Fourier-domain mode delay measurement for multimode fibers using phase detection

Optics Communications 345 (2015) 26–30

Contents lists available at ScienceDirect

Optics Communications journal homepage: www.elsevier.com/locate/optcom

Fourier-domain mode delay measurement for multimode fibers using phase detection Chan-Young Kim, Tae-Jung Ahn n Department of Photonic Engineering, Chosun University, 375 Seosuk-dong, Dong-gu, Gwangju, South Korea

art ic l e i nf o

a b s t r a c t

Article history: Received 12 December 2014 Received in revised form 19 January 2015 Accepted 20 January 2015 Available online 22 January 2015

We have proposed a powerful method based on a phase detection reflectometric technique to solve the difficulty of the small signal discrimination in the amplitude-detection method for differential modal delay measurement of multimode optical fibers (MMFs). The phase is radically shifted to π at the time delay positions among the excited modes even when the amplitudes of the peaks cannot be distinguished with the noise level. The modal dispersion of the MMF under test can be simply determined by choosing the time delay in the last phase shift in the Fourier domain. In addition, we confirmed that the phase-sensitive interferometric measurement does not need to scramble the excited modes in the fiber. We subsequently conclude that a portable modal dispersion or mode analysis equipment can be developed by using the phase-detection intermodal interferometric technique proposed here. & 2015 Published by Elsevier B.V.

Keywords: Differential mode delay measurement Mode division multiplexing Optical frequency domain reflectometry Fourier transform

1. Introduction LOCAL area network (LAN) such as computer server system interconnection usually employ multimode optical fibers (MMFs) taking advantage of their high bandwidth compared to copper cable [1]. Recently, some kind of MMFs which have several propagation modes, named few-mode fibers (FMFs), have been employed in research on mode division multiplexing (MDM) communication for higher capacity [2–8]. In MDM communications, each mode plays a role of a channel in multiplexing. Several techniques for MDM communication have been reported in research [9–20]. Some techniques require very small modal dispersion [5–7], while others require large dispersion [8]. Differential mode delay (DMD) of an MMF is typically defined as the difference in group delay between the fastest and slowest mode per unit length [21]. It is strongly related to network performance for both conventional MMF-based local communication and MDM longhaul communications [22,23]. Modal dispersion measurement methods have been reported [21]. Typically, time-of-flight methods measure the spreading pulse through the fiber by using a short pulse laser, high-speed detection system in combination with a sampling oscilloscope or real-time data acquisition [21]. It is very expensive and complicate to build the time-of-flight measurement. In addition, it is difficult to bring the measurement system to the practical field. The n

Corresponding author. E-mail address: [email protected] (T.-J. Ahn).

http://dx.doi.org/10.1016/j.optcom.2015.01.056 0030-4018/& 2015 Published by Elsevier B.V.

portable and compact modal delay measurement equipment of the MMFs is needed in the MMF-based communication applications such as conventional LANs, MDM for long-haul networks, and polymer MMF communications for vehicles. To simplify the measurement system and to improve its signal detection sensitivity and resolution, fiber interferometric modal delay measurement techniques have been reported [24–29]. First research, among these, measures modal delay using interferogram which produced between fundamental mode propagated in reference arm and every mode in fiber under test (FUT) [26]. It provided comparable or much better resolution with simple and low-cost system comparing with the time-of-flight measurement. However, environments such as temperature and/or vibration affects the system to be unstable fluctuating optical path length difference between reference arm and FUT. In order to solve the problem, we have suggested to measure a modal delay with spectral interferometric scheme included an MMF under test as a fiber interferometer itself. The excited modes in the MMF play roles of the single-mode-fiber (SMF) based different arms in an interferometer. Stable interferogram from intermodal interference among the modes provides the good accurate differential modal delay information because the intermodal interferometer has only one fiber arm in which all propagation modes experience same fluctuation on even the environment is unstable [27,28]. In addition, high sensitivity of the interferometer facilitates detection of small signal which related to a higher order mode in the fiber. Utilizing the advantage, modal dispersion of FUT can be measured even using light reflected from the fiber facet [27,28]. However,

