1. Introduction

Various techniques have been developed to measure the dispersion of optical fibers [1–4]. They are typically based on measuring either the phase response of the fiber at different wavelengths as in white-light interferometry or the group-delay response as in pulse delay and phase-shift techniques. The dispersion is then obtained by differentiation of the measured quantity. The measurement accuracy is limited by the differentiation operation which amplifies the measurement noise. In addition, in order to obtain the dispersion value at a particular wavelength, the measurements must be performed over a proper wavelength interval. Furthermore, the interferometric techniques are very sensitive to the environmental conditions and the pulse delay method requires fiber samples of tens of centimeters to kilometer length, depending on the absolute dispersion values, the precision required and the width of the pulses.

The recently developed microstructured fibers (MFs) exhibit a range of unique optical properties. For instance, their dispersion characteristics may be tailored in wide ranges making these fibers interesting for novel applications in photonic technology. In the literature, several ways to measure the dispersion properties of these fibers have been applied [5–8]. For very small-core MFs it has turned out to be particularly difficult to use, e.g., the traditional white-light interferometric technique due low coupling efficiency between the white-light source and the fiber which results in a poor signal-to-noise ratios.

2. Theory

When short pulses with wavelengths in the anomalous dispersion region of the MF are launched into the fiber at sufficient pulse energy, a strong modulation is observed in the optical spectrum of the fiber output. The modulation results from interference between a soliton wave and a dispersive field radiated when the input pulse evolves into an asymptotic soliton [9]. The same phenomenon is responsible for the presence of sharp sidebands in the pulse spectrum of soliton fiber lasers [10,11]. This effect can be advantageously used to measure the dispersion of the fiber, as the period of the modulation depends, to the first order, directly on the dispersion of the fiber. The method requires one spectral measurement to obtain the dispersion value at a given wavelength. However, since the method is based on the propagation of a soliton pulse in the fiber, the technique is limited to the anomalous dispersion region of the fiber. Also the length of the fiber is required to exceed a few dispersion lengths, in order that clear oscillations would appear in the spectrum. In this letter, we apply the method to measure the anomalous dispersion of a highly-birefringent MF and a very small-core MF.

Including only the lowest-order contributions, the propagation of optical pulses in a lossless optical fiber can be described by the nonlinear Schrödinger equation

which is written here for wavelengths falling in the anomalous dispersion region in the retarded time frame using the appropriate normalized units [12]. The parameter N is the soliton number defined by N² =γP_P$T_{\mathit{0}}^{\mathit{2}}$ /|β₂ | with T₀ being the temporal width of the input pulse, P_p its peak power, and β₂ and γ representing the group-velocity dispersion and the nonlinear coefficient of the fiber, respectively. When the effects of dispersion and nonlinearity are in balance, i.e., when N=1, the pulse propagates as a fundamental soliton which maintains its shape along propagation [12]. For input pulse parameters such that N=1±ε with ε∈[-1/2,1/2], the pulse evolves asymptotically into a fundamental soliton, and during the reshaping process, a dispersive wave is stripped off from the pulse [13]. Interference between this dispersive wave and the soliton is seen as oscillations in the output spectrum [9].

The spectral phase difference between the soliton and the dispersive field depends on both the propagation distance L and the wavelength λ₀ of the input pulse as [9]

where c is the speed of light in vacuum. By making use of the fact that the phase difference between two consecutive maxima at wavelengths λ₁ and λ₂ in the optical spectrum is equal to 2π, Δφ(λ₂ )-Δφ(λ₁ )=2π, the dispersion parameter at the pulse wavelength λ₀ can be approximated as

where L and c are expressed in m and m·s^-1, respectively. In the derivation of Eq. (3), the contributions of the tan ^-1 terms to Δϕ are assumed to cancel each other. This is a good approximation for ε≠-0.5 and fiber lengths L which exceeds two soliton periods, i.e., L>${\pi T}_{\mathit{0}}^{\mathit{2}}$ /|β ₂ |. Furthermore, in order to be able to apply Eq. (3), at least two oscillations need to be present on either side of the spectrum. This implies that the period of the oscillations should be much shorter than the width of the hyperbolic-secant shaped spectral envelope. This condition is again fulfilled for L>${\pi T}_{\mathit{0}}^{\mathit{2}}$ /|β₂ |. In practice, for the laser pulse widths in the range of hundreds of fs, the length of the fiber must typically be longer than tens of centimeters. The accuracy of the method is limited by the fact that the tan^-1 terms of Eq. (2) are neglected in the calculation of the dispersion. Numerical calculations show that the accuracy is within 10% and improves as the ratio of L to the soliton period increases.

