 mW is improved from 98.32% to 99.05% while the coefficient between 6.01 mW and 0.96 mW is improved from 62.79% to 73.51% after the filter. It means that the noise is the one of the main reasons about the distortion.

Fig. 7 The de-noised signal.

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Table 2. Similarity Coefficient

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r^{2} = \frac{{(m \sum_{i = 1}^{m} f_{i} u_{i} - \sum_{i = 1}^{m} f_{i} \sum_{i = 1}^{m} u_{i})}^{2}}{(m \sum_{i = 1}^{m} f_{i}^{2} - {(\sum_{i = 1}^{m} f_{i})}^{2}) (m \sum_{i = 1}^{m} u_{i}^{2} - {(\sum_{i = 1}^{m} u_{i})}^{2})} .

Where m is the number of the measured result; f_i and u_i represent the results of another source power and 6.01 mW, respectively.

The applied current is the same at the different power in the experiment, which is about 1.78 A measured by a digital multimeter (Agilent Co., Ltd., model U3402A). However, the detected current value in Fig. 7 at 0.96 mW is actually different from that at 3.99 mW and 6.01 mW, i.e., the effective values at 0.96 mW, 3.99 mW and 6.01 mW are about 1.67 A, 1.76 A and 1.78 A, respectively. The wavelength dependence of the Verdet constant of the sensing fiber will be the main reason of the difference. The sensing fiber belongs to magneto-optic glass medium. Thus, the Verdet constant is given by [20]

V (w) = C_{0} \cdot \frac{w^{2}}{n {(w_{0}^{2} - w^{2})}^{2}} .

Where, C₀ is constant; w is angular frequency of the incident light; w₀ is the inherent frequency of the sensing fiber; n is the effective index.

Moreover, according to the Sellmeir dispersion formula [21] (Eqs. (14)), the wavelength dependence of the Verdet constant can be expressed by Eqs. (15).

n^{2} = 1 + C_{1} \cdot \frac{1}{w_{0}^{2} - w^{2}} .

V (λ) = \frac{4 π^{2} v^{2} C_{0}}{C_{1}^{2}} \cdot \frac{{(n^{2} - 1)}^{2}}{n} \cdot \frac{1}{λ^{2}} .

Where, C₁ is constant; v is the propagation velocity of the light; $λ$ is the light wavelength.

The change of the index n is small with wavelength change. Thus, according to Eqs. (15), the Verdet constant is inversely proportional to the square of the wavelength. The central wavelength is measured by the spectrum analyzer (Agilent Co., Ltd., model 86142B). Therefore, the central wavelength at 0.96 mW, 3.99 mW and 6.01 mW is 1589.4 nm, 1556.7 nm and 1549.5 nm, respectively. Based on Eqs. (15), with the Verdet constant at 6.01 mW as a reference, the Verdet constant at 3.99 mW decreases by 0.92% while that at 0.96 mW decreases by 4.96%. In addition, with the detected current at 6.01 mW as a reference, the detected current at 3.99 mW decreases by 1.12% and the detected current at 0.96 mW decreases by 6.18%. Thus, the Verdet constant shifting induced by the center wavelength shifting will affect the performance of the sensor as the source power decline.

4. Conclusion

In this paper, we have demonstrated a polarimetric current sensor based on PDM detection. The theoretical and experimental results have been presented. Based on the results, the sensitivity of PDM detection is the double of PD detection. Moreover, in the presence of the phase modulation error, PDM detection is more suitable than PD detection. The sensor head with the heat insulation cavity can reduce the temperature-induced birefringence. Furthermore, the distortion as the source power decline is illustrated, which indicates the noise and the shifting of the Verdet constant are the main reasons. This sensor can be applied to electrochemical corrosion protection, such as the railway transit system or pipeline corrosion control.

Acknowledgments

This work is supported by the Graduate Education Innovation Project of Jiangsu Province of China (CXZZ13_0928), and the Priority Academic Program Development of Jiangsu Higher Education Institutions of China (PAPD).

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	15Hz (A)			30Hz (A)
	PD	PDM	Difference	PD	PDM	Difference
Positive	11.30	9.92	1.38	17.02	15.61	1.41
Negative	−8.53	−9.90	1.37	−14.21	−15.59	1.38

	Original Signal		De-noised Signal
	6.01mW & 3.99mW	6.01mW & 0.96mW	6.01mW & 3.99mW	6.01mW & 0.96mW
r²	98.32%	62.79%	99.05%	73.51%

	15Hz (A)			30Hz (A)
	PD	PDM	Difference	PD	PDM	Difference
Positive	11.30	9.92	1.38	17.02	15.61	1.41
Negative	−8.53	−9.90	1.37	−14.21	−15.59	1.38

	Original Signal		De-noised Signal
	6.01mW & 3.99mW	6.01mW & 0.96mW	6.01mW & 3.99mW	6.01mW & 0.96mW
r²	98.32%	62.79%	99.05%	73.51%

Polarimetric current sensor based on polarization division multiplexing detection

Abstract

1. Introduction

2. Configuration and principle

2.1 Jones matrix of optical elements

2.2 Power signal

2.3 Sensor output

2.4 Experimental setup

3. Experimental results and discussion

3.1 Sensitivity

3.2 Modulation error

3.3 Output power of light source

4. Conclusion

Acknowledgments

References and links

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Figures (7)

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Equations (17)

Optics Express