Novel dispersion compensation method for cross-coupling measurement in PM-PCF based on OCDP

Optical Fiber Technology 19 (2013) 495–500

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Optical Fiber Technology www.elsevier.com/locate/yofte

Novel dispersion compensation method for cross-coupling measurement in PM-PCF based on OCDP Jing Jin, Shu Wang ⇑, Jingming Song, Ningfang Song, Zuoming Sun, Man Jiang School of Instrumentation Science and Opto-Electronics Engineering, Beihang University, XueYuan Road No. 37, HaiDian District, Beijing, PR China

a r t i c l e

i n f o

Article history: Received 3 April 2013 Revised 3 June 2013 Available online 5 July 2013 Keywords: Polarization maintaining photonic crystal fiber OCDP Inter-mode chromatic dispersion Cross-coupling measurement Compensation algorithm

a b s t r a c t For polarization maintaining photonic crystal fiber (PM-PCF), the cross-coupling measurement can be severely affected by interferogram broadening, due to the large inter-mode chromatic dispersion. In this paper, a novel dispersion compensation method is proposed to mitigate the influence, including a frequency domain algorithm and investigation of the dispersion coefficient for a phase packet in algorithm. A numerical simulation and measurement of birefringence as a function of wavelength reveal that the dispersion coefficient in PM-PCF is much larger than that in PANDA-PM fiber. After compensation, the accuracy of coupling strength measured can be restored and spatial resolution is improved. Experiments show the compensation provides high accuracy with average relative error less than 0.31% and spatial resolution improved 4.1 times. Ó 2013 Elsevier Inc. All rights reserved.

1. Introduction Photonic crystal fibers have spurred many studies of its application in fiber sensing because of the flexibilities on design and unique optical properties, such as high nonlinearity and flat dispersion, etc. For PM-PCF, high birefringence and thermal insensitivity [1] help it find promising applications in optical interferometer for example fiber optical gyroscope (FOG). However, cross-couplings produced in winding harm the polarization performance of FOG coil. Cross-coupling pairs located symmetrically to coil’s middle point contribute to the host phase error [2].Therefore, the accurate measurement of cross-coupling distribution is necessary to improve the precision of FOG [3,4]. Optical Coherence Domain Polarimeter (OCDP) is widely used in polarization cross-coupling measurement for its high precision and spatial resolution. However, for PM-PCF, the inter-mode chromatic dispersion decreases the sensitivity and spatial resolution in longdistance measurement [5–7]. This effect is more obvious than in stress-induced high-birefringence fiber (e.g. PANDA fiber). The study on inter-mode chromatic dispersion of PM-PCF has not been reported yet. We found the inter-mode chromatic dispersion is 500 times larger than that of a PANDA-PM fiber, so interferogram broadens much notably. To mitigate the influence of inter-mode chromatic dispersion in cross-coupling measurement, dispersion compensation is necessary. ⇑ Corresponding author. E-mail addresses: [email protected], [email protected] (S. Wang). 1068-5200/$ - see front matter Ó 2013 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.yofte.2013.06.003

In traditional measurement of the dispersion coefficient [8], precise measurement of the width of interference envelope is required, as well as the parameters of light source including central wavelength and spectral width. As the parameters of light source, usually superluminescent diode (SLD), fluctuate with the drive current and temperature, the repeatability of measurement cannot be guaranteed. For time-domain compensation methods [9], Hilbert transform and nonlinear least-squares fit of white light interference envelope, which are indispensable, add to the complexity of calculation. Moreover, when the interferogram envelope does not follow a Gaussian distribution strictly, time domain method lost its effectiveness. Here, in order to solve these problems, a novel frequency-domain compensation algorithm is proposed. The analysis of interference spectrum is implemented for data processing. Then the dispersion coefficient is investigated through simulation and experiment for the first time. Finally, validity of the method including the accuracy of cross-coupling strength and system resolution is checked experimentally. 2. Theory 2.1. Principle of OCDP and inter-mode chromatic dispersion A typical OCDP system is based on the principle of white light Michelson interferometer. Fig. 1 shows the configuration. The low-coherence beam passes through the polarizer and collimates into the intrinsic axis of the fiber to populate only one eigenmode. In transmission, a fraction of beam in eigenmode is coupled into the orthogonal mode at the cross-coupling point. The beams

