Optical frequency-domain chromatic dispersion measurement method for higher-order modes in an optical fiber

We propose a new chromatic dispersion measurement method for the higher-order modes of an optical fiber using optical frequency modulated continuous-wave (FMCW) interferometry. An optical fiber which supports few excited modes was prepared for our experiments. Three different guiding modes of the fiber were identified by using far-field spatial beam profile measurements and confirmed with numerical mode analysis. By using the principle of a conventional FMWC interferometry with a tunable external cavity laser, we have demonstrated that the chromatic dispersion of a few-mode optical fiber can be obtained directly and quantitatively as well as qualitatively. We have also compared our measurement results with those of conventional modulation phase-shift method. ©2005 Optical Society of America OCIS codes: (260,2030) Chromatic dispersion; (120.3180) Interferometry References and links 1. A.H. Gnauck, L.D. Garrett, Y. Danziger, U. Levy and M. Tur, “Dispersion and dispersion-slope compensation of NZDSF over the entire C band using higher-order-mode fibre,” Electron. Lett. 36, 1946–1947 (2000). 2. R.I. Killey, V. Mikhailov, S. Appathurai, and P. Bayvel, “Investigation of nonlinear distortion in 40 Gb/s transmission with higher order mode fiber dispersion compensators,” J. Lightwave Technol. 20, 2282–2289 (2002). 3. S. Ramachandran, S. Ghalmi, S. Chandrasekhar, I. Ryazansky, M.F. Yan, F.V. Dimarcello, W.A. Reed, and P. Wisk, “Tunable dispersion compensators utilizing higher order mode fibers,” IEEE Photon. Technol. 15, 727– 729 (2003). 4. C. Dorrer and S. Ramachandran, “Self-referencing dispersion characterization of multimode structures using direct instantaneous frequency measurement,” IEEE Photon. Technol. 16, 1700-1702 (2004). 5. L. H. Jae, M. Oh, and Y. Kim, “Two-mode fiber-optic resonant ring interferometer as a sensor,” Opt. Lett. 15, 198-200 (1990). 6. A.M. Vengsarkar, W.C. Michie, L. Jankovic, B. Culshaw, and R.O. Claus, “Fiber-optic dual technique sensor for simultaneous measurement of strain and temperature,” J. Lightwave Technol. 12, 170-177 (1994). 7. C.E. Covington, J. Blake, and S.L.A. Carrara, “Two-mode fiber-optic bending sensor with temperature and stain compensation,” Opt. Lett. 19, 676-678 (1994). 8. T.-J. Chen, “Use of liquid-crystal-clad fiber as a modal filter for a two-mode fiber-optic interferometer,” Opt. Lett. 29, 2852-2854 (2004). 9. D. Menashe, M. Tur and Y. Danziger, “ Interferometric technique for measuring dispersion of high order modes in optical fibres,” Electron. Lett. 37, 1439-1440 (2001). 10. J.W. Nicholson, S. Ramachandran, S. Ghalmi, E.A. Monberg, F.V. DiMarcello, M.F. Yan, P. Wisk, and J.W. Fleming, “Electrical spectrum measurements of dispersion in higher order mode fibers,” IEEE Photon. Technol. 15, 831-833 (2003). 11. Y. Jaouën, C. Palavicini, A.-F. Obaton, C. Moreau, P. Silard, “Direct chromatic dispersion determination of higher-order mode fibers using OLCR technique,” in Proc. Conference on lasers and electro-optics 2005, CThB4, Baltimore, USA (2005). 12. D. Derickson, Fiber optic test and measurement, Hewlett-Packard professional books, Prentice Hall PTR, Upper Saddle River, New Jersey, 1998, USA, Chap. 11. #8660 $15.00 USD Received 6 September 2005; revised 25 November 2005; accepted 2 December 2005 (C) 2005 OSA 12 December 2005 / Vol. 13, No. 25 / OPTICS EXPRESS 10040 13. R. Passy, N. Gisin, J. P. von der Weid, and H. H. Gilgen, “Experimental and theoretical investigations of coherent OFDR with semiconductor laser sources,” J. Lightwave Technol. 12, 1622–1630 (1994). 14. M. Yoshida, K. Nakamura, and H. Ito, “A new method for measurement of group velocity dispersion of optical fibers by using a frequency-shifted feedback fiber laser,” IEEE Photon. Technol. Lett. 13, 227–229 (2001). 15. T.