1. Introduction

Chromatic dispersion originates from the variation of the refractive index of an optical fiber or an optical device as a function of wavelength. It is an essential characteristic for optical fibers and many other photonic devices, as it is a significant parameter that affects the bandwidth of a high speed optical transmission system through pulse broadening and nonlinear optical distortion. Additionally the chromatic dispersion slope or the second order dispersion (SOD) of a fiber is an important parameter in a wavelength division multiplexing (WDM) system. If SOD is not accurately managed with respect to all wavelengths in a WDM system, “walk off” may occur when dispersion accumulates at the highest and lowest wavelength channels, potentially causing signal distortion[1–2].

The time-of-flight method and the modulation phase shift (MPS) method are two widely used conventional chromatic dispersion measurement methods [3]. These methods were developed for long optical fiber samples. The time-of-flight method is a way to measure relative temporal delays for pulses at different wavelengths, whereas the MPS technique measures the phase delay of a modulated signal as a function of wavelength. Even though the MPS method is a standard technique adapted in most of optical fiber manufacturing companies, it has a number of drawbacks. First, the resolution of the method is restricted due to the wavelength stability and the jitter of a laser source [4]. Second, it requires a complicated experimental setup and expensive equipments such as a high-speed optical modulator, a detector and an optical tunable filter. Finally, it cannot be used for determining the chromatic dispersion of a short length optical fiber.

For the dispersion measurement of a short length fiber, white light interferometry (WLI) is normally used, where a sample fiber is put in one arm of an interferometer, and a cross-correlation interferogram is obtained with a broadband light source. Two different approaches were normally used to obtain an interferogram: temporal WLI and spectral WLI. In temporal WLI an interferogram is measured as a function of time by moving one arm of an interferometer with a constant speed. The chromatic dispersion of a sample fiber can be acquired by doing Fourier transformation of the interferogram. Many different versions of this technique have been proposed due to its simple experimental setup and high resolution in dispersion measurement [5–8]. Even though most of problems in temporal WLI such as the external effects of temperature and air fluctuations during experiments have been solved by Shellee et al. [6], a temporal WLI system is considered to be vulnerable to external noise sources associated with the scanning arm of an interferometer. Spectral WLI has many advantages over the temporal WLI and is used in various optical measurement applications such as dispersion measurement for photonic devices [9–12], and depth resolved optical imaging [13–17]. This can be inherently possible due to parallel data acquisition for spectral interferogram. Measurement time to obtain a spectral interferogram is usually less than 500 ms by using an optical spectrum analyzer. Because there is no moving part during a measurement in its experimental setup, the spectral WLI is invulnerable to external environments. However, there is a major limitation in dispersion measurements by using the spectral WLI; reliable dispersion measurement results can be obtained only near the zero-dispersion wavelength of a sample [18, 19].

In this paper, we present a powerful new technique to measure the chromatic dispersion and the SOD of a short length fiber sample by using a spectral interferometer with a broadband light source and an optical spectrum analyzer (OSA). Due to an effective phase retrieval technique used in our spectral WLI, we have demonstrated that the chromatic dispersion of a sample fiber can be obtained at wavelengths far from the ZDW of a sample. The basic operational principle of our proposed wavelength domain interferometry is quite similar to that of Fourier domain optical coherence tomography (FD-OCT), which is a very hot topic in OCT [20–23]. In an FD-OCT system people are only interested in retrieving amplitudes of interference fringes in wavelength domain. Asour proposed method provide a very accurate way for retrieving phase as well as amplitude of interference fringes, we expect that the basic principles we provide can be applied in error analysis of FD-OCT or in phase-sensitive OCT system.

2. Theory

Our proposed method is basically a spectral WLI with a light emitting diode as a white light source and a spectrometer as a parallel detection system. When a sample fiber is put in one arm of a Mach-Zehnder interferometer, we measure cross-correlation interferogram in the frequency-domain. The measured spectrogram contains the spectral phase information of a sample as a function of optical frequency. The frequency dependent spectral interferogram can be generally written as

where f is optical frequency, 〈|E(f)|²〉 is the spectral intensity of a broadband source, a ² is relative optical power for the transmitted optical signal through a tested fiber, ϕ(f) is relative phase between a reference signal and a transmitted signal through a tested fiber, and < > denotes an ensemble average. The relative phase can be expressed as ϕ(f)=β(f)·L-β ₀·L ₀ where β(f) is the propagation constant of transmitted light in a tested fiber, L is the length of the tested fiber, β ₀ is the propagation constant in vacuum, L ₀ is the length of the reference arm in an interferometer. When c is the speed of light in vacuum, n(f) is the refractive index of a test fiber, we have β(f)≡2π/c·n(f)·f and β ₀·L ₀=2π/λ·L ₀=2πτ ₀·f where have used a relation λf=c between optical frequency f and optical wavelength λ. Delay time associated with the reference arm of an interferometer is defined as τ ₀≡L ₀/c. This can be controlled arbitrarily by adjusting a translation stage in the reference arm of an interferometer. Then, the relative phase can be expressed as

If we take the derivative of the relative phase with respect to optical frequency, we obtain

where we have used

v_g(f) is the group velocity of light in a sample fiber, and τ_g(f) is the group delay after transmitting through a given sample fiber whose length is L. The chromatic dispersion coefficient D(λ) of a fiber is the variation in the group delay with respect to wavelength per unit length of a fiber, D(λ)=(1/L)·∂τ_g(λ)/∂λ. In our experiment, we retrieve phase ϕ(f) directly from a spectral interferogram 〈I(f)〉 and calculate the group delay τ_g(f), dispersion coefficient D(λ), and second order dispersion coefficient dD(λ)/dλ of a tested fiber from the relative phase function.