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2.2. Influence of dispersion in cross-coupling distribution measurement and compensation algorithm The absolute value of coherence function at single coupling point is present by

Z  jcd ðDNb l  dÞj ¼ 

þ1

SðkÞ expfi½k0 d  DbðkÞlg dk

1

Z

þ1

1

  SðkÞ dk ð4Þ

where S(k) is the spectrum of light source, l is the distance between the cross-coupling point and output facet of the test fiber, and d is OPD of two arms in Michelson interferometer. In our system, a SLD emitting at 1550 nm is used as low-coherent light source. The spectral density function follows a Gaussian shape

" # 2 1 ðk  k0 Þ SðkÞ ¼ pffiffiffiffiffiffiffi exp  2 2 p Dk 2 Dk

ð5Þ

where Dk = (2p/c)Dm, where Dm is the 3 dB bandwidth. Substituting (5) into (4), the coherence function turns into Fig. 1. Configuration of polarization cross-coupling measurement OCDP system. 2 1=4

jcd ðDNb l  dÞj ¼ ð1 þ n Þ entering the tandem Michelson interferometer contain a combination of two modes. Due to the difference of group velocity between the two modes, optical path difference (OPD) is produced. The scanning mirror in the interferometer compensates the OPD and the output interferogram is read out by a photo diode (PD), transformed by a data acquisition module (DAQ) and processed in computer during the scanning. As high birefringence(hi-bi) eliminates the degeneracy of HE11 mode, differences exist in group velocity and chromatic dispersion which correspond to the differential group delay (DGD) and intermode chromatic dispersion respectively. The inter mode chromatic dispersion is defined as [10]

DDðkÞ ¼ Df ðkÞ  Ds ðkÞ

ð1Þ

8 " # 9 < 2ðDN l  dÞ 2 = b exp  pffiffiffiffiffiffiffiffiffiffiffiffiffiffi : 1 þ n2 Lc ;

ð6Þ

where n characterizes the accumulation of dispersion along the fiber, n = (Dk/k)2  2pc  l  DD. Inter-mode chromatic dispersion dep ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4 creases interference contrast by a factor of 1 þ n2 , and the spatial pffiffiffiffiffiffiffiffiffiffiffiffiffiffi resolution declines by 1 þ n2 times as the distance l increases. The interferogram of single cross-coupling point in time-domain is given by

Icoup ðsÞ ¼ C  hEx ðtÞ  Ey ðt þ sÞi Z þ1 Z þ1  0 0 0  0 ¼Ch Eðk Þ exp i½k ct þ bx ðk Þ dk E ðkÞ 1 1    exp i½kcðt þ sÞ þ by ðkÞ dki Z þ1 jEðkÞj2 exp½iDbðkÞl exp½ikcs dk ¼C 1

where f and s denote the fast and slow eigenmodes. A two-beam interferometer with dispersive differential propagation constant Db(k) is useful to model the polarization-maintaining fiber. We represent Db(k) in a Taylor series about central wave vector k0 to the second order [11]. 2

2

DbðkÞ ¼ kDnb þ ðk  k0 ÞDNb þ pcðk  k0 Þ =k0  DD þ   

ð2Þ

where c is the light speed in free space, Dnb is modal birefringence, and DNb is the group birefringence. DD is the component of the second order item. Although high order (after second) dispersion causes interferogram deformation such as asymmetry, it is negligible in the contribution to interferogram broadening and the loss of interference contrast compared with the second order item. The DGD, expressed as dDb(k)/dk, is considered as the first order dispersion. Combining with (2), DD is the variation in the DGD with respect to wavelength in the fiber. It is also deduced as the quadratic derivatives of effective modal birefringence Dneff, 2