C.-Y. Kim, T.-J. Ahn / Optics Communications 345 (2015) 26–30

the returned signals of the higher order modes in the round-trip interferometric modal delay measurement system is not enough to measure modal dispersion accurately. It is difficult to distinguish the small returned signal to the background noises and to determine the last peak in the Fourier domain. In the paper, we have proposed phase-sensitive modal delay measurement based on intermodal interferometric technique to improve the discrimination of the returned signals related to mode beatings. The phase of an interference component between two modes among all propagating modes is radically shifted at the certain time corresponding to modal delay. Modal delay can be easily determined even when the excited modes have noise-like small intensities.

2. Theory Intermodal interferometric modal delay measurement technique is composed of a broadband light source, a spectrum analyzer, and a simple fiber interferometer, i.e. MMF under test [30]. Modal interference components can be achieved by taking an inverse Fourier transform of a measured interferogram (E (f )) in the spectral domain coming from the intermodal interference among the excited modes in the MMF. Generally, Fourier transformation of the interfered signal gives the real and imaginary parts which denote A and B, respectively. The relation is mathematically expressed by [30]

(1)

Generally, the amplitude ((A2 + B2)1/2) of the Fourier transform was used for determination of the last peak in the Fourier domain corresponding to the maximum modal delay of the MMF. Here we use the phase term (tan−1(B /A)) to discriminate the last peak even in noises. We simply confirm the phase change caused by the intermodal interference between two modes that have different optical path through the MMF. Summation of those two electrical field can be described to ... Note that f indicates optical frequency of the light, t is a time, and Δt is an arriving time difference between two modes. The real and imaginary parameter of inverse Fourier transform of the electric field can be described as follows:

A=

1 cos π (fmax + fmin )(t − Δt) π (t − Δt)

(

(

)

)

× sin π (fmax − fmin )(t − Δt) + g (t) 1 B= sin π (fmax + fmin )(t − Δt) π (t − Δt)

(

(

Fig. 1. Real and imaginary coordinate.

Eq. (5)), the sign of the function A becomes positive as well. On the other hand, the sign of the function B becomes negative and positive when the time t comes from a negative (Eq. (6)) and positive (Eq. (7)) infinite value, respectively (Fig. 1).

lim A = ( − )( + )( − ) = ( + )

t → Δt −

lim A = ( + )( + )( + ) = ( + )

t → Δt +

lim B = ( − )( − )( − ) = ( − )

t → Δt −

lim B = ( + )( + )( + ) = ( + )

F −1{E(f )} = A + jB

)

× sin π (fmax − fmin )(t − Δt) + h(t)

27

t → Δt +

(4) (5)

(6) (7)

While the function A has plus sign for both positive and negative approaching, the function B has opposite sign due to the different approaching. In consideration of the phase term (tan−1(B /A)), the left hand side of the phase at the reference Δt is minus value and the right hand side is plus value, as shown in Fig. 2. In other words, the phase of the modal interference between two modes radically changes at the time delay between two modes. Therefore, we can consider that the positions of the phase changes in Fourier domain are corresponding to modal delays among the excited modes in the MMF. The last phase change is determined to the maximum modal delay between the fastest and slowest mode.

3. Experiments and results

(2)

) (3)

Note that fmax and fmin are respectively the upper and lower integration range in the spectrum of the broadband source. The functions of g (t) and h(t) are independent to Δt . Here we consider the phase shift of the modal interference at t = Δt . In Eqs. (2) and (3), both A and B are three different term as function of t − Δt . The t − Δt term will be negative or positive value due to the value of t , so the A and B term also have negative or positive value determined by the sign of the t − Δt value. Eqs. (4)–(7) show the determined sign of the A and B terms. The function A is composed of multiplying an inverse function, a cosine function, and a sine function, whereas the function B is composed of multiplying an inverse function, a sine function, and the other sine function. In Eq. (4), the first, second, and third terms of the function A become negative, positive, and negative value, respectively and then the sign of the function A is determined to be positive when time t is approaching to Δt from a negative infinite value. When time t is approaching to Δt from a positive infinite value (as described in