-J. Ahn, S. Moon, Y. Youk, Y. Jung, K. Oh, and D.Y. Kim, “Mode analysis and modal delay measurement of a few modes fiber by using optical frequency domain reflectometry based on 1550 nm TLS,” in Proc. Conference on lasers and electro-optics 2005, JthE5, Baltimore, USA (2005). 16. T.-J. Ahn, S. Moon, Y. Youk, Y. Jung, K. Oh, and D.Y. Kim, “New optical frequency domain differential mode delay measurement for a multimode optical fiber,” Opt. Express 13, 4005-4011 (2005). http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-11-4005 17. T. -J. Ahn and D. Y. Kim, "High-resolution differential mode delay measurement for a multimode optical fiber using a modified optical frequency domain reflectometer," Opt. Express 13, 8256-8262 (2005), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-20-8256 18. U. Glombitza and E. Brinkmeyer, “Coherent frequency-domain reflectometry for characterization of singlemode integrated-optical waveguides,” J. Lightwave Technol. 11, 1377–1384 (1993). 19. J.M. Senior, Optical fiber communications: Principles and practice, Prentice Hall Europe, 1992. 20. T.-J. Ahn, J. Y. Lee, and D.Y. Kim, “Suppression of nonlinear frequency sweep in an optical frequency domain reflectometer by using Hilbert transformation,” Appl. Opt. (to be published) 21. L. Jeunhomme and J.P. Pocholle, “Selective mode excitation of graded index optical fibers,” Appl. Opt. 3, 463468 (1978) 22. K. Shimizu, T. Horiguchi and Y. Koyamada, “Measurement of Rayleigh backscattering in single-mode fibers based on coherent OFDR employing a DFB laser diode.” IEEE Photon. Technol. Lett. 3, 1039–1041 (1991). 23. T. Okoshi, Optical fibers, Academic press, inc., San Diego, California, 1982. 24. C.E. Kerbage, B.J. Eggleton, P.S. Westbrook and R.S. Windeler, “Experimental and scalar beam propagation analysis of an air-silica microstructure fiber,” Opt. Express 7, 113-122, (2000). http://www.opticsexpress.org/abstract.cfm?URI=OPEX-7-3-113 25. K. Oh, S. Choi, Y. Jung, and J.W. Lee, “Novel hollow optical fibers and their applications in photonic devices for optical communications,” J. Lightwave Technol., 23, 524-532 (2005).


Introduction
Few-mode optical fibers (FMFs), also called higher order mode fibers, have been versatilely used in many applications for dispersion compensation in long-haul transmission systems [1][2][3][4], environmental sensors for strain, temperature, and hydrostatic pressure [5][6][7], and modal filters [8].The mode analysis and the chromatic dispersion measurement of FMFs are important to improve the performance of the optical devices and sensors based on FMFs.In particular, characterizing the chromatic dispersions and the dispersion slopes of each mode in a FMF is required for an effectively dispersion compensating FMF design.The chromatic dispersions of higher-order modes and their associated measurement techniques have been the subject of many recent studies [9][10][11].The group delay and dispersion of higher order modes in FMFs were obtained by measuring optical fringe spacing of an intermodal interferometer at different source wavelengths [9], but this measurement has a length limitation of a few meters of a sample fiber due to finite spectral bandwidths in tunable lasers used in this technique.In another method, the chromatic dispersions of the higher order modes of a long fiber in the order of several km length were evaluated by measuring wavelength-dependent side-lobe spacing in electrical spectrum using an intermodal interferometer [10].These measurement techniques can only determine the relative dispersion of the higher order mode of a higher order mode fiber which supports two modes [9,10].In order to overcome this drawback, Y. Jaouën et al. have recently suggested a phase-sensitive optical low-coherence reflectometer (OLCR) technique for chromatic dispersion measurement of FMFs [11].However, in this method, high quality phase measurement for each mode is required to measure the dispersion of a higher order mode fiber accurately.
