Quasi-distributed birefringence dispersion measurement for polarization maintain device with high accuracy based on white light interferometry

Zhangjun Yu; Jun Yang; Yonggui Yuan; Chuang Li; Shuai Liang; Lu Hou; Feng Peng; Bing Wu; Jianzhong Zhang; Zhihai Liu; Libo Yuan

doi:10.1364/OE.24.001587

1. Introduction

As an intrinsic Mach-Zehnder interferometer, polarization maintain (PM), or rather high birefringence, components have various applications in the fields of microwave photonic filters [1], interferometric fiber optic sensors [2]. The presence of birefringence dispersion (BD), which is the differential chromatic dispersion of the two eigenmodes, will decrease the sensitivity and spatial resolution for distributed sensor [3, 4 ] and may lead to reflection spectrum distortion for microwave photonic filters [5]. There are two types of white light interferometry (WLI), spectral [6, 7 ] and temporal [8, 9 ], to measure the BD for PM components. Spectral WLI can carry out broad spectral range measurement of BD using supercontinuum source, but it is not suitable for distributed measurement of BD for complicated devices. Temporal WLI that typically requires a Gaussian source, has the capability for distributed measurement by lateral force method for PM fibers [9] or splice points induced crosstalk for other PM components.

There are several temporal WLI to measure the BD for PM components. Okamoto and Hosaka [10] measured the BD of PM fiber by subtracting the chromatic dispersion of its two eigenmodes. This technique employs commercial instrument to obtain chromatic dispersion, which is quite convenient. The accuracy however is not sufficient for PM fiber nowadays with less BD. The technique for BD measurement proposed by Tang et al. [9] holds the advantages of high sensitivity and accuracy, and demonstrates the distributed measurement ability. This technique however is able to obtain the BD merely at central wavelength, while the BD is wavelength dependent. Flavin et al. [8] presented a method for BD measurement that deals with the phase spectrum of interferogram using polynomial fitting method based on ordinary least square (OLS) algorithm. Unavoidable temporal white Gaussian noise will erode the measurement results, and the signal-to-noise ratio (SNR) of interferogram is related to the coupling intensity of splice points [11] which is a fixed number. When making up a packaged PM device, manufacturers expect less coupling intensity [12] of splice points between operating component and its pigtails, which will thus make it difficult for measurement of BD with high accuracy.

In this paper, we propose a novel white light interferometric technique to measure the quasi-distributed BD for a PM device with high accuracy. The noise in conventional temporal WLI based on polynomial fitting technique is analyzed and results show that the variance of phase spectrum error is inversely proportional to the power spectrum of interferogram. Accordingly, the weighted least square (WLS) method, which specially deals with heteroscedastic model [13] that has unequal error variance, should be utilized to estimate the dispersion coefficient from phase spectrum and then the local BD curve for a interferogram could be reconstructed. Besides, using power spectrum of interferogram as weight value instead of employing iterative operation to carry out WLS method, the technique we proposed is more accurate and timesaving. Combining local BD curves of several interferograms, we obtain the quasi-distributed BD of every part for a packaged Y-waveguide with two 1m-long PM pigtails.

2. Theory

2.1 Quasi-distributed birefringence dispersion measurement

The experiment setup for recording of a series of temporal interferograms from which we characterize the quasi-distributed BD in the multifunctional integrated waveguide modulator (MFIWM) is illustrated in Fig. 1 . Passing through a coupler, an isolator and a polarizer in turn, the broadband light will be coupled in the fast and slow axes of the MFIWM under test, respectively. Subsequently, the output light from these two axes will be coupled in a single mode fiber using an analyzer. Finally, a Mach-Zehnder interferometer, which is considered as an tunable birefringent device for simplicity, is exploited to obtain interferograms by varying the optical path difference between the probe and reference arms. The MFIWM under test consists of input extend fiber (PMF1), input pigtail (PMF2), LiNbO₃ Y-waveguide chip, output pigtail (PMF3) and output extend fiber (PMF4). These two extend fibers are actually pigtails of the polarizer and the analyzer, respectively. The splice points A, B, C, D between these components all can result in a pair of interferograms because of 1st order polarization coupling. In addition, the polarization extinction capability of Y-waveguide chip will also lead to a pair of interferograms, and the fiber Mach-Zehnder interferometer will arise a singleinterferogram. All these interferograms are shown in Fig. 2 . We can see the full width at half maximum (FWHM) of interferograms C and D are narrow and the FWHM of other interferograms located left or right of them are gradually increasing. This is because of the positions of interferograms C and D are close to the zero BD area, given the BD of interferometer is negative, and negative or positive BD will both broaden the interferograms.