3. Experiments and results

Figure 1 shows the schematic diagram of our proposed experimental setup to measure chromatic dispersion in an optical fiber. The configuration was based on a fiber Mach-Zehnder interferometer with a broadband light source and an optical spectrum analyzer (OSA). A light emitting diode (LED) is used as a broadband light source with a center wavelength of 1550 nm and a 3-dB bandwidth of 50 nm. A 3-dB fiber coupler is used to split light into two different paths. A fiber polarization controller (PC) was used to obtain the maximum visibility in an interferometric fringe pattern. A test fiber is inserted into one arm of the interferometer and the other arm has free space propagation with two fiber collimators. The length of the free space delay line can be adjusted by a translating stage driven with a micrometer on which a fiber collimator was mounted. As can be seen in Eq. (2), the modulation period in the measured interferogram can be varied by changing path length in the reference arm. A 55 cm-long single mode fiber (SMF, SMF28) from Corning Corporation and dispersion shifted fiber (DSF) from Sumitomo Electric were used as sample fibers. Two transmitted optical signals are combined to form a cross-correlation interferogram. The optical interference signal in the spectral domain was measured with an OSA and transferred to a personal computer for numerical processing. The chromatic dispersion of a sample fiber can be obtained by measuring the wavelength dependent phase in the spectral interferogram.

 

Fig. 1. Experimental setup for the chromatic dispersion measurement of an optical fiber

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Figure 2 shows a measured spectral interference pattern for a sample fiber. Figure 2(a) is the normalized spectral interferogram obtained by the OSA, and Fig. 2(b) is the closeup view of Fig. 2(a) near 1550 nm wavelength. From Eq. (1), the phase difference between adjacent positive fringe peaks becomes 2π. All peak positions were calculated directly from the spectral fringe interferogram. A polynomial curve fit was used to determine the wavelength of each peak. After finding the wavelength of each positive oscillation peak, we have generated a discrete relative phase function ϕ(λ) with respect to wavelength. Then, the calculated relative phase function was converted from the wavelength domain into the frequency domain to obtain ϕ(f) in Eq. (2). Note that the frequency axis in the converted function is not regular while wavelength difference between data points in ϕ(λ) is constant. We have used a cubic spline fitting processing to recalculate a regularly spaced phase function ϕ(f) in the frequency domain [17]. Figure 3(a) shows the recalculated phase spectra and relative group delay spectra in the frequency domain. Black square dots are the calculated equally spaced relative phase function. A red line is a fitting curve with a least square algorithm for a simple 3rd order Taylor expansion form:

 

Fig. 2. (a). A measured spectral interferogram obtained by an OSA and (b). its close-up view at about 1550 nm wavelength

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Note that in Fig. 3(a), the group delay τ_g(f) of a fiber is proportional to the first derivative of the calculated ϕ(f) and the dispersion D(f) of a fiber depends on the second derivative of ϕ(f). Blue empty squares In Fig. 3(a) are calculated from the first order derivative of the relative phase function, which is

with the same parameters obtained for the relative phase function of Eq. (5). The zerodispersion frequency f ₀ is the frequency at which the group delay is minimum, and it can be calculated to be f ₀=-ϕ₂/ϕ₃ from Eq. (6). Therefore, we can calculate the zerodispersion frequency or the zero-dispersion wavelength of a sample fiber as long as we find ϕ₂, ϕ₃ coefficients with high accuracy by doing a least square curve fitting process for measured phase spectrum data ϕ(f). When we replace f with c/λ in Eq. (6) and take the first order derivative with respect to λ, we have an expression for the chromatic dispersion coefficient of a sample fiber

Figure 3(b) shows the chromatic dispersion coefficient D(λ) of a sample fiber obtained with our proposed method. It also shows the second order dispersion coefficient dD(λ)/dλ which signifies the dispersion slop of the sample fiber. Red empty dots are measurement results by a conventional MPS method for a 1km length of same fiber. The measurement result has in good agreement with those of the conventional measurement method. We have also measured the chromatic dispersion and the second order dispersion of another optical fiber sample: DSF (Sumitomo Electric). The length of the sample fiber was only 550 mm. Figure 4(a) is the measured spectral interferogram obtained by an OSA. It was normalized by its maximum intensity. Figure 4(b) is the close-up view of Fig. 4(a) near 1550 nm wavelength. A 3rd polynomial curve fitting method was used to determine the wavelength of each peak in Fig. 4(b). Figure 5(a) shows the calculated phase spectra and relative group delay spectra for the DSF sample. Black empty squares are equally spaced phase function. A fitted function with third order polynomials is drawn with a red line. The relative group delay of the DSF sample is calculated from the first order derivative of the fitting curve for the relative phase function, and it is drawn with empty

