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Modal delay and modal bandwidth measurements of bi-modal optical fibers through a frequency domain method
Optical Fiber Technology 55 (2020) 102145
Contents lists available at ScienceDirect
Optical Fiber Technology journal homepage: www.elsevier.com/locate/yofte
Modal delay and modal bandwidth measurements of bi-modal optical fibers through a frequency domain method
T
Kangmei Li⁎, Xin Chen, Snigdharaj K. Mishra, Jason E. Hurley, Jeffery S. Stone, Ming-Jun Li Corning Incorporated, Corning, NY 14831, USA
A R T I C LE I N FO
A B S T R A C T
Keywords: Optical fiber communications Data center Fiber modal bandwidth Bi-modal fibers
We propose a simple and robust frequency domain method for measuring modal delay and bandwidth of bimodal optical fibers. An analytical transfer function model is formulated showing excellent agreement with experimental results for relatively short fibers. Using the model, a full set of information can be extracted, including modal delay and modal bandwidth under any launch conditions. As a result, one can obtain a worstcase modal bandwidth that can gauge the fiber modal bandwidth under general conditions. In addition, the frequency domain measurement method and the analytical model are validated through the excellent agreements with the time domain measurement results. The analytical model is also generalized for longer fiber lengths when additional degradation effects become significant to alter the behavior of the transfer function. Through the detailed study, we show that the simple frequency domain measurement method as facilitated by the analytical model can deliver a full set of modal delay and modal bandwidth information that otherwise requires more complex method of differential mode delay measurements.
1. Introduction Standard single-mode fibers have been widely used for single mode (SM) transmission in the wavelength window between 1260 and 1650 nm. Below the cable cutoff wavelength of 1260 nm, the fiber is no longer single-mode, in which case the LP11 mode also exists. Recent reports show that a single-mode fiber can have high modal bandwidth for short reach transmission using 850 nm VCSEL transceivers [1,2]. Specifically, graded-index single-mode fibers, which are bi-modal fibers supporting both LP01 and LP11 modes around 850 nm, were used for transmission using a SM VCSEL over 1.5 km at 25 Gb/s [2]. Such transmission system could be attractive as a potentially cost-effective high bandwidth solution for data center applications and future highspeed short distance communications. As reported, the system reach highly depends on the modal bandwidth of the fiber. In this paper, we focus on exploring a simple and robust modal bandwidth measurement method for this type of bi-modal fibers. The commonly used modal bandwidth measurement method is designed for multimode fibers (MMFs), which measures differential mode delay (DMD) of the fibers in the time domain over various radial offset locations across the fiber core [3], and the modal bandwidth is calculated using several assumed laser launch conditions. The method is suitable for MMFs with a large number of mode groups when Gaussian
⁎
approximation for the transfer function (TF) is held. Although the DMD method is valid for few-mode and bi-modal fibers, it is not optimal for the current study involving bi-modal fibers with smaller cores. Bandwidth measurement study specifically tailored for bi-modal fibers exists, for example, a method based on wavelength scanning has been reported in Ref. [4]. However, the method requires preparation of fibers in multiple different lengths in order to get the wavelength dependence of modal delay, and the impact of launch condition on the actual modal bandwidth is not clarified. On the other hand, a frequency domain bandwidth measurement method has been defined [5] and used for measuring bandwidth of MMFs [6]; however, the bandwidth obtained is limited to one specific launch condition used in the measurement and it is not straightforward to obtain modal bandwidths for general launch conditions. To overcome these limitations, in this paper, we propose a new bandwidth measurement method focusing on bimodal fibers, which is applicable for the fibers studied in Refs. [1,2]. We develop an analytical model to describe the transfer function of a bimodal fiber under any launch condition. With the analytical model, we only need to measure the transfer function with one launch condition. By analyzing the measured transfer function facilitated by the model, we can obtain the full information including the modal delay and modal bandwidth for arbitrary launch conditions. In Section 2, we describe the frequency domain method for measuring the modal bandwidth of the
Corresponding author. E-mail address: [email protected] (K. Li).
https://doi.org/10.1016/j.yofte.2020.102145 Received 15 November 2019; Accepted 9 January 2020 1068-5200/ © 2020 The Authors. Published by Elsevier Inc. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/BY-NC-ND/4.0/).
Optical Fiber Technology 55 (2020) 102145
K. Li, et al.
function. The conventional definition of fiber bandwidth is the frequency when the transmission drops by 3 dBo (optical dB, defined by 10·log10(x)) or 6 dBe (electrical dB, defined by 20·log10(x)) from the zero-frequency point. However, the measured transfer function can vary with launch conditions; as an example, Fig. 1(b) shows two measurements of the same bi-modal fiber at 850 nm under different launch conditions. We observe that the transfer function exhibits a periodic behavior, which might be due to the bi-modal nature of the fiber. To get a more thorough understanding of the features, we formulate an analytical model to describe the transfer function in Section 3. 3. Transfer function of bi-modal fibers In this Section, we develop a model to analyze the transfer function. Through the model we find that the modal delay and relative power ratio between the two fiber modes can be extracted directly from the transfer function. Based on the extracted parameters, the transfer function under any launch conditions can be calculated and a worstcase modal bandwidth can be defined. 3.1. Transfer function formula and analyses
Fig. 1. (a) Experimental setup for measuring fiber modal bandwidth in the frequency domain. VNA: vector network analyzer; IM: intensity modulator; FUT: fiber under test; PD: photodetector. (b) Measured transfer functions of the same bi-modal fiber with two different launch conditions.