Fig. 1 Experiment setup of the measurement system.

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Fig. 2 The output signal of the measurement system. The four pairs of interferograms A, A’, B, B’, C, C’, D, and D’ are first order coupling crosstalk induced by the four splice points A, B, C, and D in Fig. 1, respectively. Polarization extinction capability of the Y-waveguide chip leads to the interferograms Y and Y’. The interferogram M is primarily resulted by the interference of excited mode in the probe arm and in the reference arm. The interferograms between B and Y, as well as interferograms between B’ and Y’, are second order coupling crosstalk which are beyond this paper.

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One can employ fast Fourier transform (FFT) technique and a simple program to obtain the amplitude spectrum and unwrapped phase spectrum from an interferogram. Ignoring terms more than three order, the phase spectrum $φ (ω)$ can be represented by the Taylor expansion about the source central frequency $ω_{0}$ as [14]

φ (ω) = φ (ω_{0}) + L \cdot [LB (ω - ω_{0}) + \frac{1}{2} SBD {(ω - ω_{0})}^{2} + \frac{1}{6} TBD {(ω - ω_{0})}^{3}]

where

ω

is the angular frequency,

L

is the length of device under test, and the dispersion coefficients are the linear birefringence (LB), the second order birefringence dispersion (SBD), and the third order birefringence dispersion (TBD), respectively. The second derivative of phase spectrum

φ (ω)

with respect to wavelength is the BD of interferogram

D (λ) = (- 2 π c / λ^{2}) (d^{2} φ / d λ^{2}) / L

, the BD of interferogram therefore can simplicity be expressed with dispersion coefficient as

D (λ) = - [SBD + TBD (\frac{2 π c}{λ} - \frac{2 π c}{λ_{0}})] \cdot \frac{2 π c}{λ^{2}}

where

λ_{0}

is the central wavelength of source and

c

is the velocity of light in vacuum. Consequently, the quasi-distributed BD of every component in the MFIWM can be calculated as [11]

{\begin{array}{l} Δ D_{l 1} = (D_{M} - D_{A}) / l_{1} = (D_{A'} - D_{M}) / l_{1}, \\ Δ D_{l 2} = (D_{A} - D_{B}) / l_{2} = (D_{B'} - D_{A'}) / l_{2}, \\ Δ D_{l 3} = (D_{D} - D_{C}) / l_{3} = (D_{C'} - D_{D'}) / l_{3}, \\ Δ D_{l 4} = (D_{M} - D_{D}) / l_{4} = (D_{D'} - D_{M}) / l_{4}, \\ Δ D_{Y} = (D_{B} - D_{M} - D_{Y} + D_{C}) / l_{Y} = (D_{Y'} - D_{B'} - D_{C'} + D_{M}) / l_{Y} \end{array}

where

Δ D_{l 1},

Δ D_{l 2},

Δ D_{l 3}

Δ D_{l 4}

and

Δ D_{Y}

are the quasi-distributed BD of PMF1, PMF2, PMF3, PMF4 and Y-waveguide chip, respectively;

D_{A},

D_{A^{'}},

D_{B},

D_{B^{'}},

D_{C},

D_{C^{'}},

D_{D},

D_{D^{'}},

D_{Y},

D_{Y^{'}},

and

D_{M}

are the local BD of interferograms A, A’, B, B’, C, C’, D, D’, Y, Y’, and M, respectively.

2.2 Dispersion coefficient estimation with weighted least square method

The discrete phase spectrum obtained by means of FFT technique can be expressed as

φ = {[\begin{matrix} φ (ω_{1}) & φ (ω_{2}) & ... & φ (ω_{k}) \end{matrix}]}^{T}

where

{[\cdot]}^{T}

represents to matrix transposition,

φ (ω_{i})

is the phase value at differential angular frequency

ω_{j}

(we replace

Ω_{j} = ω_{j} - ω_{0}

with

ω_{j}

for simplicity), and

k

is the sample channel number of effective phase spectrum. Without regard to dispersion with order higher than three, one can employ the WLS algorithm to fit the discrete phase spectrum to 3rd order polynomial curve that expressed as