In this paper, we propose a novel measurement technique for chromatic dispersions of all excited modes in optical fibers using an optical frequency-modulated continuous-wave (FMCW) interferometry with a tunable laser and a simple interferometer.Generally, an FMCW interferometry has been used to evaluate several characteristics of single mode optical fibers, such as fault positions, chromatic dispersion, and polarization mode dispersion [12][13][14].
Recently, we have reported an optical frequency-domain method to analyze the mode structure and the differential mode delay (DMD) of an MMF using an FMCW interferometry [15][16][17].There we have mentioned the possibility of identifying higher order modes in an FMF and their chromatic dispersion measurement through the analysis of a measured beating spectrum in an FMCW interferometry.In order to verify our idea, we have prepared a FMF which shows three LP guiding modes.By doing repeated experiments with a sample FMF while changing the center wavelength of the FMCW interferometic measurement, we have demonstrated that the chromatic dispersions of higher order modes can be determined very effectively and accurately with our proposed optical frequency domain technique.

Theory
The traditional FMCW interferometer system consists of a Michelson interferometer and a tunable laser source (TLS), in which a swept center frequency is utilized instead of a movable delay line which is used in a conventional optical low-coherence reflectometer (OLCR).In general, the usage of this FMCW interferometry is to determine the positions of irregularities in a single mode optical fiber (SMF) with high sensitivity and high spatial resolution.Frequency-swept coherent light is launched into each mode of a fiber under test (FUT), and transmitted lights from each mode of a FUT are interfered with another light from the reference arm of the interferometer.When the powers of these interfered lights are detected using a slow photo-detector, beating oscillations with frequencies proportional to time delays corresponding to optical path differences between the lights from the FUT and the light from the reference arm is generated.In case of a multimode optical fiber, each mode travels with a different propagation constant, and has a different group velocity.Therefore relative time delay, which corresponds to each mode in an MMF, can be measured with an FMCW interferometry [15][16][17].The chromatic dispersion of each mode in an MMF can be also obtained by measuring wavelength-dependent time delay in a FMCW interferometer by spanning the center wavelengths of the tunable laser.The wavelength-dependent a.c.detector voltage of an interfered signal in a FMCW interferometry is generally described by [18] { } where U m and ϕ m are the constant amplitude and phase of an interfered signal between the fundamental mode of the reference fiber and the m-th order mode of the FUT, respectively.N is the number of the excited modes in a multimode optical fiber.λ is the center wavelength of a tunable laser source.A frequency-tuning rate γ(λ) is a settable constant value of our tunable laser source.It represents the light frequency change per unit time.Relative group delay Δτ m is the difference between the time delay of the m-th excited mode of an MMF and the fundamental mode of a SMF as the reference fiber.It is simply given by where τ m and τ R are, respectively, wavelength-dependent time delays of m-th excited mode in an MMF and the fundamental mode of an SMF.f m (λ) is the beating frequency corresponding m-th excited mode obtained through the FFT of a measured beating signal.It is determined by dividing a beating frequency with the frequency-tuning rate at a certain wavelength λ in Eq. (2).Chromatic dispersion of the excited modes in a sample fiber, D m (λ), as a function of wavelength is obtained from a differential of time delay(τ m ) w.r.t.wavelength and dividing it by the length of the sample fiber (L), and is formulated as [19] ) where D R (λ) and ΔD m (λ) are, respectively, the chromatic dispersion of an SMF as a reference fiber and the relative chromatic dispersion difference between the m-th mode of a FMF and the fundamental mode of a reference fiber.Here we assume that fiber lengths of the FUT and the reference fiber are nearly the same.If we know the dispersion of a SMF used in the reference arm, the chromatic dispersions of all excited modes in a MMF can be determined by Eq. (3).After frequency-swept light from the TLS is split using a 3 dB optical coupler, fifty percent of the optical power goes into an auxiliary Michelson type fiber interferometer, as shown in a dashed box in Fig. 1.The auxiliary interferometer is used to monitor the nonlinearity of the frequency sweep rate γ as a function of time.By analyzing time-varying phase of a beating signal in the auxiliary interferometer with the help of Hilbert transformation compensation method, most of errors associated with the nonlinear frequency sweep rate of our TLS were effectively removed [20].A bulk Mach-Zehnder interferometer is used as the main interferometer, where the remaining optical power is split once again using a 3 dB fiber coupler.A wave propagated through a FUT is combined with another wave from the reference arm by using a bulky beam splitter, and they generate interference signals at a photo-detector.A homemade few mode fiber (FMF), of which the core diameter and the index difference were about 8 μm and 0.026, respectively, was used as a sample fiber under test.