DDðkÞ ¼

ds k d Dneff ¼ ; c dk2 dk

ð3Þ

¼ C  F T ðSðkÞ exp½iDbðkÞlÞ

ð7Þ

Here h i denotes ensemble average, s is time delay, FT means Flourier transform and C is scale factor of Fourier transform. This formula demonstrates that interferogram influenced by dispersion is equivalent to the Flourier transform of spectrum distribution multiplied by a phase packet Db(k)l. In cases that differential propagation constant Db losses dependence on k, Icoup(s) is just the Fourier transform of light spectrum which means the influence of dispersion is eliminated. So the dispersion compensation can be achieved by introducing a phase compensation packet ucom = Db(k)l,

Icom ðsÞ ¼ C  F T ðSðkÞ exp½iDbðkÞl  exp½iucom Þ ¼ C  F T ðF 1 T ðI coup =CÞ  exp½iucom Þ

ð8Þ

To re-construct the interferogram of coupling point (Icom), the inverse Flourier transform of the Icoup should be multiplied by a phase compensation packet to offset the phase shift accumulated by dispersion. Interferogram Icoup(s) and the dispersion coefficient DD are necessary. As the spectral interferogram S(k) acts as a intermediate item in the algorithm, the interference in spectral-domain is investigated preliminarily to transform the Icoup into suitable form for numerical calculation. 2.3. Spectrum analysis for compensation

where s is the DGD in unit [ps/km]. The wavelength-dependence of effective modal birefringence is closely associated with the dispersion.

The spectral density of a coupling’s spectral interferogram Scoup can be described as [12]

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Fig. 5. SEM image of the PM-PCF(x – fast axis, y – slow axis). Fig. 2. Normalized spectral interferogram of single coupling.

Table 1 Parameters of PM-PCF. Values

Beatlength @ 1550 nm Attenuation @ 1550 nm Cross-talk Material Large/small hole diameter Pitch Cladding/coating diameter

1.96 mm 3.8 dB/km <28 dB/100 m @ 1550 nm Pure silica 6.924/3.049 lm 5.414 lm 125/250 lm

by OCDP system only covers the interferogram when OPD is positive. Hence, symmetry operation is necessary to prepare the suitable Icoup(s). Fourier inversion and multiplication by phase packet are carried out sequentially. Finally, Icom(s) can be retrieved from a Fourier transform. Fig. 4 shows the flow chart of compensation algorithm.

Fig. 3. Light intensity signal of a single coupling point.

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Scoup ðz; kÞ ¼ Sf ðkÞ þ Ss ðkÞ þ 2 Sf ðkÞSs ðkÞRe½iDbðkÞz

Parameters

ð9Þ

where Sf(k) and Ss(k) are respectively the spectral density of polarized light in fast and slow mode. We locate the moving stage of OCDP to the position where the optical paths of two modes generated at a single coupling are just balanced. The output spectrum is recorded by an Optical Spectrum Analyzer (Agilent 4396B) with wavelength resolution set as 0.5 nm. The normalized spectral interferogram is shown in Fig. 2. We firstly transform the spectral interferogram Scoup(k) to Scoup(k) in k domain. The Fourier transform conducted on Scoup(k) yields the interferogram of a single coupling point Icoup(s) in time-domain; the result is displayed in Fig. 3. Two band lopes located symmetrically to the central maximum correspond to OPD in positive and negative. Practically, the raw data recorded

3. Experiment The dispersion of PM-PCF reflects the variation of birefringence with wavelength, so the simulation and measurement of birefringence is implemented. 3.1. Simulation and measurement of birefringence The PM-PCF investigated in this paper is fabricated by Yangtze Optical Electronic Co., Ltd. (PM-125-02), of which the cross-section is shown in Fig. 5. The parameters of PM-PCF are listed in Table 1. Birefringence of PM-PCF is simulated by the CUDOS MOF Utilities, a software package based on multipole expansion. The dispersion is analyzed through the calculation of effective index of fundamental mode according to (3). An infinitely large solid

Fig. 4. Flow chart of compensation algorithm.