φ = W \cdot ω \cdot B + ε

here

\begin{array}{l} ω = [\begin{matrix} 1 & ω_{1} & ω_{1}^{2} & ω_{1}^{3} \\ 1 & ω_{2} & ω_{2}^{2} & ω_{2}^{3} \\ ⋮ & ⋮ & ⋮ & ⋮ \\ 1 & ω_{k} & ω_{k}^{2} & ω_{k}^{3} \end{matrix}], W = [\begin{matrix} w_{1} & 0 \\ w_{2} \\ ⋱ \\ 0 & w_{k} \end{matrix}], \\ B = {[\begin{matrix} B_{0} & B_{1} & B_{2} & B_{3} \end{matrix}]}^{T}, ε = {[\begin{matrix} ε_{1} & ε_{2} & \dots & ε_{k} \end{matrix}]}^{T} \end{array}

where

ε_{j} (j = 0, 1, \dots, k)

and

w_{j}

is the error and weight value of every phase channel, respectively. The optimal value of weight value is

w_{j} = 1 / σ_{ε j}^{2}

[13], where

σ_{ε j}^{2}

is the variance of

ε_{j}

.

B_{i} (i = 0, 1, 2, 3)

is the i ^th coefficient of fitting phase curve and represent initial phase, LB, SBD/2, TBD/6, respectively.

If the interferogram is subject to white Gaussian noise, the effective phase spectrum obtained by FFT technique will be eroded. In addition, the variance of phase spectrum error is [15]

σ_{ε}^{2} (ω) = \frac{1}{S \cdot {[a (ω)]}^{2}}

where

a (ω)

is the normalized amplitude spectrum of interferogram,

S

is the SNR of

{[a (ω)]}^{2}

, i.e. Spectral SNR. The weight value therefore can be set as

w_{j} = {[a (ω_{j})]}^{2}

, or equivalently normalized power spectrum, for simplicity. Moreover, the weighted least squares estimator of dispersion coefficient matrix B is

{\hat{B}}_{wls} = {(ω^{T} W ω)}^{- 1} ω^{T} W \cdot φ

And the covariance matrix of estimator

{\hat{B}}_{wls}

is [13]

V = \frac{1}{S} {(ω^{T} W ω)}^{- 1}

where the diagonal elements of covariance matrix

V

, are the variance of

B_{i}

in turn, i.e.

V_{i, i} = σ_{B i}^{2}

. Accordingly, the variance of measurement error for SBD and TBD is

σ_{SBD}^{2} = 4 V_{3, 3}

and

σ_{TBD}^{2} = 36 V_{4, 4}

, respectively. On the other hand, if OLS method is utilized to estimate the dispersion coefficient matrix

B

, in other word let the weight value matrix

W

be equal to unit matrix

I

, the estimator will be

{\hat{B}}_{ols} = {(ω^{T} ω)}^{- 1} ω^{T} \cdot φ

, and the corresponding covariance matrix is

V_{ols} = \frac{1}{S} {(ω^{T} ω)}^{- 1} ω^{T} W^{- 1} ω {(ω^{T} ω)}^{- 1}

where the meaning of diagonal elements of

V_{ols}

are the same as Eq. (9). We can know from [16] that the the matrix

V_{ols} - V

is nonnegative definite, therefore all the diagonal elements of matrix

V_{ols} - V

are

\geq 0

. Accordingly, OLS estimator is no longer a minimum-variance estimator for the unequal variance model described as Eq. (5) when the source spectrum is not ideal flat-top, given the variance of WLS estimator is less than it at least. Whereas the WLS method is equivalent to OLS method when the normalized amplitude spectrum of interferogram is constant, because the weight value matrix degenerate to unit matrix. However, the temporal interferometry typically employs Gaussian source for its symmetry between time and frequency domain. Consequently, the WLS method will be more suitable than OLS method since it is still a minimum-variance estimator for Gaussian source.

3. Experiments and results

3.1 Simulation for dispersion coefficient measurement

For given broadband source and Spectral SNR value $S$ , one can utilizing Eqs. (9) and (10) to calculate the theoretical variance of SBD and TBD for WLS and OLS method, respectively. The measurement error variance of WLS and OLS method are both inversely proportional to spectral SNR, and this relationship is verified by simulation experiment. The interferogram in simulation experiment is with SBD of 100 fs² and TBD of 10⁴ fs³, which is added by white Gaussian noise with different noise level. The BD of interferogram with noise is characterized by WLS and OLS method, respectively. Figure 3 shows the experiment results measured via 1000-time simulations, which demonstrates the relationship between noise power and the standard deviation of SBD and TBD. Notice that the noise power is in time domain, whereas the $S$ in Eq. (7) is spectral SNR. The noise power is proportional to spectral SNR [15] andwe don’t care about what the exact factor of proportionality is. As depicted in Fig. 3, the WLS method is roughly an order of magnitude improvement than OLS method in standard deviation of TBD and SBD at spectral SNR of 50 dB to 65 dB. When the noise is >50 dB, the OLS method is seriously affected by outliers and no longer meet the linear relationship.