Experiments and results
A dashed box at the bottom of Fig. 1 shows a butt coupling setup used between the 3 dB coupler and the FUT.Offset launching with a lateral displacement of 6 μm between the launching fiber and the FMF was adopted in order to have all modes in the FMF to be excited [21].A fiber polarization controller (PC) was placed at the reference arm of the interferometer in order to obtain the highest visibility in an interference signal detected by a photodiode (PD).The transmitted light through the reference SMF was focused with a lens at the PD whose detection area and bandwidth are 1 mm 2 and of 125 kHz, respectively.It was interfered with lights corresponding to the excited modes of a FMF by using a beam splitter as shown in Fig. 1.The sizes and positions of the two beams from the reference SMF and the test MMF were carefully adjusted to obtain the best visibility of interference signal data.The centers of two beams on the surface of the PD were offset slightly to obtain nonzero overlap integrals between the fundamental mode and the higher order odd modes.Low frequency beating signals detected with a PD were acquired by using a PD and a data acquisition board with a bandwidth of 1.2 MHz.Spectral peaks of the measured beating signal were obtained by Fourier transformation of the signal.The frequency components in the beating signal divided by the frequency tuning rate γ(λ) of a TLS correspond to the relative temporal delays or the relative group delays between the fundamental mode of a SMF in the reference arm of the interferometer and the guiding modes of a FUT in the test arm of the interferometer.The length of the SMF in the reference arm was almost the same as the length of FUT, which makes the frequency of the beating signal very low.This could decrease the phase noise in the measured beating signal [21].Here we used 40 m length of FMF as a sample fiber and 49 m length of SMF as a reference fiber.Figure 2 shows the relative group delays of three excited modes in our FMF measured at three different center wavelengths of 1524, 1540, and 1556 nm, respectively, using our FMCW interferometer method.The intensity is normalized to the maximum peak intensity of each measurement.The temporal resolution of this interferometer is about 13 ps as a full width half maximum of the peak in the beat spectrum.There are three peaks corresponding to the three guiding modes of the FMF in each measurement at three different wavelengths.We have verified that the FMF supports three guided modes corresponding to LP01, LP11, and LP21 modes.This was checked by observing the spatial beam profiles of the excited modes in the FMF by using an objective lens, an infrared Videcon camera (C2741-03, Hamamasu Inc.) and a 1550 nm laser source, as shown in the inset of Fig. 2. As cutoff wavelength of a higher order mode is shorter than that of a lower order mode, the positions of the three guiding modes in time domain can be easily identified by monitoring the bending loss of each mode in a sample fiber.Regardless of wavelength, the LP 01 mode is the one that arrives at the end of the fiber earlier than any other modes.Relative group delay of the LP 01 mode of the FMF was not changed much when the center wavelength of the 2 nm tuning TLS is varied from 1524 to 1556 nm.It is because the group delay of LP 01 mode in the FMF is almost as much as that of the fundamental mode of the SMF, which is used as a reference fiber in our experiment.Strictly speaking, relative group delay of the LP 01 mode has varied somewhat as the center wavelength is changed and it is used to estimate the relative dispersion of the LP 01 mode.The group delay of the LP 11 mode has increased with wavelength showing its positive dispersion in the FMF.On the contrary, the last peak shown in Fig. 2 moves to the left side as the center wavelength of the TLS is increased.This implies that the LP 21 mode has a negative dispersion.In addition, relative powers in the higher order modes of the FMF, LP 11 and LP 21 , are decreased with increasing wavelength.It is due to wavelength-dependent bending loss for higher order modes in a multimode fiber [19].