3.1. Proof of concept Superluminescent light diode (EXS8510-2411, EXALOS Inc.) operating at the wavelength of 853 nm with 56 nm FWHM is used as a wide spectral light source. We used mini-spectrometer (HR4000, Ocean Optics Inc.) to analyze the spectral intensity of the interfered signal instead of a conventional optical spectrum analyzer. Its measurable wavelength is ranging from 753 to 932 nm and it provides the resolution of up to 0.05 nm at 850 nm wavelength. It enables the construction of a portable modal delay measurement or mode analysis equipment for the field tests.

Fig. 2. Phase change with respect to time delay.

28

C.-Y. Kim, T.-J. Ahn / Optics Communications 345 (2015) 26–30

Fig. 5. Schematic diagram of modal delay measurement. Fig. 3. Schematic diagram of Michelson interferometer for the proof of concept.

Firstly, we confirm the phase shift in Section 2. Fig. 3 shows a schematic diagram of Michelson interferometer designed to make a path difference between two beams assuming two modes for the proof of concept demonstration. Two divided beams at the beam splitter (BS) are reflected from M1 and M2 mirrors and then they combine into the spectrometer. It assumes that the path difference of two beams corresponds to the path difference between two modes in the MMF. The acquired sinusoidal interferogram in spectral domain by the min-spectrometer is Fourier transformed to both amplitude and phase with respect to time. Both the amplitude and phase radically change at the time delay between two modes. The amplitude-based interferometric measurement is typically used for the modal dispersion determination [28]. Here we propose the phase-based interferometric measurement to provide a good performance in peak discrimination. Phase is abruptly changing to π-shift at the time (corresponding to the arrival time difference between two beams) when the amplitude becomes the maximum. We measured the spectral intensity of the interferogram in the wavelength range from 840 nm to 880 nm. Fig. 4(a) shows the spectral intensity in the wavelength domain of the interferogram from the Michelson interferometer. The interferogram disappears when one arm is blocked. The frequency of the sinusoidal interferogram is consistent with the time delay between two beams with path length difference. Fig. 4(b) and (c) shows the intensity (square of amplitude) and phase of the inverse Fourier transform of the measured interferogram in Fig. 4(a), respectively. An abrupt intensity peak due to the interference at the time delay of 1 ps in

Fig. 4. Comparison of measurement results according to interference. (a) Spectral intensity, (b) amplitude and (c) phase.

Fig. 4(b) and the phase is also shifted to π at the same point of 1 ps as shown in Fig. 4(c). After the interfered point of 1 ps, the phase becomes 2π as same of the initial phase. Therefore, it turns out that the discrimination of the π-phase shift points is determined to modal delay of the MMF under test. It is expected that the phase shift is still detectable even when it is difficult to distinguish the small amplitude signal to noises in the Fourier domain. 3.2. Comparison of phase and amplitude Fig. 5 is a schematic diagram of modal dispersion measurement for MMFs. We simply replaced Michelson interferometer (in Fig. 3) with MMF (CPC6, Corning Inc.) under test. The manufacturer provides the bandwidth of 1363 MHz km at operating wavelength of 850 nm. Here, a mode scrambler (MS, FM-1, Newport Inc.) was used to excite all possible modes in the MMF. Modal delay is related to the time difference per unit length between the fastest and slowest mode. The excited modes are propagating along the MMF and then a part of the modes with weak intensity reflect from the end of the MMF. Generally in field tests, the reflectometric configuration is advantageous to test the MMF installed in a local area network. Fig. 6(a) shows the logarithm-scaled intensity and the phase with respect to time delay of the inverse Fourier transform of the interferogram. The phase is new factor that makes measurement more clearly compared to the previously reported paper. Measuring the dispersion, it is difficult to discriminate the last intensity peak even in log-scale due to very weak mode intensity, comparable to the noise

Fig. 6. Intermodal dispersion of MMF (a) intensity, (b) phase, and (c) phase (enlarged).