When one optical pulse is launched into a bi-modal fiber, the output can be considered as “two” pulses, resulting from the different propagation constants of the two fiber modes, as depicted in Fig. 2(a). The modal delay time τ between the output pulses can be expressed as τ = |τ2 − τ1| = τ0 ∙L , where τ1 and τ2 are the times of flight of the two modes at the output of the FUT relative to the time of launch, and τ0 is a normalized modal delay time (modal delay per unit length) and L is the length of the FUT. The commonly used unit for τ0 is ps/m or ns/km. In some cases, the “two” pulses associated with the two modes partially overlap with each other due to a small difference of propagation constants. Assume that the input pulse is described by Hin (t ) in the time domain and Hin (f ) in the frequency domain, which is basically the Fourier transform of Hin (t ) ; the output is Hout (t ) and Hout (f ) in the time and frequency domain, respectively; then the transfer function can be described as
fiber under test. In Section 3, we present the detailed formalism of the analytical transfer function for bi-modal fibers based on the observation obtained in Section 2. We show how the modal delay and modal bandwidth under any given launch condition can be extracted from the formalism. Experimental measurements are conducted to compare the model with measured transfer functions. In Section 3, we further validate the frequency domain method and the analytical model using time domain measurements. In Section 4, we show that the frequency domain measurement method can be generalized to multiple wavelengths to obtain the wavelength with zero modal delay and peak modal bandwidth. In Section 5, the analytical formalism is generalized to take into account additional factors that affect the modal bandwidth in longer fibers. Finally, a brief summary is presented in Section 6.
S21 =
Hout (f ) Hin (f )
2. Frequency domain modal bandwidth measurement method
(1)
When a narrow linewidth light source is used, the chromatic dispersion effect is negligible, and the output can be simply described as:
The frequency domain bandwidth measurement method is illustrated in Fig. 1(a). A narrow linewidth continuous-wave light source at the wavelength of interest is intensity modulated with the modulation frequency controlled by the vector network analyzer (VNA). The launch condition into the fiber under test (FUT) is regulated via a mode conditioner. The optical signal is converted back to electric signal through the photodetector (PD) to obtain the transfer function, which is the transmission over a range of modulation frequency. Fiber modal bandwidth can be extracted from the measured transfer
Hout (t ) = a1 ∙Hin (t − τ1) + a2 ∙Hin (t − τ2)
(2)
where a1 and a2 are the output powers or relative output powers in each mode compared to the input. The Fourier transform of the output pulse can be described by
Hout (f ) = a1 ∙Hin (f ) ∙exp(−i∙2πfτ1) + a2 ∙Hin (f ) ∙exp(−i∙2πfτ2)
(3)
The transfer function can then be written as:
Fig. 2. (a) A pulse splits into two pulses after propagating through a bi-modal fiber, due to the modal delay between mode 1 and mode 2. (b) Useful information that can be extracted from the transfer function. 2
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determined. When c = 1/3, the bandwidth BW = 1/2τ . When c ≥ 1/3, the product of modal bandwidth and modal delay can be described as
S21 = |a1 ∙exp(−i∙2πfτ1) + a2 ∙exp(−i∙2πfτ2)| a12
=
+
a22
+ 2a1 a2cos(2πfτ )
(4)
BW ∙τ =
This transfer function is applicable to any two-mode cases, regardless of the input pulse shape. The transfer function can be re-written in logarithmic scale as using electrical definition:
S21 = 20·log10[ 1 + c 2 + 2c∙cos(2πfτ ) ] + d
1 2c − 3 − 3c 2 ⎞ ∙arccos ⎛ 2π 8c ⎠ ⎝ ⎜
⎟
(9)
and plotted in Fig. 3(d). The bandwidth-delay product increases with decreasing c . The minimum bandwidth-delay product is achieved when c = 1, i.e. when both modes are equally excited, with the minimum bandwidth
(5)
where c = a2 / a1 is the relative power ratio, and d is related to the optical loss after the fiber under test is inserted. In deriving the transfer function Eq. (5), we have utilized the time domain terminology, i.e. assuming an optical pulse is launched into the fiber and it propagates through the fiber to become two split optical pulses. However, the resulting transfer function is independent of the optical pulse or any specific parameters associated with an optical pulse. The transfer function describes the frequency response of the bimodal fiber for a given launch condition. Therefore, the derived transfer function can be used to fit the measured transfer function obtained by the frequency domain bandwidth measurement method. The transfer function Eq. (4) is plotted schematically in Fig. 2(b) from which the following information is extracted:
f3dB _min =
1 3τ
(10)
This means that a worst-case modal bandwidth can be defined simply related to the modal delay, independent of the launch condition. Unlike the DMD test, only one measurement is needed here, from which the modal delay can be extracted to determine the worst-cast modal bandwidth. Therefore, the method here dramatically simplifies the fiber bandwidth measurement. 3.3. Comparison with experimental results
related to the relative power ratio c.