We have measured the relative group delays of the guiding modes in the FMF as function of wavelength by monitoring the peak spectral component of each mode, and are shown in Fig. 3(a).We have used our optical frequency-domain technique with the experimental setup shown in Fig. 1.The center wavelength of the TLS was scanned from 1520 nm to 1580 nm with 2 nm step size.For each center wavelength, the wavelength sweeping range of the TLS for each measurement was about 2 nm.It shows that the relative group delay of the LP 01 mode does not change much as a function of wavelength.The variations in the relative group delays of LP 11 and LP 21 modes are nearly the same but in the opposite direction.The relative group delay of LP 11 mode increases while that of LP 21 mode decreases as a function of wavelength.These two higher order modes meet at around 1575 nm wavelength, which means that the two modes have the same group velocity at this wavelength and reach the end of a fiber at the same time.We have theoretically calculated the group delays of a typical step-index FMF to verify our experimental measurement result in Fig. 3(a).This is also used to identify and sequence the observed modes in Fig. 3(a).The group delay (τ ) for a step-index multimode fiber is formulated as [23] [ ] ( ) where the parameter n 1 , called the group index, is the ratio of the speed of light to the group velocity of a plane wave in the core material.Parameter y is the difference between the material dispersion in core and cladding.The values, c and Δ are, respectively, the well known speed of light in vacuum and index difference between the core and the cladding of a step index fiber.The parameter x as functions of the normalized frequency (v) is defined as [23].
where k and β are a propagating constant and phase constant, respectively and where In this simulation, the index difference and the core radius of a step index fiber are set to 0.034 and 4 μm, respectively, which makes this step index fiber to support three LP modes.We assumed that n 1 is constant and y is zero.Group delays of this step-index fiber were calculated by using Eq. ( 4) for the three supporting modes as a function wavelength and are shown in Figure 3(b).In this simulation, the relative group delay of LP 01 mode is almost identical with the measured result of LP 01 mode shown in Fig. 3(a).Whereas there exist some discrepancies between the measured and the calculated relative group delays for LP 11 mode.We believe that this is because the refractive index profile of the FMF sample we have used is off from an ideal step-index profile, while our analytic solution for the group delay of LP 11 mode is for an ideal step-index fiber.The relative group delay of LP 21 mode decreases with wavelength, which is similar to the measured result of Fig. 3(a).Here, the group delay of the LP 01 mode of an SMF is used as a reference to the calculation of each group delay for higher order modes in a FMF.The crossing point between LP 11 and LP 21 modes in Fig. 3(b) matches that of Fig. 3(a) at around 1575 nm wavelength.We did a third order polynomial fitting for the data shown in Fig. 3(a).The relative chromatic dispersions of the excited modes in the FMF with the fiber length of 40 m were obtained by using the differential of the fitted curve and are shown in Fig. 4(a).To estimate the actual chromatic dispersion of the excited mode, we also measured the chromatic dispersion (D R ) of the reference SMF by using a commercial optical dispersion analyzer (Agilent 86038A), which is based on the conventional modulation phase-shift method.Measured chromatic dispersion of a 49 m long SMF by using the conventional modulation phase shift method is shown in Fig. 4(b).Using Eq. ( 4), the chromatic dispersions of the excited modes were calculated by adding the chromatic dispersion of the SMF to the measured relative chromatic dispersion of the FMF and are plotted in Fig. 4(a).Both LP 01 and LP 11 modes show positive dispersions and dispersion slopes, while the dispersion and the dispersion slope of LP 11 are much larger than those of LP 01 mode.Since LP 21 mode shows very large negative dispersion and negative dispersion slope, it can be used as a dispersion compensating fiber in long haul optical transmission systems [1][2][3][4].