C.-Y. Kim, T.-J. Ahn / Optics Communications 345 (2015) 26–30

29

method. In the paper, we have proposed a powerful method based on a phase detection reflectometric technique to solve the difficulty of the small signal discrimination in the amplitude-detection method. The phase is radically shifted to π at the time delay positions among the excited modes even when the amplitudes of the peaks cannot be distinguished with the noise level. The modal dispersion of the MMF under test can be simply determined by choosing the time delay in the last phase shift in the Fourier domain. In addition, we confirmed that the phase-sensitive interferometric measurement does not need to scramble the excited modes in the fiber. We subsequently conclude that a portable modal dispersion or mode analysis equipment can be developed by using the phase-detection intermodal interferometric technique proposed here. The technique will be useful to apply for many industrial applications such as a conventional MMF-based LAN, mode division multiplexing using a few-mode fiber, optical sensing and so on. Fig. 7. Comparison of modal delay according to mode scrambling grade.

Acknowledgment level. On the other hand, the obtained phase is discretely changing until the fastest and slowest modes are interfered and the phase does not change in the region of no intermodal interference, as shown in Fig. 6(b). The last π-phase shift position can be easily determined to 0.566 ps/m as the maximum modal delay in Fig. 6(c) in enlarged scale. In recent report, the bandwidth expressed by bit-length (BL) product of the MMF can be estimated by inverse of the maximum modal delay at the time of the last intensity peak or phase shift [26]. The bandwidth of the MMF tested here is determined to 1325 MHz km which is consistent with 1363 MHz km provided from the manufacturer and measured by the traditional bandwidth measurement. The error is only 2.8%. Thus, it turns out that phase-sensitive modal delay measurement based on intermodal interference provides unparalleled performance in discrimination of the last peak corresponding to the modal delay that limits the fiber bandwidth. 3.3. Proof of intermodal interference discernment The mode scrambler (MS) in Fig. 5 is always needed in amplitude detection based modal delay interferometric measurements because the higher modes weakly excite when the input light is launching into the center of the MMF [26]. The MS enables the excitation of multiple modes in the fiber and the modes are interfered with each other. The highest mode possible to be excited in the fiber makes the last interfered peak amplitude with the fundamental mode. Therefore, the modal delay measured without the MS is generally smaller than that measured with the MS in case of the amplitude detection technique. We assume that the MS is not necessary in the phase detection technique which can recognize the interference between weak intensity modes. For the proof of concept demonstration, the maximum modal delays were measured in the phase detection technique with six different pitch grades of the MS. Fig. 7 indicates the averaged value and the standard deviation of the measured modal delay with respect to the mode scrambling grades in the MS. The values in the mode scrambling grades from 1 to 4 are similar to that without the MS. Thus, it turns out that the modal delay can be determined by the phase-sensitive interferometric measurement method without scrambling modes in the fiber. 4. Conclusion Modal dispersion of MMFs has been measured with not only timeof-flight method but also the amplitude-detection interferometric

This research was financially supported by Chosun University (2011), South Korea.

Appendix A. : Expansion of inverse Fourier transform for conceptual interferometer Here, we have derived Eq. (A2) as below. Electric fields of two different optical modes with the same amplitude and a different time delay (Δt ) are combined and interfered as propagating along the MMF. Total electric field can be expressed by

Etot = e j2πft + e j2πf (t +Δt)

(A1)

where f is the optical frequency of the light and j is complex conjugate. Inverse Fourier transformation of Eq. (A1) is corresponding to time delay distribution of the excited modes in the MMF. F −1{Etot (f )} = g (Δt) f

= =

∫f max (e j2πft + e j2πf (t +Δt))e−j2πf Δtdf min fmax

∫f

(e j2πf (t −Δt) + e j2πft )df

min

= =

1 ⎡e j2πf (t −Δt)⎤ fmax + 1 ⎡e j2πft ⎤ fmax ⎦f ⎦f 2πt ⎣ j2π (t − Δt) ⎣ min min 1 ⎡e j2πfmax (t − Δt) − e j2πfmin (t −Δt)⎤ ⎦ j2π (t − Δt) ⎣ 1 ⎡ j2πfmax t j2πfmin⎤ e + −e ⎦ 2πt ⎣