The transfer function Eq. (5) is used to fit the experimental data obtained from 1-km fiber samples as shown in Fig. 4. The blue and red dots in Fig. 4(a) represent the measured transfer functions of a standard step-index single-mode fiber at 850 nm with two different launch conditions, and the solid curves are the fitted results based on the analytical model. The model fits the experimental results with excellent agreements. The modal delay is extracted to be 1.83 ns/km, yielding a bandwidth of 0.18 GHz·km, which is consistent with previous report [7]. Fig. 4(b) depicts the measured and fitted transfer functions of a graded-index single-mode fiber at 850 nm, from which the modal delay is found to be 0.023 ns/km, corresponding to a bandwidth of 14.3 GHz·km. This illustrates that a graded-index fiber can have a bandwidth much higher than a step-index fiber. The agreement between the experimental data and the fitting also suggests that the transfer function follows the assumption of the analytical formalism and mode coupling is negligible in the fiber.
3.2. Impact of launch condition on modal bandwidth
4. Comparison with time domain measurements
As described above, the modal bandwidth of the fiber depends on the launch condition, which determines how the two modes are excited. The launch condition can be described directly by the power ratio c between the two modes and can also be related to extinction ratio. Using Eq. (8), the extinction ratio as a function of the relative power ratio c can be plotted as shown in Fig. 3(a); for a bi-modal fiber case, c can take values within the range of 0 ≤ c ≤ 1. As can be seen, the extinction ratio increases with the relative power ratio; when c < 1/3, ER < 6dBe (or 3dBo) , and a conventional modal bandwidth cannot be
To further verify the frequency domain modal bandwidth measurement method as well as the analytical model, we compare the results with time domain measurement results. In the time domain measurements, the light source is a wavelength stabilized diode with a center wavelength of 850.73 nm and full width at half maximum (FWHM) linewidth of 0.02 nm. With such linewidth, the chromatic dispersion effect is negligible. The CW laser source is further modulated by an intensity modulator to generate an optical pulse with FWHM pulse width around 95 ps. Shown in Fig. 5(a) is the time domain
1. The modal delay can be obtained either from the first minimum at
f1 =
1 2τ
(6)
or the oscillation period at
f0 =
1 τ
(7)
2. The optical loss is related to the maximum of the transfer function as labelled in Fig. 2(b). 3. The extinction ratio (ER) can be described as
|1 − c| ⎞ ER = 20∙log10 ⎛ ⎝ |1 + c| ⎠
(8)
Fig. 3. (a) Extinction ratio as a function of the launch condition (relative power ratio c); (b) The product of modal bandwidth and modal delay as a function of the launch condition (relative power ratio c). 3
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τ0 = 0.2053ns/km . Comparing the model delay value with the time domain measurement result, it is found the difference is only 1.3%. A transfer function can also be calculated using the Fourier transform of the measured pulses using Eq. (1), where the reference pulse is the input pulse and the pulse passing through the FUT is the output pulse; it is shown in Fig. 5(b) in orange color. The curve follows the same trend as the frequency domain measured transfer function curve, with the maximum, minimum frequencies and the oscillation period to be the same. The difference in modulation depth is attributed to the launch condition difference as described in Eq. (5). Actually, taking into account the power ratio of 0.578 and the 208 ps separation between the two split pulses in FUT (see Fig. 5(a)), the predicted modal bandwidth using Eq. (9) is 1.709 GHz. The extracted 3dBo bandwidth from the orange curve in Fig. 5(b) is 1.709 GHz.km. The two values are identical. The excellent agreement of the two bandwidth measurement methods validates our frequency domain method as well as the analytical model. Additionally, our model can also deal with fibers with high bandwidth. One example illustrated in Fig. 5(c) indicates that a complete separation of two pulses cannot be seen due to the small modal delay (and hence high bandwidth). By fitting the transfer function using our analytical model, a model delay and worst-case bandwidth can be easily obtained, in this case the model delay is 0.0547 ns/km, corresponding to a worst-case bandwidth of 6.094 GHz·km. 5. Multiple wavelength bandwidth measurements to determine the peak wavelength
Fig. 4. Measured and fitted transfer functions at 850 nm with two different launch conditions for (a) a step-index fiber; (b) a graded-index fiber.
The method above can be applied to multiple wavelengths to obtain more information. Fig. 6(a) shows the measured transfer functions of the graded-index fiber in Fig. 4(b) at several different wavelengths from 800 nm to 880 nm. The transmitted signals are normalized so that they start at the same value. The extracted modal delays at different wavelengths are marked by the blue circles in Fig. 6(b), in which we plot the absolute values of the modal delay as measured and found that the delay bounces up after reaching a minimum. However, as we learned from numerical simulation, the actual modal delay between the two modes can change from positive to negative, depending on the relative propagation speed of the two modes. Based on this understanding, we plot the modal delay with the relative sign information incorporated, as indicated by the blue filled dots in Fig. 6(b). It is found that the modal delays follow a smooth curve, so that we apply a second-order
Fig. 5. Comparison of frequency-domain method and time-domain DMD method. (a) reference pulse (blue) and pulse after passing FUT_A (red) in time domain measurements; (b) normalized transfer function based on the Fourier transform of time domain method (orange) and frequency domain measured transfer function (green) for FUT_A; (c) reference pulse (blue) and pulse after passing FUT_B (red) in time domain measurements; (d) normalized transfer function based on the Fourier transform of time domain method (orange) and frequency domain measured transfer function (green) for FUT_B. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
measured pulses with and without the 1-km-long fiber under test (FUT). A reference pulse is first obtained without the FUT as shown in the blue curve. After inserting the FUT into the setup, the pulse splits into two as shown in the red curve, resulting from the different propagation speeds of the two fiber modes. As indicated, the separation of the two pulses is around 208 ps, indicating a modal delay of τ0 = 0.208ns/km . The transfer function of the same fiber is measured with the frequency domain method described in Section 2 and 3, illustrated by the green curve in Fig. 5(b), from which we can obtain the frequency of the first minimum to be f1 = 2.435GHz, and hence the modal delay
Fig. 6. (a) Measured transfer functions of a graded-index fiber at multiple wavelengths. (b) Extracted absolute values of modal delays (blue circles), modal delays with relative signs (blue dots) and the polynomial fitting curve (solid curve) as a function of wavelength. (c) Extracted (blue dots) and fitted (solid curve) worst-case bandwidths. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.) 4