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cladding with index of 1.45 is assigned. To ensure the precision of the order 1012, the truncation order M of Bessel function expansions is set as 7. At each wavelength, two effective refractive indices representing fast and slow fundamental modes are found. Fig. 6.1 plots the effective refractive indices of x-(fast axis) and y(slow axis) polarized modes vs wavelength. Effective refractive indices of x- and y-axis are fitted with cubic polynomial curves to calculate the dispersion coefficient. Simulated modal birefringence of PM-PCF is 7.85  104 at 1550 nm. As the mode field is confined in the core area, material dispersion of SiO2 should be taken into consideration. Both waveguide dispersion DW and material dispersion DSiO2 contribute to chromatic dispersion in each axis Di = DW + DSiO2 (i = x, y). The index of SiO2 as a function of wavelength is described by Sellmeier equation [13]. DSiO2 is 21.9 ps/(kmnm) at 1550 nm. The chromatic dispersion curve is drawn in Fig. 6.2. Inter-mode chromatic dispersion coefficient, yielded by Dx  Dy, is 1.192 ps/(km nm). Wavelength scanning experiment setup is designed to obtain the birefringence of PM-PCF under broad wavelength range. The schematic is shown in Fig. 7. A supercontinum source (700–1700 nm) was used as the light source. A PM-PCF length of 27 cm was under test. Linearly polarized light was launched into PM-PCF without aligning with intrinsic axis to excite both modes. The angle of polarizer with the axis of PM-PCF was set as 45° to achieve maximal interference contrast.

Fig. 7. Schematic of wavelength scanning setup.

Two modes propagated along the axes and hence, phase delay was caused. The light beam at output end passed through an analyzer and finally sent into OSA. A SMF-28 fiber between analyzer and OSA served as a mode filter, inhibiting high-order mode. The phase difference Du between two adjacent peak and valley of the normalized white-light spectrum located at wavelengths kj and kj+1 should be ±p.

  Dnðkjþ1 Þ Dnðkj Þ ¼ p Duðkjþ1 Þ  Duðkj Þ ¼ 2pL  kjþ1 kj

ð10Þ

Specially, since the simulation shows that phase birefringence increases with wavelength, the right side of function (10) should be negative. The white light transmission spectrum is not enough to determine Dn(kj+1) uniquely unless we know the modal birefringence Dn at kj as the initial value. The periodicity in output spectrum (see Fig. 8) results from the variation of D/ with k. Although the phase delay depends on both linear and circular birefringence, circular phase delay is so small in polarization maintaining fiber, the total beat-length differing by

Fig. 6.1. Effective refractive index of x- and y-axis. Fig. 8. Broad-band transmission spectrum of a 27 cm PM-PCF.

Fig. 6.2. Chromatic dispersion of x- and y-axis.

Fig. 9. Birefringence measured and simulated as the function of wavelength.

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Fig. 10. Configuration of experiment setup.

Table 2 Parameters for OCDP system. Parameters

Optical path interval

Sampling rate

Scanning speed

Scanning range

Central frequency

3 dB spectral width of SLD

Central wavelength of SLD

Values

2.4 lm

100000

6 mm/s

250 mm

15483 Hz

36.5 nm

1558.12 nm

less than 1%, that the result shows essentially the birefringence as the function of wavelength. In calculation, nb = 7.9  104 at 1550 nm is used as the initial value (given by datasheet), birefringence at all wavelength can be analyzed. The result yields the curve in Fig. 9. In Fig. 9, the slopes of the measurement and simulation curves are identical closely. The inter-mode chromatic dispersion coefficient is 1.188 ps/(km nm) based on measured birefringence. The result matches the simulation well. In this paper, the dispersion is measured at room temperature (25 °C). As the dispersion is closely associated with the effective modal birefringence in (3), the temperature dependent birefringence coefficient dDn/dT for PM-PCF is 1.24  108/°C [14], about 40 times less than that in PANDA fiber. PM-PCF’s extra-low thermal sensitivity and geometry-induced birefringence well withstand the influence of pressure and temperature fluctuation on its dispersion coefficient. 3.2. Cross-coupling measurement Fig. 11.1. Dispersion effect in PM-PCF.