⎡ cos(2πf (t − Δt)) ⎤ max ⎥ ⎢ ⎢ +j sin(2πfmax (t − Δt)) ⎥ 1 ⎥ ⎢ = j2π (t − Δt) ⎢ − cos(2πfmin (t − Δt)) ⎥ ⎥ ⎢ ⎣ −jsin(2πfmin (t − Δt)) ⎦ +

⎤ ⎡ 1 ⎢ cos(2πfmax t) + j sin(2πfmax t) ⎥ 2πt ⎢⎣ − cos(2πfmin t) − j sin(2πfmin t) ⎥⎦

(A2)

Here, the second term of Eq. (A2) is not function of Δt . Hence, it can be ignored in case of considering relation between g (Δt) and Δt . Thus,

30

C.-Y. Kim, T.-J. Ahn / Optics Communications 345 (2015) 26–30

⎡ cos(2πf (t − Δt)) ⎤ max ⎢ ⎥ ⎢ sin(2 (t − Δt)) ⎥ j f + π 1 max ⎢ ⎥ g (Δt) ≈ j2π (t − Δt) ⎢ − cos(2πfmin (t − Δt)) ⎥ ⎢ ⎥ ⎣ −jsin(2πfmin (t − Δt)) ⎦ ⎧ ⎡ cos(2πf (t − Δt)) ⎤ ⎫ max ⎪⎢ ⎥⎪ ⎪ ⎢ − cos(j2πf (t − Δt)) ⎥ ⎪ ⎦⎪ ⎪⎣ min 1 ⎨ ⎬ = ⎤⎪ j2π (t − Δt) ⎪ ⎡ sin(2πf ( )) t t − Δ max ⎥⎪ ⎪ + j⎢ ⎢ ⎥ ⎪ ⎩ ⎣ −sin(2πfmin (t − Δt)) ⎦ ⎪ ⎭ ⎧ ⎡ f ⎤ ⎫ + fmin ⎪ −2 sin⎢2π max (t − Δt)⎥ ⎪ ⎢⎣ ⎥⎦ ⎪ ⎪ 2 ⎪ ⎪ ⎤ ⎪ ⎪ ⎡ fmax − fmin (t − Δt)⎥ ⎪ ⎪ sin⎢2π ⎢ ⎥ 2 ⎣ ⎦ ⎪ ⎪ 1 ⎬ ⎨ = ⎡ f ⎤⎪ j2π (t − Δt) ⎪ + fmin max (t − Δt)⎥ ⎪ ⎪ +2j cos⎢2π ⎢⎣ ⎥⎦ ⎪ 2 ⎪ ⎪ ⎪ ⎡ ⎤ ⎪ ⎪ ⎢ fmax − fmin ⎥ t t sin 2 ( ) π − Δ ⎪ ⎪ ⎢⎣ ⎥⎦ 2 ⎭ ⎩ =

1 cos⎡⎣π (fmax + fmin )(t − Δt)⎤⎦ π (t − Δt) 1 sin⎡⎣π (fmax − fmin )(t − Δt)⎤⎦ + j π (t − Δt) sin⎣⎡π (fmax + fmin )(t − Δt)⎤⎦ sin⎣⎡π (fmax − fmin )(t − Δt)⎤⎦

= A + jB

(A3)

Therefore, we obtained the real and imaginary parameters as A and B of the inverse Fourier transformation of Eq. (A1), respectively.