Optical Fiber Technology 55 (2020) 102145
K. Li, et al.
assume an input pulse width of σ0, the fitted modal delay value is independent of this assumption. To further verify this generalized model, we prepare a 1-km sister fiber, which is next to the 7-km fiber when the fiber is drawn. The measured and modeled transfer functions of which are plotted in Fig. 7(b), yielding a modal delay of 0.0547 ns/km, deviating by only 1.83% from the 7-km fiber value. This difference can be attributed to the minor variation of the fiber and/or measurement error. Similarly, we can extract the modal delay information from this model either using the first local minimum of the transfer function, 1 f1 = 2τ , which is related only to the modal delay, or using the period of
Fig. 7. Experimental (blue) and modeled (magenta) transfer function of a 7-km long fiber; (b) experimental (blue) and modeled (magenta) transfer function of a 1-km long sister fiber. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
1
the small oscillations in the transfer function, f0 = τ . Therefore, we can also extract the modal delay from the transfer function, exactly like the previous short fiber case.
polynomial fitting to obtain the modal delay as a function of wavelength. Excellent agreement with the experimental data is achieved as illustrated by the solid line in Fig. 6(b). The peak wavelength, defined as the wavelength when the modal delay is minimum (or the modal bandwidth is maximum), can hence be determined from the fitting; in this case, it is the wavelength at which the modal delay crosses zero. As indicated by the dashed vertical line, the peak wavelength of this fiber is 841 nm. The worst-case bandwidths are calculated by Eq. (10) for both experimental and fitted results as plotted in Fig. 6(c), where the peak wavelength of 841 nm is indicated by the dashed vertical line. We note here in Ref. [4] a different method of measuring modal delay of bi-modal fibers was presented, which is based on wavelength scanning method. The method also relies on the relationship between the modal delay and group index difference of the two modes [8], which can be obtained by analyzing the oscillation period when scanning the wavelength. Since this method relies on the observing the oscillation period due to interference between the two fiber modes, which is highly related to the length of the fiber, the fiber length used in the test needs to be carefully chosen depending on the modal delay value. In addition, to get the modal delay dependence on wavelength, the fiber needs to be prepared with multiple different lengths, which is cumbersome to perform. The method introduced in the current paper can perform all the measurement with one fixed length, so it is simpler and more robust.
7. Summary In summary, we use a frequency domain modal bandwidth measurement method to obtain the transfer functions of bi-modal fibers and formulate a simple analytical model to describe the behaviors of measured transfer functions. The model allows one to extract the modal delay based on one single transfer function measurement, regardless of the launch condition. The transfer function and hence modal bandwidth with arbitrary launch condition can be calculated, in particular, a minimum or worst-case bandwidth can be defined, which is solely related to the modal delay. The measured transfer function agrees very well with the analytical transfer function. The analytical model is further validated using time domain measurements both with the direct measurement of the modal delay between two split pulses from the two modes and the calculated transfer function. The measurement method can also be applied to multiple wavelengths, from which the modal bandwidth or modal delay as a function of wavelengths can be fitted using polynomial functions with good agreements to experiments. The fitting allows us to extract additional information, specifically, the peak wavelength, a gauge how well the fiber is made relative to the targeted laser wavelength. The analytical model is also generalized to take into account effects such as polarization mode dispersion that occur in longer fibers when they become more observable. The modal delay obtained from longer fiber is in excellent agreement with the delay value from shorter fiber.
