## Abstract

We present a white-light spectral interferometric technique employing a low-resolution spectrometer for measurement of the dispersion of the group and phase modal birefringence in an elliptical-core optical fiber over a wide spectral range. The technique utilizes a tandem configuration of a Michelson interferometer and the optical fiber to record a series of spectral interferograms and to measure the equalization wavelengths as a function of the optical path difference in the Michelson interferometer, or equivalently, the wavelength dependence of the group modal birefringence in the optical fiber. Applying a polynomial fit to the measured data, the wavelength dependence of the phase modal birefringence can also be determined.

© 2003 Optical Society of America

## 1. Introduction

Polarization-maintaining elliptical-core optical fibers have attracted considerable interest for a number of applications, including sensing of various physical quantities employing interferometric techniques. For these applications, it is important to know the dispersion of the phase and group modal birefringence in the sensing fiber. Several methods have been developed to measure the dispersion of birefringence in optical fibers over a wide spectral range. A wavelength scanning technique can be applied to either short [1] or long fibers [2]. A standard technique of time-domain tandem interferometry [3] uses processing of either a single interferogram [4] or a series of interferograms at different wavelengths [5, 6] recorded in a tandem interferometer.

Recently, a new measurement technique employing a low-resolution spectrometer at the output of a tandem configuration of a Michelson interferometer and a two-mode optical fiber has been used to measure the intermodal dispersion in circular-core [7] and elliptical-core [8] optical fibers. In comparison with the standard time-domain tandem interferometry our technique of spectral-domain tandem interferometry [9] uses a series of the recorded spectral interferograms to resolve the so-called equalization wavelengths [7, 8, 9] at which the overall group optical path difference (OPD) is zero and finally to obtain the wavelength dependence of the intermodal group OPD in a two-mode optical fiber.

In this paper, the technique of white-light spectral interferometry is extended to measure the dispersion of birefringence in an elliptical-core optical fiber over a wide spectral range. First, measuring the equalization wavelengths as a function of the OPD in a Michelson interferometer, the wavelength dependence of the group modal birefringence in the optical fiber is obtained. Applying a polynomial fit to the measured data, the wavelength dependence of the phase modal birefringence is also determined. The both wavelength dependences are used to compare the theoretical spectral interferogram with the recorded one and to confirm good agreement between theory and experiment.

## 2. Experimental method

Consider a tandem configuration of a nondispersive Michelson interferometer and an optical fiber under test of length *z* (see Fig. 1), which supports a guiding of the slow and fast polarization eigenmodes characterized by the wavelength-dependent propagation constants *β*_{s}
(λ) and *β*_{f}
(λ). The spectral intensity *I*(**R**,*Δ*
_{M}; λ) recorded by a spectrometer at the output of the tandem configuration at the position vector **R** in the transverse observation plane can be expressed in the following way [9]:

$$\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\times \mathrm{cos}[\Delta {\beta}_{s\phantom{\rule{.2em}{0ex}}f}\left(\lambda \right)z\pm \left(\frac{2\pi}{\lambda}\right){\Delta}_{M}]\},$$

where *I*
_{0}(**R**, λ) is the reference spectral intensity, *Δ*
_{M} is the OPD in the Michelson interferometer, *V*(**R**; λ) is a visibility term due to the wavelength-dependent overlap of both polarization eigenmodes, Δ*β*_{sf}
(λ)=*β*_{s}
(λ)-*β*_{f}
(λ) is the wavelength-dependent difference between the propagation constants of both polarization eigenmodes, ${\Delta}_{\mathit{\text{fs}}}^{\mathrm{g}}$(*z*; λ) is the wavelength-dependent differential group OPD in the optical fiber and Δλ_{R} denotes the width of the spectrometer response function. Equation (1) was derived assuming a Gaussian response function of the spectrometer and only the effect of the first-order modal dispersion. Suitable adjustment of the OPD *Δ*
_{M} in the Michelson interferometer means that the spectral interference fringes are resolved in the vicinity of the so-called equalization wavelength [8, 7] λ_{0} governed by the relations *Δ*
_{M}=-${\Delta}_{\mathit{\text{sf}}}^{\mathrm{g}}$(*z*; λ_{0}) and *Δ*
_{M}=${\Delta}_{\mathit{\text{sf}}}^{\mathrm{g}}$ (*z*; λ_{0}). Thus, the OPD *Δ*
_{M} adjusted in the interferometer and measured as a function of the equalization wavelength λ_{0} gives directly the spectral dependence of the differential group OPD ${\Delta}_{\mathit{\text{sf}}}^{\mathrm{g}}$(*z*; λ_{0}) or the group modal birefringence *G*(λ_{0})=${\Delta}_{\mathit{\text{sf}}}^{\mathrm{g}}$(*z*; λ_{0})/*z* in the optical fiber. Because the group modal birefringence *G*(λ) is related to the phase modal birefringence *B*(λ) via the expression *G*(λ)=*B*(λ)-λd*B*(λ)/dλ=-λ^{2}d[*B*(λ)/λ]/dλ, we can obtain the relative wavelength dependence of the phase modal birefringence. It can be combined with the known value at one specific wavelength to obtain absolute values of the wavelength dependence of the phase modal birefringence *B*(λ).