A=

1 cos(π (fmax + fmin )(t − Δt)) π (t − Δt) × sin(π (fmax − fmin )(t − Δt))

B=

(A4)

1 sin(π (fmax + fmin )(t − Δt)) π (t − Δt) × sin(π (fmax − fmin )(t − Δt))

(A5)

References [1] IBM, August 2011, Planning for Fiber Optic Link, [Online]. Available: 〈http:// www-01.ibm.com/support/docview.wss? uid ¼ isg2769b4853ca2e2ff085256aea004d42ad〉. [2] V.A.J.M. Sleiffer, Y. Jung, V. Veljanovski, R.G.H. van Uden, M. Kuschnerov, H. Chen, B. Inan, L.G. Nielsen, Y. Sun, D.J. Richardson, S.U. Alam, F. Poletti, J. K. Sahu, A. Dhar, A.M.J. Koonen, B. Corbett, R. Winfield, A.D. Ellis, H. de Waardt, 73.7 Tb/s (96  3  256-Gb/s) mode-division-multiplexed DP-16QAM transmission with inline MM-EDFA, Opt. Express 20 (2012) B428. [3] M. Salsi, C. Koebele, G. Charlet, S. Bigo, Mode division multiplexed transmission with a weakly-coupled few-mode fiber, in: Proceedings of OFC/NFOEC, Los Angeles, CA, 2012, pp. 1–3. [4] R. Ryf, S. Randel, A.H. Gnauck, C. Bolle, A. Sierra, S. Mumtaz, M. Esmaeelpour, E. C. Burrows, R. Essiambre, P.J. Winzer, D.W. Peckham, A.H. McCurdy, R. Lingle, Mode-division multiplexing over 96 km of few-mode fiber using coherent 6  6 MIMO processing, J. Lightwave Technol. 30 (2012) 521. [5] S. Randel, R. Ryf, A. Sierra, P.J. Winzer, A.H. Gnauck, C.A. Bolle, R.-J. Essiambre, D.W. Peckham, A. McCurdy, R. Lingle, 6  56-Gb/s mode-division multiplexed transmission over 33-km few-mode fiber enabled by 6  6 MIMO equalization, Opt. Exp. 19 (2011) 16697.