6. Generalization of the analytical formalism The testing done in Section 3 was for fiber samples with relatively short length (1 km). In other testing, we have prepared the fiber samples with much longer lengths (over 6 km). We observed that the transfer function of long fibers behaves differently. While the overall transfer function still shows oscillating behavior, the overall profile decays with increasing frequency. We believe the decay is due to pulse broadening effects using time domain terminology, which can result from polarization mode dispersion or chromatic dispersion effect. For longer fibers, such broadening effects should be considered; for example, the blue curve in Fig. 7(a) shows the measured transfer function of a 7-km fiber at 850 nm, and the overall profile decays with increasing frequency, which is presumably due to pulse broadening effects. As-
References [1] M. Li, X. Chen, K. Li, J.E. Hurley, J. Stone, Optical fiber for 1310nm single-mode and 850nm few-mode transmission, Proc. SPIE 10945, Broadband Access Communication Technologies XIII, 1094503, (2019). [2] K. Li, X. Chen, J.E. Hurley, J.S. Stone, M. Li, High data rate few-mode transmission over graded-index single-mode fiber using 850 nm single-mode VCSEL, Opt. Express 27 (2019) 21395–21404. [3] FOTP-220 (TIA-455-220-A), Differential mode delay measurement of multimode fiber in the time domain (2003). [4] R. Ryf, S. Randel, A.H. Gnauck, C. Bolle, A. Sierra, S. Mumtaz, M. Esmaeelpour, E.C. Burrows, R.-J. Essiambre, P.J. Winzer, D.W. Peckham, A.H. McCurdy, R. Lingle Jr., Mode-division multiplexing over 96 km of few-mode fiber using coherent 6X6 MIMO processing, J. Lightwave Technology 30 (4) (2012) 521–531. [5] IEEE Standard 802TIA/EIA 455–203, Launched power distribution measurement procedure for graded-index multimode fibre transmitters. [6] X. Chen, S.R. Bickham, J.E. Hurley, H. Liu, O.I. Dosunmu, M. Li, 25 Gb/s transmission over 820 m of MMF using a multimode launch from an integrated silicon photonics transceiver, Opt. Express 22 (2) (2014) 2070–2077. [7] C.-H. Cheng, C.-C. Shen, H.-Y. Kao, D.-H. Hsieh, H.-Y. Wang, Y.-W. Yeh, Y.-T. Lu, S.W.H. Chen, C.-T. Tsai, Y.-C. Chi, T.-S. Kao, C.-H. Wu, H.-C. Kuo, P.-T. Lee, G.-R. Lin, 850/940-nm VCSEL for optical communication and 3D sensing, Opto-Electron. Adv. 1 (3) (2018) 180005. [8] X. Chen, J.E. Hurley, M. Li, R.S. Vodhanel, Effects of multi-path interference (MPI) on the performance of transmission systems using fabry-perot lasers and short bend insensitive jumper fibers in optical fiber, Communication Conference and National Fiber Optic Engineers Conference, Paper NWC5, (2009).
( ) t2
suming Gaussian input pulse Hin (t ) = exp − 2σ 2 , the generalized 0
transfer function can be written as:
S21 = 20·log10 ⎡ ⎢ ⎣ +d
|σ1·exp (−2π 2σ12 f 2 ) + c·σ2·exp (−2π 2σ22 f 2 )·exp(−i2πτf )| ⎤ ⎥ σ0·exp (−2π 2σ02 f 2 ) ⎦ (11)
where σ1 and σ2 are the output pulse widths of the two modes. The model agrees with the experimental data very well, as shown in Fig. 7 (a), yielding a modal delay of 0.0537 ns/km. Note that although we
5
Contents lists available at ScienceDirect
Optical Fiber Technology journal homepage: www.elsevier.com/locate/yofte
Modal delay and modal bandwidth measurements of bi-modal optical fibers through a frequency domain method
T
Kangmei Li⁎, Xin Chen, Snigdharaj K. Mishra, Jason E. Hurley, Jeffery S. Stone, Ming-Jun Li Corning Incorporated, Corning, NY 14831, USA
A R T I C LE I N FO
A B S T R A C T
Keywords: Optical fiber communications Data center Fiber modal bandwidth Bi-modal fibers
We propose a simple and robust frequency domain method for measuring modal delay and bandwidth of bimodal optical fibers. An analytical transfer function model is formulated showing excellent agreement with experimental results for relatively short fibers. Using the model, a full set of information can be extracted, including modal delay and modal bandwidth under any launch conditions. As a result, one can obtain a worstcase modal bandwidth that can gauge the fiber modal bandwidth under general conditions. In addition, the frequency domain measurement method and the analytical model are validated through the excellent agreements with the time domain measurement results. The analytical model is also generalized for longer fiber lengths when additional degradation effects become significant to alter the behavior of the transfer function. Through the detailed study, we show that the simple frequency domain measurement method as facilitated by the analytical model can deliver a full set of modal delay and modal bandwidth information that otherwise requires more complex method of differential mode delay measurements.
1. Introduction Standard single-mode fibers have been widely used for single mode (SM) transmission in the wavelength window between 1260 and 1650 nm. Below the cable cutoff wavelength of 1260 nm, the fiber is no longer single-mode, in which case the LP11 mode also exists. Recent reports show that a single-mode fiber can have high modal bandwidth for short reach transmission using 850 nm VCSEL transceivers [1,2]. Specifically, graded-index single-mode fibers, which are bi-modal fibers supporting both LP01 and LP11 modes around 850 nm, were used for transmission using a SM VCSEL over 1.5 km at 25 Gb/s [2]. Such transmission system could be attractive as a potentially cost-effective high bandwidth solution for data center applications and future highspeed short distance communications. As reported, the system reach highly depends on the modal bandwidth of the fiber. In this paper, we focus on exploring a simple and robust modal bandwidth measurement method for this type of bi-modal fibers. The commonly used modal bandwidth measurement method is designed for multimode fibers (MMFs), which measures differential mode delay (DMD) of the fibers in the time domain over various radial offset locations across the fiber core [3], and the modal bandwidth is calculated using several assumed laser launch conditions. The method is suitable for MMFs with a large number of mode groups when Gaussian
⁎
approximation for the transfer function (TF) is held. Although the DMD method is valid for few-mode and bi-modal fibers, it is not optimal for the current study involving bi-modal fibers with smaller cores. Bandwidth measurement study specifically tailored for bi-modal fibers exists, for example, a method based on wavelength scanning has been reported in Ref. [4]. However, the method requires preparation of fibers in multiple different lengths in order to get the wavelength dependence of modal delay, and the impact of launch condition on the actual modal bandwidth is not clarified. On the other hand, a frequency domain bandwidth measurement method has been defined [5] and used for measuring bandwidth of MMFs [6]; however, the bandwidth obtained is limited to one specific launch condition used in the measurement and it is not straightforward to obtain modal bandwidths for general launch conditions. To overcome these limitations, in this paper, we propose a new bandwidth measurement method focusing on bimodal fibers, which is applicable for the fibers studied in Refs. [1,2]. We develop an analytical model to describe the transfer function of a bimodal fiber under any launch condition. With the analytical model, we only need to measure the transfer function with one launch condition. By analyzing the measured transfer function facilitated by the model, we can obtain the full information including the modal delay and modal bandwidth for arbitrary launch conditions. In Section 2, we describe the frequency domain method for measuring the modal bandwidth of the
Corresponding author. E-mail address: [email protected] (K. Li).