Optical signal in the fundamental mode of an SMF is transformed into a higher order mode (like LP 21 , LP 02 etc.) in a FMF with negative dispersion by using mode converters such as long-period fiber gratings [3].After propagating a certain distance along the higher order mode of a FMF with negative dispersion, the signals are converted back to the fundamental mode of an SMF.The length of this dispersion compensating FMF needs to be properly chosen to compensate the positive dispersion effect of an SMF used in a transmission system.A proper negative dispersion slope is required to have simultaneous dispersion compensation in all channels of a wavelength domain multiplexed (WDM) transmission system [1].Therefore, it is important to measure the wavelength-dependent dispersion value and dispersion slope of higher order modes in the application of dispersion management and compensation across the whole transmission band of a WDM system.In particular, our optical frequency domain dispersion measurement method for higher order modes offers full dispersion information of all excited modes in an optical fiber.It requires a sample fiber length of a few meters.Large positive or negative dispersions in LP 11 and LP 21 modes can be used for optical pulse compression applications as well.In order to test the accuracy of our proposed method, the chromatic dispersion of the LP 01 mode of the sample FMF was obtained by the conventional modulation phase-shift method.It is measured after annihilating all higher order modes in the FMF by bending the fiber large enough.It was not possible to measure the dispersion of a higher order mode with the conventional modulation phase shift method because of mixed effects from different modes in a FMF.As shown in Fig. 4(b), the maximum difference between the chromatic dispersions measured by the conventional method and our optical frequency-domain method is less than 1 ps/km-nm at all operating wavelengths near 1550 nm.
Our optical frequency domain dispersion measurement method has several advantages for the measurement of the characteristics of higher order modes in a MMF.One is that it can give the information of the dispersions of all excited modes simultaneously with very high accuracy.Second major advantage of our method is that it can measure a sample fiber with large variety of its length.By increasing the frequency tuning rate of a laser source, we can measure a sample fiber with few meters which can not be measured by a conventional modulation phase shift method.We can also measure a few kilometer length sample fiber, which can not be measured with an interferometric method, by increasing the frequency tuning range of a laser source.Phase noise in measured data for a long fiber sample due to the finite laser frequency tuning range can be easily overcome by using a long reference fiber whose length is compatible to the sample fiber length.
In our measurement scheme, dispersions are obtained directly and intuitively by measuring the time delays of the modes with respect to wavelength.Another advantage is a simple experimental set-up consisting of a tunable laser source and a simple interferometer.Other methods need to measure the chromatic dispersion of the LP 01 mode of a sample FMF with a conventional phase modulation method to obtain the dispersion of a higher order mode for each sample FMF [9,10].However, our method does not need to measure the dispersion of the LP 01 for each sample FMF.Since the dispersion information of an SMF is well known, if we use a standard SMF as a reference fiber in our measurement setup, no further reference dispersion measurements would be required.Furthermore, several high performance commercial tunable lasers have been developed recently in many different wavelengths such that the measurable wavelength can be extended easily from infrared to visible light making it possible to measure dispersion of a multimode fiber from visible to near infrared wavelengths with our proposed method.In addition, our technique can be applied to the mode analysis for various specialty optical fibers such as photonic crystal fibers [24], hollow optical fibers [25] with many modes.We also expect this optical frequency domain measurement technique can be used for quantifying the environmental changes in the traditional intermodal interferometric sensors [5][6][7].