[6] R. Ryf, A. Sierra, R. Essiambre, S. Randel, A. Gnauck, C.A. Bolle, M. Esmaeelpour, P.J. Winzer, R. Delbue, P. Pupalaikis, A. Sureka, D. Peckham, A. McCurdy, R. Lingle, Mode-equalized distributed raman amplification in 137-km few-mode fiber, in: Proceedings of 37th ECOC, Geneva, 2011, pp. 1–3. [7] E. Ip, N. Bai, Y.-K. Huang, E. Mateo, F. Yaman, M.-J. Li, S. Bickham, S. Ten, J. Linares, C. Montero, V. Moreno, X. Prieto, V. Tse, K.M. Chung, A. Lau, H.-y. Tam, C. Lu, Y. Luo, G.-D. Peng, G. Li, 88  3  112-Gb/s WDM transmission over 50 km of three-mode fiber with inline few-mode fiber amplifier, in: Proceedings of 37th ECOC, Geneva, 2011, pp. 1–3. [8] M. Salsi, C. Koebele, D. Sperti, P. Tran, H. Mardoyan, P. Brindel, S. Bigo, A. Boutin, F. Verluise, P. Sillard, M. Bigot-Astruc, L. Provost, G. Charlet, Modedivision multiplexing of 2  100 Gb/s channels using an LCOS-based spatial modulator, J. Lightwave Technol. 30 (2012) 618. [9] N. Hanzawa, K. Saitoh, T. Sakamoto, T. Matsui, S. Tomita, M. Koshiba, Asymmetric parallel waveguide with mode conversion for mode and wavelength division multiplexing transmission, in: Proceedings of OFC/NFOEC, Los Angeles, CA, 2012, pp. 1–3. [10] J. Carpenter, B.C. Thomsen, T.D. Wilkinson, Degenerate mode-group division multiplexing, J. Lightwave Technol. 30 (2012) 3946. [11] A. Li, X. Chen, A.A. Amin, W. Shieh, Fused fiber mode couplers for few-mode transmission, IEEE Photon. Technol. Lett. 24 (2012) 1953. [12] H.S. Chen, A.M.J. Koonen, LP01 and LP11 mode division multiplexing link with mode crossbar switch, Electron. Lett. 48 (2012) 1222. [13] A.M.J. Koonen, H. Chen, H.P.A. van den Boom, O. Raz, Silicon photonic integrated mode multiplexer and demultiplexer, IEEE Photon. Technol. Lett. 24 (2012) 1961. [14] J.D. Love, N. Riesen, Single-, few-, and multimode Y-junctions, J. Lightwave Technol. 30 (2012) 304. [15] D. Askarov, J.M. Kahn, Design of transmission fibers and doped fiber amplifiers for mode-division multiplexing, IEEE Photon. Technol. Lett. 24 (2012) 1945. [16] M.-J. Li, B. Hoover, S. Li, S. Bickham, S. Ten, E. Ip, Y.-K. Huang, E. Mateo, Y. Shao, T. Wang, Low delay and large effective area few-mode fibers for mode-division multiplexing, in: Proceedings of 17th OECC, Busan, 2012, pp. 495–496. [17] S. Randel, A. Sierra, S. Mumtaz, A. Tulino, R. Ryf, P. Winzer, C. Schmidt, R. Essiambre, Adaptive MIMO signal processing for mode-division multiplexing, in: Proceedings of OFC/NFOEC, Los Angeles, CA, 2012, pp. 1–3. [18] S. Akhtari, P.M. Krummrich, Optical amplifier with rotationally symmetrical pump modes for enhanced mode multiplexing, IEEE Photon. Technol. Lett. 24 (2012) 2097. [19] F. Ferreira, D. van den Borne, P. Monteiro, H. Silva, Crosstalk optimization of phase masks for mode multiplexing in few mode fibers, in: Proceedings of OFC/NFOEC, Los Angeles, CA, 2012, pp. 1–3. [20] C.-Y. Kim, I.-S. Song, T.-J. Ahn, Differential mode group delay measurement for fewmode fiber using phase-sensitive intermodal spectral domain interferometer, in: proceedings of PGC, Singapore, 2012, pp. 1–3. [21] Differential mode delay measurement of multimode fiber in the time domain, TIA-455-220-A, 2003. [22] S.O. Kasap, Dielectric waveguides and optical fibers, in: Optoelectronics and Photonics: Principles and Practices (International Edition), 1st ed., Prentice Hall PTR, 2003, pp. 50–106. [23] S.O. Arik, D. Askarov, J.M. Kahn, Effect of mode coupling on signal processing complexity in mode-division multiplexing, J. Lightwave Technol. 31 (2013) 423. [24] T.-J. Ahn, S. Moon, Y. Youk, Y. Jung, K. Oh, D. Kim, New optical frequency domain differential mode delay measurement method for a multimode optical fiber, Opt. Express 13 (2005) 4005. [25] T.-J. Ahn, D. Kim, High-resolution differential mode delay measurement for a multimode optical fiber using a modified optical frequency domain reflectometer,, Opt. Express 13 (2005) 8256. [26] T.-J. Ahn, D.S. Choi, H.-S. Jeong, J.S. Park, W. Shin, Reflection-type monitoring method for differential mode delay at the 850-nm band, in: Proceedings of 15th OECC, Sapporo, 2010, pp. 304–305. [27] T.-J. Ahn, S. Moon, S. Kim, K. Oh, D.Y. Kim, J. Kobelke, K. Schuster, J. Kirchhof, Frequency-domain intermodal interferometer for the bandwidth measurement of a multimode fiber, Appl. Opt. 45 (2006) 8238. [28] C.-Y. Kim, T.-J. Ahn, Fourier-domain modal delay measurements for multimode fibers optimized for the 850-nm band in a local area network, IEICE Trans. Commun E96-B (2013) 2840. [29] J.W. Nicholson, A.D. Yablon, S. Ramachandran, S. Ghalmi, Spatially and spectrally resolved imaging of modal content in large-mode-area fibers, Opt. Express 16 (2008) 7233–7243. [30] E. Hecht, Transfer functions, in: Optics, 4th ed., Adison Wesley, UK, 2001, pp. 550–556.