https://doi.org/10.1016/j.yofte.2020.102145 Received 15 November 2019; Accepted 9 January 2020 1068-5200/ © 2020 The Authors. Published by Elsevier Inc. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/BY-NC-ND/4.0/).
Optical Fiber Technology 55 (2020) 102145
K. Li, et al.
function. The conventional definition of fiber bandwidth is the frequency when the transmission drops by 3 dBo (optical dB, defined by 10·log10(x)) or 6 dBe (electrical dB, defined by 20·log10(x)) from the zero-frequency point. However, the measured transfer function can vary with launch conditions; as an example, Fig. 1(b) shows two measurements of the same bi-modal fiber at 850 nm under different launch conditions. We observe that the transfer function exhibits a periodic behavior, which might be due to the bi-modal nature of the fiber. To get a more thorough understanding of the features, we formulate an analytical model to describe the transfer function in Section 3. 3. Transfer function of bi-modal fibers In this Section, we develop a model to analyze the transfer function. Through the model we find that the modal delay and relative power ratio between the two fiber modes can be extracted directly from the transfer function. Based on the extracted parameters, the transfer function under any launch conditions can be calculated and a worstcase modal bandwidth can be defined. 3.1. Transfer function formula and analyses
Fig. 1. (a) Experimental setup for measuring fiber modal bandwidth in the frequency domain. VNA: vector network analyzer; IM: intensity modulator; FUT: fiber under test; PD: photodetector. (b) Measured transfer functions of the same bi-modal fiber with two different launch conditions.
When one optical pulse is launched into a bi-modal fiber, the output can be considered as “two” pulses, resulting from the different propagation constants of the two fiber modes, as depicted in Fig. 2(a). The modal delay time τ between the output pulses can be expressed as τ = |τ2 − τ1| = τ0 ∙L , where τ1 and τ2 are the times of flight of the two modes at the output of the FUT relative to the time of launch, and τ0 is a normalized modal delay time (modal delay per unit length) and L is the length of the FUT. The commonly used unit for τ0 is ps/m or ns/km. In some cases, the “two” pulses associated with the two modes partially overlap with each other due to a small difference of propagation constants. Assume that the input pulse is described by Hin (t ) in the time domain and Hin (f ) in the frequency domain, which is basically the Fourier transform of Hin (t ) ; the output is Hout (t ) and Hout (f ) in the time and frequency domain, respectively; then the transfer function can be described as
fiber under test. In Section 3, we present the detailed formalism of the analytical transfer function for bi-modal fibers based on the observation obtained in Section 2. We show how the modal delay and modal bandwidth under any given launch condition can be extracted from the formalism. Experimental measurements are conducted to compare the model with measured transfer functions. In Section 3, we further validate the frequency domain method and the analytical model using time domain measurements. In Section 4, we show that the frequency domain measurement method can be generalized to multiple wavelengths to obtain the wavelength with zero modal delay and peak modal bandwidth. In Section 5, the analytical formalism is generalized to take into account additional factors that affect the modal bandwidth in longer fibers. Finally, a brief summary is presented in Section 6.