A 3.7-m length PM-PCF and a 19.8-m length PANDA-PM fiber were connected through splicing in the experiment. A SLD source at 1550 nm with a short coherence length about 32 lm was coupled into a polarizer to make sure only one mode excited and then collimated into the slow axis of PANDA-PM fiber. To create a coupling point, an 8° mismatched angle was designed in alignment. The coupling mode was generated at the fusion point and propagated along the PM-PCF. The theoretical coupling strength corresponding to an angular mismatch h is given by

h ¼ 20 log½tanðhÞ

ð11Þ

The coupling strength of 8° is 17.04 dB. Then output interferogram shows the interferogram broadening due to the large dispersion of PCF. To compare with the dispersion effect in PANDA-PM fiber, the SLD source was designed to couple into the opposite port of FUT (fiber under test). The coupling mode was generated at the same fusion point and propagated through an inverse optical path, accumulating the dispersion along the PANDA fiber instead. The intermode chromatic dispersion of PANDA-PM fiber is 0.0014 ps/ (km nm) [15], and the interferogram broadening can be neglected. Configuration of experiment setup is shown in Fig. 10. Two sections of fiber were connected according to a low loss fusion splicing method mentioned in [16]. The splicer is supplied by Vtran Co. Ltd. The axis of fiber was aligned with the polarizer by monitoring a polarization extinction ratio meter (Santec PEM320). Parameters of OCDP system is presented in Table 2. Figs. 11.1 and 11.2 show the interferogram influenced by dispersion effect in PM-PCF and PANDA-PM fiber respectively. The measured coupling strengths are 23.81 dB and 17.09 dB respectively. The 1/e widths are marked. The interferogram broadens notably in PM-PCF. As the dispersion effect is negligible in PANDA fiber, so the value 17.09 dB is much closer to theoretical

Fig. 11.2. Dispersion effect in PANDA fiber.

coupling strength. The adjacent couplings near the central maximum (OPD = 0) are produced by self-coherent phenomenon of SLD source. 4. Results According to the dispersion coefficient 1.188 ps/(km nm) of PMPCF obtained in wavelength scanning experiment, Figs. 12.1 and 12.2 shows the results before and after compensation. The 1/e

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5. Conclusion For PM-PCF, notable interferogram broadening affects the precision of cross-coupling measurement by OCDP, due to intermode chromatic dispersion. The dispersion is found to be much larger than that of stress-induced hi-bi fiber. A frequency domain compensation method has been proposed and verified for mitigating the influence. The method includes a numerical algorithm and measurement of dispersion coefficient for the offset-phase packet in the algorithm. The dispersion coefficient is measured by wavelength scanning method. In addition, the novel compensation method provides advantage over traditional ones. No parameters of light source are required precisely and curve fitting procedure is saved. Further, the method is not affected by interferogram asymmetry. These advantages show the potential for the inspection of PM-PCF coils in FOG’s application. Acknowledgements

Fig. 12.1. Before compensation.

The author would like to thank David Yang for insightful discussions and technical support operating OCDP. This work was supported by the National Natural Science Foundation of China under Grant 61007040. References

Fig. 12.2. After compensation.

Table 3 Compensated strength and relative error. Mismatch (degree)

Theoretical strength (dB)

Measured strength (dB)

Compensated strength (dB)

Relative error (%)

2 4 6 8 10

29.14 23.11 19.57 17.04 15.07

35.84 29.71 26.41 23.81 21.59

29.24 23.18 19.62 17.09 15.12

0.34 0.32 0.28 0.29 0.32

widths of interferogram are 150.5 lm and 36.7 lm respectively, decreasing 4.1-fold. The spatial resolution LR of OCDP system is recovered from 19.05 cm to 4.64 cm, giving nb = 7.9  104 in the relationship LR = Lc/nb. After compensation, the coupling strength is rectified to h = 17.086 dB which exists relative error less than 0.29% with the theoretical value. Repetitive experiments were conducted with mismatched angle ranging from 2° to 10°. The results are listed in Table 3. The average relative error is 0.31%.

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