Fig. 3 Relationship between noise power and the measurement error of TBD (black) and SBD (blue) with OLS method (dash line) and WLS method (solid line) for 1000 sets simulation data, respectively. Inset: zoom out to show the results of OLS method at high noise power from −45 dB to −30 dB.

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3.2 Error variance of birefringence dispersion for interferograms

The measurement error variance of BD can be calculated by utilizing Eq. (2) as

σ_{D}^{2} (λ) = {(\frac{2 π c}{λ^{2}})}^{2} [4 V_{3,3} + 36 (\frac{2 π c}{λ} - \frac{2 π c}{λ_{0}}) V_{4, 4}],

given the SBD and TBD are statistical independent. The packaged Y-waveguide was measured for 100 times with the technique proposed in this paper. The experiment setup and the MFIWM under test are demonstrated in Fig. 1. The broadband source we used is a super luminescent diode (SLD) emitting at 1550 nm with FWHM of approximately 50 nm. A distributed feedback (DFB) laser is used to eliminate the influence of mechanical vibration of scanning mirror. The 2/98 coupler C1 and photo diode PD1 followed the SLD are used for monitoring the source. The splitting ratio of coupler C2 and C3 both are 50:50, and actually C2 and C3 are wavelength division multiplexers. Photo diode PD2 and PD3 are followed by a differential circuit. The length of PMF1, PMF2, PMF3, PMF4 and Y-waveguide chip in the MFIWM under test are 15.36 m, 1.41 m, 0.71 m, 20.16 m and 0.02 m, respectively. In addition, all the four PM fibers are made by YOFC. The product type of PMF1 and PMF4 are PMF 1310/125-18/25-Y, PMF2 is PMF 1550/125-18/250-Y, and PMF3 is PMF 1550/80-18/165-Y.

Figure 4 demonstrates the measurement repeatability of interferograms M and Y’, and corresponding theoretical error profile with almost the same spectral SNR for each interferogram. Using FFT technique, we obtain the spectral SNR of M and Y’ interferogram of approximate 60 dB and 30 dB, respectively, and the central wavelength of them is about 1552 nm. For interferogram M, the WLS method is roughly a fact of 5 improvement than OLS method from Fig. 4(a) and (b). The improvement less than the simulation value (nearly an order of magnitude) and the large deviations between theoretical and experimental results away central wavelength in Fig. 4(b) perhaps due to the ripple of light source which is not white Gaussian noise (it can be seen from Fig. 2). The region of ripple varies with the properties of the working medium in SLD and its cavity length, and the amplitude of ripple is approximately 60 dB less than the corresponding interferogram [17]. We will possibly obtain better results if utilizing the sidelobe-free SLD in [17]. For interferograms Y and Y’, which are not affected by the ripple of first order interferograms induced by SLD source, the WLS method holds indeed roughly an order of magnitude improvement than OLS method.

Fig. 4 Comparison of measurement error variance between theory and 100 sets experimental data. (a) Measurement error of interferogram M with OLS method (red solid line) and corresponding theoretical error curve with spectral SNR of 60 dB (blue dash line), (b) error of M with WLS method and corresponding 60 dB theoretical error, (c) error of Y’ with OLS method and corresponding 30 dB theoretical error, (d) error of Y’ with WLS method and corresponding 30 dB theoretical error.

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3.3 Measurement results of birefringence dispersion for the interferometer

It has already been stated that the interferogram M is caused by the interferometer, the BD of interferogram M is therefore the BD of the Mach-Zehnder interferometer in Fig. 1 in which the length difference of single-mode fiber for two arms is ~0.12 m. According to Eq. (3), the measurement accuracy of BD for PMF1, PMF4, and Y waveguide is bound up with that for the interferometer. Although the spectral domain (SD) WLI is not appropriate for distributed measurement of BD for complicated devices, we could utilize it to measure the BD of interferometer, and compare the measurement result with the Flavin’s OLS method and our WLS method that are both based on temporal WLI. In order to use the SD method proposed by Lee and Kim in [7] to measure the BD of interferometer, we merely require splicing the point S and E in Fig. 1 directly, without the polarizer and analyzer of course, and employing an optical spectrum analyzer (YOKOGAWA AQ6370C) in stead of the differential signal acquisition device to obtain the channel spectrum, i.e. the spectral interference pattern. We adjust the SNR to ~60 dB via varying the power of light source and measure the interferometer 100 times as well as the temporal methods for the sake of consistency.