S21 =
Hout (f ) Hin (f )
2. Frequency domain modal bandwidth measurement method
(1)
When a narrow linewidth light source is used, the chromatic dispersion effect is negligible, and the output can be simply described as:
The frequency domain bandwidth measurement method is illustrated in Fig. 1(a). A narrow linewidth continuous-wave light source at the wavelength of interest is intensity modulated with the modulation frequency controlled by the vector network analyzer (VNA). The launch condition into the fiber under test (FUT) is regulated via a mode conditioner. The optical signal is converted back to electric signal through the photodetector (PD) to obtain the transfer function, which is the transmission over a range of modulation frequency. Fiber modal bandwidth can be extracted from the measured transfer
Hout (t ) = a1 ∙Hin (t − τ1) + a2 ∙Hin (t − τ2)
(2)
where a1 and a2 are the output powers or relative output powers in each mode compared to the input. The Fourier transform of the output pulse can be described by
Hout (f ) = a1 ∙Hin (f ) ∙exp(−i∙2πfτ1) + a2 ∙Hin (f ) ∙exp(−i∙2πfτ2)
(3)
The transfer function can then be written as:
Fig. 2. (a) A pulse splits into two pulses after propagating through a bi-modal fiber, due to the modal delay between mode 1 and mode 2. (b) Useful information that can be extracted from the transfer function. 2
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determined. When c = 1/3, the bandwidth BW = 1/2τ . When c ≥ 1/3, the product of modal bandwidth and modal delay can be described as
S21 = |a1 ∙exp(−i∙2πfτ1) + a2 ∙exp(−i∙2πfτ2)| a12
=
+
a22
+ 2a1 a2cos(2πfτ )
(4)
BW ∙τ =
This transfer function is applicable to any two-mode cases, regardless of the input pulse shape. The transfer function can be re-written in logarithmic scale as using electrical definition:
S21 = 20·log10[ 1 + c 2 + 2c∙cos(2πfτ ) ] + d
1 2c − 3 − 3c 2 ⎞ ∙arccos ⎛ 2π 8c ⎠ ⎝ ⎜
⎟
(9)
and plotted in Fig. 3(d). The bandwidth-delay product increases with decreasing c . The minimum bandwidth-delay product is achieved when c = 1, i.e. when both modes are equally excited, with the minimum bandwidth
(5)
where c = a2 / a1 is the relative power ratio, and d is related to the optical loss after the fiber under test is inserted. In deriving the transfer function Eq. (5), we have utilized the time domain terminology, i.e. assuming an optical pulse is launched into the fiber and it propagates through the fiber to become two split optical pulses. However, the resulting transfer function is independent of the optical pulse or any specific parameters associated with an optical pulse. The transfer function describes the frequency response of the bimodal fiber for a given launch condition. Therefore, the derived transfer function can be used to fit the measured transfer function obtained by the frequency domain bandwidth measurement method. The transfer function Eq. (4) is plotted schematically in Fig. 2(b) from which the following information is extracted:
f3dB _min =
1 3τ
(10)
This means that a worst-case modal bandwidth can be defined simply related to the modal delay, independent of the launch condition. Unlike the DMD test, only one measurement is needed here, from which the modal delay can be extracted to determine the worst-cast modal bandwidth. Therefore, the method here dramatically simplifies the fiber bandwidth measurement. 3.3. Comparison with experimental results
related to the relative power ratio c.
The transfer function Eq. (5) is used to fit the experimental data obtained from 1-km fiber samples as shown in Fig. 4. The blue and red dots in Fig. 4(a) represent the measured transfer functions of a standard step-index single-mode fiber at 850 nm with two different launch conditions, and the solid curves are the fitted results based on the analytical model. The model fits the experimental results with excellent agreements. The modal delay is extracted to be 1.83 ns/km, yielding a bandwidth of 0.18 GHz·km, which is consistent with previous report [7]. Fig. 4(b) depicts the measured and fitted transfer functions of a graded-index single-mode fiber at 850 nm, from which the modal delay is found to be 0.023 ns/km, corresponding to a bandwidth of 14.3 GHz·km. This illustrates that a graded-index fiber can have a bandwidth much higher than a step-index fiber. The agreement between the experimental data and the fitting also suggests that the transfer function follows the assumption of the analytical formalism and mode coupling is negligible in the fiber.
3.2. Impact of launch condition on modal bandwidth
4. Comparison with time domain measurements
As described above, the modal bandwidth of the fiber depends on the launch condition, which determines how the two modes are excited. The launch condition can be described directly by the power ratio c between the two modes and can also be related to extinction ratio. Using Eq. (8), the extinction ratio as a function of the relative power ratio c can be plotted as shown in Fig. 3(a); for a bi-modal fiber case, c can take values within the range of 0 ≤ c ≤ 1. As can be seen, the extinction ratio increases with the relative power ratio; when c < 1/3, ER < 6dBe (or 3dBo) , and a conventional modal bandwidth cannot be
To further verify the frequency domain modal bandwidth measurement method as well as the analytical model, we compare the results with time domain measurement results. In the time domain measurements, the light source is a wavelength stabilized diode with a center wavelength of 850.73 nm and full width at half maximum (FWHM) linewidth of 0.02 nm. With such linewidth, the chromatic dispersion effect is negligible. The CW laser source is further modulated by an intensity modulator to generate an optical pulse with FWHM pulse width around 95 ps. Shown in Fig. 5(a) is the time domain
1. The modal delay can be obtained either from the first minimum at
f1 =
1 2τ
(6)
or the oscillation period at
f0 =
1 τ
(7)
2. The optical loss is related to the maximum of the transfer function as labelled in Fig. 2(b). 3. The extinction ratio (ER) can be described as
|1 − c| ⎞ ER = 20∙log10 ⎛ ⎝ |1 + c| ⎠
(8)
Fig. 3. (a) Extinction ratio as a function of the launch condition (relative power ratio c); (b) The product of modal bandwidth and modal delay as a function of the launch condition (relative power ratio c). 3
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τ0 = 0.2053ns/km . Comparing the model delay value with the time domain measurement result, it is found the difference is only 1.3%. A transfer function can also be calculated using the Fourier transform of the measured pulses using Eq. (1), where the reference pulse is the input pulse and the pulse passing through the FUT is the output pulse; it is shown in Fig. 5(b) in orange color. The curve follows the same trend as the frequency domain measured transfer function curve, with the maximum, minimum frequencies and the oscillation period to be the same. The difference in modulation depth is attributed to the launch condition difference as described in Eq. (5). Actually, taking into account the power ratio of 0.578 and the 208 ps separation between the two split pulses in FUT (see Fig. 5(a)), the predicted modal bandwidth using Eq. (9) is 1.709 GHz. The extracted 3dBo bandwidth from the orange curve in Fig. 5(b) is 1.709 GHz.km. The two values are identical. The excellent agreement of the two bandwidth measurement methods validates our frequency domain method as well as the analytical model. Additionally, our model can also deal with fibers with high bandwidth. One example illustrated in Fig. 5(c) indicates that a complete separation of two pulses cannot be seen due to the small modal delay (and hence high bandwidth). By fitting the transfer function using our analytical model, a model delay and worst-case bandwidth can be easily obtained, in this case the model delay is 0.0547 ns/km, corresponding to a worst-case bandwidth of 6.094 GHz·km. 5. Multiple wavelength bandwidth measurements to determine the peak wavelength
Fig. 4. Measured and fitted transfer functions at 850 nm with two different launch conditions for (a) a step-index fiber; (b) a graded-index fiber.