3. Experiments

We have applied the technique to measure the dispersion of two MFs. For this purpose, the 80-MHz pulse train from a Ti:Sapphire laser (Spectra Physics/Tsunami Lite) operating in the 720–860 nm wavelength range was coupled into the MF. The laser produces linearly polarized pulses with a T₀ of approximately 60 fs. The spectrum of the pulses after propagation through the MF was recorded using an optical spectrum analyzer (Ando/AQ6315). The input power was gradually increased until oscillations were observed in the spectrum at the output of the MF. The first fiber under test was a 4.5-m long highly-birefringent MF (Crystal Fibre A/S/NL·PM·700) with an elliptical core of dimensions 1.5×2.4 µm². By inserting a half-wave plate in the beam path, we could rotate the linear polarization of the input pulses and measure the dispersion along both principal axes. A typical experimental spectrum obtained at the output of the MF as well as the dispersion deduced from the data for both axes are presented in Fig. 1. For comparison, the dispersion was also measured with a low-coherence white-light interferometer using an unpolarized source. In that case, the dispersion values were obtained from the phase of the Fourier transform of the interferogram [2] by fitting a sixth-order polynomial to the phase data over the wavelength range of interest. The dispersion profile was then obtained by double-differentiation of the phase. The measured phase changes induced by the fiber represent an average for the two orthogonal polarizations and, therefore, yield a polarization-averaged value for the dispersion. The measurement error for the interferometeric measurement is estimated to be around 10 %. The dispersion values are displayed in Fig. 1 as a solid line and fall nicely in between the fast and slow axis dispersion values obtained with the spectral modulation technique.

 

Fig. 1. Measured dispersion values for the highly-birefringent MF. The red squares and blue triangles represent the values obtained using the spectral modulation technique. The solid line indicates the measurement results obtained with white-light interferometry. Inset: typical spectrum recorded at the fiber output.

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As another example, we measured the dispersion of a 1.5-m long MF with a small-core diameter of 1-µm (Crystal Fibre A/S/NL·1.0·600). Spectra of the pulses at the output of the fiber for increasing input power are shown in Fig. 2. As the power is increased, the envelope of the spectrum broadens and the amplitude of the oscillation is enhanced. At higher power levels the central part of the spectrum becomes asymmetric due to Raman scattering which shifts the central peak to the red. For this reason, the laser should be operated at a power level for which the intra-pulse Raman scattering is negligible.

We repeated the measurement for four wavelengths within the tuning range of the laser. The obtained dispersion values are given in Fig. 3.

 

Fig. 2. Experimental spectra recorded at the output of the small-core MF for increasing input power from bottom to top. For clarity, an arbitrary vertical offset has been added to each curve. The peak power of the input pulses is of the order of a few watts.

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Fig. 3. Measured dispersion values for the 1-µm core MF.

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To gain further confidence in the experimental observations, we simulated the output spectrum, using Eq. (1), for pulses propagating in a 2-m long MF when ε=-0.2, -0.1, 0, 0.1 and 0.2. Figure 4 illustrates the results. In the simulations, the envelope of the input pulse is assumed to be of the form (1+ε)sech(t/T₀ ), and the dispersion, the nonlinear coefficient and T₀ are chosen to be 100 ps/nm·km, 100 W^-1·km^-1, and 60 fs, respectively. The spectra exhibit similar characteristics to those presented in Fig. 2. For an increasing input power, the envelope of the spectrum broadens, which corresponds to the narrowing of the pulse in the time domain. Indeed, the asymptotic soliton takes the form (1+2ε)sech[(1+2ε)t/T₀ ] [12]. Moreover, the amplitude of the oscillations on the wings of the spectra is seen to decrease as |ε| decreases. For ε=0 the pulse launched into the fiber is of the fundamental soliton form, and, consequently, no oscillations are observed.

 

Fig. 4. Simulated spectrum at the output of a 2-m long MF for various ε-values. For clarity, an arbitrary vertical offset has been added to each curve.

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Furthermore, to test the robustness of the technique, we replaced in the simulations the initial hyperbolic-secant pulse shape by a Gaussian shape of the form (1+ε)exp(-t ²/2$T_{\mathit{0}}^{\mathit{2}}$ ). The pulse propagation through a 4-m long MF was then simulated for various values of ε and T₀ . The simulations were repeated for two different dispersion values (40 and 100 ps/nm·km, respectively). The dispersion was then estimated from the simulated output spectrum using Eq. (3). The accuracy of the method was evaluated by comparing the dispersion values fed to the simulation to the values retrieved using Eq. (3). The discrepancy was found to be of the order of 10% for fiber lengths exceeding four soliton periods. For the Gaussian pulse this distance is longer than for the sech-shaped pulse. Further increase of the fiber length results in a better accuracy. The analysis shows that the method is rather insensitive to the form of the input pulse. We also investigated what effect various optical components inserted in between the laser and the fiber have on the measured result. For this purpose, we imposed a frequency chirp on the input pulse and evaluated the measurement error. The effect was found to stay within 5% for a chirp corresponding to a doubling of the input pulse width. Increasing the fiber length again results in a better accuracy.

4. Conclusion

In conclusion, we have developed a simple and robust technique to measure the anomalous dispersion of microstructured fibers. It is based on the spectral modulation of a short laser pulse when it reshapes into a soliton propagating along the fiber. To a good approximation, the dispersion of the fiber is related to the period of the spectral oscillations, and is, to the first order, independent of the power and temporal width of the input pulse. The method is particularly applicable to the characterization of small-core and birefringent MFs, allowing for direct measurement of the dispersion at the desired wavelength from a single pulse.