 

Fig. 3. (a). Regularly spaced relative phase and calculated relative group delay in the frequency domain and (b). The chromatic dispersion coefficient and the second order dispersion coefficient of a sample fiber with our proposed method

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blue circles. As the dispersion zero frequency is inside of our measurement range, there exists an extremum point in the relative group delay.

 

Fig. 4. (a). A measured spectral interferogram measured by an OSA and (b). its close-up view near 1550 nm wavelength.

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The calculated chromatic dispersion coefficient D(λ) of the DSF is shown in Fig. 5 (b) with a black line. Red empty dots are few discrete measurement results obtained by a commercially available dispersion measurement system based on the MPS method. It shows that these two different measurement results are in good agreement with each other. In the commercially available measurement system, they have used a five-term Sellmeier equation to fit the group delay as a function of wavelength near the zero dispersion wavelength of the DSF sample [24, 25], whereas our proposed method used third order polynomial fitting equation in the frequency domain. There exists a small variation in measurement results for the commercial dispersion measurement system depending on the selection of fitting program. The second order dispersion coefficient dD(λ)/dλ obtained with our proposed method is drawn with a blue solid line in Fig. 5(b). The dispersion slope at 1550 nm wavelength with both methods was found to be the same: 0.066 ps/nm²/km. This confirms that our proposed method can determine the second order dispersion coefficient with a reliable accuracy.

 

Fig. 5. (a). The calculated phase spectra, and its polynomial fitting curve. (b) The chromatic dispersion coefficient and the second order dispersion coefficient of a DSF sample.

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Our propose measurement method is a very simple and effective to directly retrieve phase information on the measured spectral interferogram without a complicated signal processing procedure. It uses only a basic principle that phase difference between adjacent positive fringe peaks from the measured spectral interferogram is 2π. We believe that our proposed method is one of the most effective and simplest method for measuring relative group delay, chromatic dispersion and SOD simultaneously. As our measurement method is based on spectral domain measurement technique with a spectrum analyzer, the signal to noise ratio can be achieved at least 80 dB [26] such that a reliable interferogram or an accurate relative phase function can be achieved over a wide range of spectrum for a given broad band light source. In our experiment, a reliable interferogram data was obtained over 200 nm wavelength range when the full width half maximum of our LED light source was only about 50 nm. When a very wide band optical source such as a supercontinuum source is used, we expect to obtain the dispersion information of a sample over 400 nm wavelength range with a single measurement.

Our spectral interferometric measurement setup consists of two fiber-type couplers, and only one arm of the interferometer has a polarization controller. The polarization controller is to obtain the maximum visibility in an interferogram data measured by an OSA. We believe that our proposed method is not sensitive with respect to the polarization mode dispersion of a sample or the visibility change of an interferogram as a function of wavelength. This is because our measurement method measures the relative phase, and the variation of fringe visibility in a spectral interferogram does influence the relative phase of the interferogram very little. For our measurements, we measure the spectral fringe pattern within 500 ms in order to reduce any environmental effect such as temperature fluctuation and polarization state variation during measurement, which might decrease the accuracy of our measurement. In generally, the dispersion of a strongly dispersive material is hard to be measured with an interferometric technique at wavelengths far from the ZDW of a sample because the interference fringes become too dense to be resolved. However, as can be seen in Eq (2) or Eq. (3) the spectral modulation density in our proposed method can be easily adjusted by changing the length of the reference arm of our interferometer or τ_o. Therefore, our proposed method can be used to measure a sample with wide range dispersion.

3. Conclusion

Even though spectral WLI has been around for many years in various optical measurement applications, there is a considerable limitation in applying this technique to chromatic dispersion measurement when a sample is a strongly dispersive material whose ZDW is off the spectral range of a broad band light source used in the experiment. Therefore reliable dispersion measurement results can be obtained only near the ZDW point of a sample. In this paper, we have demonstrated that this limitation can be overcome by processing data in frequency domain with 3rd order polynomial fitting equation given in Eq. (5). We have presented a novel chromatic dispersion measurement and second order dispersion measurement techniques for an optical fiber or short length optical devices in detail. By using a simple Mach-Zehnder interferometer, we have directly obtained the phase spectra from a measured spectral interferogram. We have demonstrated the validity of the proposed method to determine chromatic dispersion and dispersion slope of two single mode optical fibers: an SMF and a DSF. This measurement method is simple, rapid (< 500 ms), highly sensitive (< 80 dB) and accurate for determining the chromatic dispersion and the second order dispersion of a sample regardless of the ZDW of a sample.