## 3. Experimental configuration

The experimental setup used to measure the dispersion of the group modal birefringence in an elliptical-core optical fiber is shown in Fig. 1. It consists of a white-light source, a 20 W quartz tungsten halogen (QTH) lamp, an aperture with a collimating lens, a bulk-optic nondispersive Michelson interferometer with a beamsplitter and a compensating plate of the same thickness, and with a micropositioner connected to one of the mirrors, a polarizer, a microscope objective, an optical fiber under test, an analyzer, micropositioners at both ends of the optical fiber, a miniature fiber optic spectrometer S2000 with an A/D converter, and a personal computer. The fiber optic spectrometer S2000 (Ocean Optics, Inc.) of an asymmetric crossed Czerny-Turner design with the input and output focal lengths of 42 and 68 mm, respectively, has a spectral operation range from 350 to 1000 nm and contains a diffraction grating with 600 lines per millimeter, a 2048-element linear CCD-array detector with a Schott glass long-pass filter, a collection lens, and a read optical fiber. The wavelength of the spectrometer is calibrated so that a third-order polynomial relation between pixel number and wavelength is used.

Measurement of the dispersion of the group modal birefringence was performed for one sample of an elliptical-core optical fiber having length *z*=7.17 m, a cutoff wavelength for the *even* LP_{11} mode equal to 780 nm, a core-cladding refractive index difference of 0.026 at a wavelength of 630 nm (the core is doped with GeO_{2} of 18 mol% concentration and the cladding is made of fused silica) and core dimensions of 3.26×1.14 *µ*m. The polarizer and the analyzer were oriented at 45° relative to the fiber eigenaxes. Optical power reaching the CCD array from both polarization eigenmodes was optimized by adjusting an excitation of the optical fiber under test and by aligning its output with the input of the read optical fiber. The spectral resolution of the fiber optic spectrometer S2000 was limited in our case by the effective width of the light beam from the read optical fiber. We used the read optical fiber with a 50 *µ*m core diameter which results in a Gaussian response function with the width Δλ_{R}=2.7 nm.

## 4. Experimental results and discussion

After optimizing excitation and detection conditions to assure the highest visibility of spectral interference fringes, the spectral interferograms were recorded for the OPDs *Δ*
_{M} in the Michelson interferometer adjusted with a step of 20 *µ*m. We have revealed that the equalization wavelengths can be resolved over the spectral range approximately from 525 to 805 nm when the OPD *Δ*
_{M} in the Michelson interferometer varies from 1876 to 2796 *µ*m. An example of the recorded spectrum (markers) obtained for the OPD *Δ*
_{M}=2376 *µ*m is shown Fig. 2. We can clearly resolve the spectral interference fringes in the vicinity of the equalization wavelength λ_{0}=686.55 nm.

The measured group modal birefringence *G*(λ_{0}) in the elliptical-core optical fiber determined for respective equalization wavelengths λ_{0} is shown by markers in Fig. 3. The solid line in the same figure represents the group modal birefringence *G*(λ) obtained from the values -*G*(λ_{0})/${\mathrm{\lambda}}_{0}^{2}$ fitted to a fifth-order polynomial. The polynomial order is sufficiently high because the fit is characterized by a correlation factor as high as 0.99999. The corresponding absolute phase modal birefringence *B*(λ), with *B*(λ)/λ represented by a sixth-order polynomial, is shown in Fig. 4. It was obtained by combining the relative phase modal birefringence *B*(λ) with one absolute value *B*=2.58×10^{-4} measured at wavelength of 632.8 nm using a method of time-domain tandem interferometry [5, 6]. The precision of the group modal birefringence measurement by our technique is affected mainly by the precisions of adjusting the OPD *Δ*
_{M} and determining the equalization wavelength λ_{0}. In our case, the OPD is adjusted with a precision of better than 1 *µ*m and the equalization wavelength is determined with an error of 0.32 nm so that the group modal birefringence is measured with a precision of better than 0.1%.

Knowledge of the polynomial representations of both the phase modal birefringence *B*(λ) and the group modal birefringence *G*(λ) enables us to compare theoretical (1) and measured intensity distribution within the spectral interferograms. This procedure is illustrated in Fig. 2, in which the recorded spectrum is compared with the calculated spectrum given by Eq. (1). The fitting of the recorded spectral interferogram to the theoretical one was performed by a least-squares method, which gives the precise value of the OPD *Δ*
_{M}=2376.035 *µ*m and the spectral fringe visibility *V*(**R**; λ_{0})=0.50. We see from Fig. 2 very good agreement between theory and experiment.

## 5. Conclusions

We used a white-light spectral interferometric technique employing a low-resolution spectrometer to measure the dispersion of the phase and group modal birefringence in an elliptical-core optical fiber over a wide spectral range (525 to 805 nm). From a series of the spectral interferograms the equalization wavelengths as a function of the OPD in the Michelson interferometer, and thus the wavelength dependence of the differential group OPD in the optical fiber was obtained.We applied a polynomial fit to the measured data to obtain the wavelength dependences of the group and phase modal birefringence. We used both measured functions to compare the theoretical spectral interferograms to the recorded ones. They showed very good agreement. We demonstrated the applicability of the white-light spectral interferometric technique for dispersion characterizing of optical fibers guiding two polarization modes over a wide spectral range.

The described method offers high measurement precision of 0.1% achieved with simple and cost-effective instrumentation. It allows for simultaneous measurements of the group and phase modal birefringence in a wider spectral band, which can be further extended by applying the CCD array operating in another spectral range.

## Acknowledgments

This research was partially supported by the Grant Agency of the Czech Republic (project No. 202/01/0077).

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