Conclusion
We have presented a new powerful chromatic dispersion measurement method for analyzing the excited modes of an optical fiber using a FMCW interferometer system.We have directly, quantitatively, and qualitatively measured the chromatic dispersions of three excited modes, LP 01 , LP 11 , and LP 21 in a homemade few-mode optical fiber by using this method.We have verified that LP 01 and LP 11 modes have positive dispersion, while LP 21 mode has negative dispersion, which can be used as a dispersion compensator.We have also demonstrated the possibility of using our frequency-domain measurement method with a FMCW interferometer as a powerful tool for mode analysis in a FMF.

Fig. 1 .Figure 1
Fig. 1.Experimental set-up for the optical frequency domain chromatic dispersion measurements of a multimode fiber.(TLS : tunable laser source, PD : photo-diode, PC : polarization controller, FUT : fiber under test, BS : beam splitter)Figure1is a schematic diagram of our experimental set-up to measure the chromatic dispersions of excited modes in a FMF using a transmission-type FMCW interferometer.An Agilent TLS 81640A tunable light source (TLS) was used with tuning range from 1520-1580 nm and tuning span (Δλ) of 2 nm as shown in the dotted box of Fig.1.The wavelength-tuning rate was set to 5 nm/s or γ = 625 GHz/s at 1550 nm wavelength defined as the slope of the swept instantaneous frequency as a function of time, which is illustrated like a sawtooth waveform in Fig.1.The optical power of the TLS was kept at 2 mW during the frequency tuning process.Beating signals were acquired using a data acquisition board with a triggering signal generated each time when the frequency sweep began.After frequency-swept light from the TLS is split using a 3 dB optical coupler, fifty percent of the optical power goes into an auxiliary Michelson type fiber interferometer, as shown in a dashed box in Fig.1.The auxiliary interferometer is used to monitor the nonlinearity of the frequency sweep rate γ as a function of time.By analyzing time-varying phase of a beating signal in the auxiliary interferometer with the help of Hilbert transformation compensation method, most of errors associated with the nonlinear frequency sweep rate of our TLS were effectively removed[20].A bulk Mach-Zehnder interferometer is used as the main interferometer, where the remaining optical power is split once again using a 3 dB fiber coupler.A wave propagated through a FUT is combined with another wave from the reference arm by using a bulky beam splitter, and they generate interference signals at a photo-detector.A homemade few mode fiber (FMF), of which the core diameter and the index difference were about 8 μm and 0.026, respectively, was used as a sample fiber under test.A dashed box at the bottom of Fig.1shows a butt coupling setup used between the 3 dB coupler and the FUT.Offset launching with a lateral displacement of 6 μm between the

Fig. 2 .
Fig. 2. Measured relative group delays for the excited modes of a FMF with respect to the fundamental mode of an SMF in three different wavelengths of 1524, 1540, and 1556 nm.

Fig. 3 .
Fig. 3. (a) Observed relative group delays of the excited modes in a FMF with respect to wavelength of light source using an OFDR and (b) Calculated relative group delay of three excited modes in a uniform-core fiber as function of wavelength.
parameter Q consists of m-th order Bessel function J m (u) as a function of u and modified Bessel function K m (w) as a function of w.It is expressed as[23]

Fig. 4 .
Fig. 4. (a) Chromatic dispersions of the excited modes in the FMF with respect to wavelength of light source and (b) dispersions of the fundamental mode in the FMF and SMF measured with modulation phase-shift method and our optical frequencydomain method