Figure 5(a) demonstrates the measurement results of BD for the interferogram with Flavin’s OLS method, WLS method, and Lee’s SD method, in which the maximum deviation of these three methods is less than $5 \times 10^{- 5}$ ps/nm that is approximately equivalent to a single- mode fiber with length of 3 mm. Figure 5(b) shows the corresponding measurement errors calculated with 100 sets data, respectively. Actually, the measurement error curves of OLS method and WLS method are identical with the experimental results of interferogram M in Fig. 4(a) and (b). It is demonstrated that the measurement error of WLS method is better than that of SD method. The measurement error of SD method is primarily induced by the phase noise in the process of data acquisition, and the suppression of phase noise is beyond this paper. Whereas, the phase noise induced at the time of data acquisition in WLS method is calibrated with a DFB laser. Besides, the phase noise induced by the amplitude noise of raw data via FFT technique is suppressed by the WLS method.

Fig. 5 Comparison of the measurement results of BD for the interferometer. (a) Results of BD for the interferometer measured with two temporal methods and a spectral domain method. (b) The measurement error of these three methods calculated with 100 sets data. The error curves of OLS method and WLS method are actually the same as the experimental error curves in Fig. 4(a) and (b), respectively.

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3.4 Measurement results of quasi-distributed birefringence dispersion for packaged device

One can obtain the quasi-distributed BD of every component under test from Eq. (3) after measuring the local BD of every labeled interferogram in Fig. 2. It is demonstrated in Fig. 6 that the measurement results for quasi-distributed BD of the four PM fibers. PMF1, PMF3 and PMF4 have the similar dispersion characteristics, whereas the PMF2 much differs from them. We have mentioned before that the zero BD point is located in the vicinity of interferograms C and D, so the interferograms on the right side of M have larger dispersion than those on the left. Figure 6 also compares the results calculated from left and right interferograms, which confirms our proposed method can measure positive and negative BD that quite close to zero. It is shown in Fig. 6(b) that PMF2 has large 3rd order BD, and if we don’t distinguish the BD of PMF2 from Y-waveguide chip, it will greatly influences the measurement result of BD for Y-waveguide chip. Besides, varying the length of pigtails when using the device in practice will change the dispersion characterization of this whole device. Figure 7 depicts the comparison of measurement results for distributed BD of Y-waveguide with and without the two pigtails.

Fig. 6 Measurement results for quasi-distributed BD of the four PM fibers. (a) PMF1, PMF3, and PMF4; (b) PMF2. Where, left means the curve is calculated by the data of interferograms on the left hand of interferogram M as shown in Fig. 2, and right means the right hand of M.

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Fig. 7 Measurement results for quasi-distributed BD of Y-waveguide chip with and without two pigtails of it.

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4. Conclusion

In conclusion, we propose a novel white light interferometric technique for quasi-distributed BD measurement with high accuracy based on WLS method. The higher accuracy than OLS method is because of WLS method is more suitable for unequal error variance model. Furthermore, employing power spectrum of interferogram as weight value is more accurate and cancels iterative operation which is time-consuming. The quasi-distributed BD of a packaged Y-waveguide with two ~1 m-long PM pigtails is measured with this technique. The measurement results show the BD of pigtails are comparable with 0.02 m-long Y-waveguide chip, thus distinguishing them becomes very important.

Acknowledgment

This word is supported by the National Natural Science Foundation of China (61422505, 61227013, 61307104, 61405044); The Program for New Century Excellent Talents in University (NCET-12-0623); National Key Scientific Instrument and Equipment Development Project (No. 2013YQ040815); and the Specialized Research Fund for the Doctoral Program of Higher Education (No. 20122304110022).

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Quasi-distributed birefringence dispersion measurement for polarization maintain device with high accuracy based on white light interferometry

Abstract

1. Introduction

2. Theory

2.1 Quasi-distributed birefringence dispersion measurement

2.2 Dispersion coefficient estimation with weighted least square method

3. Experiments and results

3.1 Simulation for dispersion coefficient measurement

3.2 Error variance of birefringence dispersion for interferograms

3.3 Measurement results of birefringence dispersion for the interferometer

3.4 Measurement results of quasi-distributed birefringence dispersion for packaged device

4. Conclusion

Acknowledgment

References and links

Cited By

Figures (7)

Equations (11)

Optics Express