The method above can be applied to multiple wavelengths to obtain more information. Fig. 6(a) shows the measured transfer functions of the graded-index fiber in Fig. 4(b) at several different wavelengths from 800 nm to 880 nm. The transmitted signals are normalized so that they start at the same value. The extracted modal delays at different wavelengths are marked by the blue circles in Fig. 6(b), in which we plot the absolute values of the modal delay as measured and found that the delay bounces up after reaching a minimum. However, as we learned from numerical simulation, the actual modal delay between the two modes can change from positive to negative, depending on the relative propagation speed of the two modes. Based on this understanding, we plot the modal delay with the relative sign information incorporated, as indicated by the blue filled dots in Fig. 6(b). It is found that the modal delays follow a smooth curve, so that we apply a second-order
Fig. 5. Comparison of frequency-domain method and time-domain DMD method. (a) reference pulse (blue) and pulse after passing FUT_A (red) in time domain measurements; (b) normalized transfer function based on the Fourier transform of time domain method (orange) and frequency domain measured transfer function (green) for FUT_A; (c) reference pulse (blue) and pulse after passing FUT_B (red) in time domain measurements; (d) normalized transfer function based on the Fourier transform of time domain method (orange) and frequency domain measured transfer function (green) for FUT_B. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
measured pulses with and without the 1-km-long fiber under test (FUT). A reference pulse is first obtained without the FUT as shown in the blue curve. After inserting the FUT into the setup, the pulse splits into two as shown in the red curve, resulting from the different propagation speeds of the two fiber modes. As indicated, the separation of the two pulses is around 208 ps, indicating a modal delay of τ0 = 0.208ns/km . The transfer function of the same fiber is measured with the frequency domain method described in Section 2 and 3, illustrated by the green curve in Fig. 5(b), from which we can obtain the frequency of the first minimum to be f1 = 2.435GHz, and hence the modal delay
Fig. 6. (a) Measured transfer functions of a graded-index fiber at multiple wavelengths. (b) Extracted absolute values of modal delays (blue circles), modal delays with relative signs (blue dots) and the polynomial fitting curve (solid curve) as a function of wavelength. (c) Extracted (blue dots) and fitted (solid curve) worst-case bandwidths. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.) 4
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assume an input pulse width of σ0, the fitted modal delay value is independent of this assumption. To further verify this generalized model, we prepare a 1-km sister fiber, which is next to the 7-km fiber when the fiber is drawn. The measured and modeled transfer functions of which are plotted in Fig. 7(b), yielding a modal delay of 0.0547 ns/km, deviating by only 1.83% from the 7-km fiber value. This difference can be attributed to the minor variation of the fiber and/or measurement error. Similarly, we can extract the modal delay information from this model either using the first local minimum of the transfer function, 1 f1 = 2τ , which is related only to the modal delay, or using the period of
Fig. 7. Experimental (blue) and modeled (magenta) transfer function of a 7-km long fiber; (b) experimental (blue) and modeled (magenta) transfer function of a 1-km long sister fiber. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
1
the small oscillations in the transfer function, f0 = τ . Therefore, we can also extract the modal delay from the transfer function, exactly like the previous short fiber case.
polynomial fitting to obtain the modal delay as a function of wavelength. Excellent agreement with the experimental data is achieved as illustrated by the solid line in Fig. 6(b). The peak wavelength, defined as the wavelength when the modal delay is minimum (or the modal bandwidth is maximum), can hence be determined from the fitting; in this case, it is the wavelength at which the modal delay crosses zero. As indicated by the dashed vertical line, the peak wavelength of this fiber is 841 nm. The worst-case bandwidths are calculated by Eq. (10) for both experimental and fitted results as plotted in Fig. 6(c), where the peak wavelength of 841 nm is indicated by the dashed vertical line. We note here in Ref. [4] a different method of measuring modal delay of bi-modal fibers was presented, which is based on wavelength scanning method. The method also relies on the relationship between the modal delay and group index difference of the two modes [8], which can be obtained by analyzing the oscillation period when scanning the wavelength. Since this method relies on the observing the oscillation period due to interference between the two fiber modes, which is highly related to the length of the fiber, the fiber length used in the test needs to be carefully chosen depending on the modal delay value. In addition, to get the modal delay dependence on wavelength, the fiber needs to be prepared with multiple different lengths, which is cumbersome to perform. The method introduced in the current paper can perform all the measurement with one fixed length, so it is simpler and more robust.
7. Summary In summary, we use a frequency domain modal bandwidth measurement method to obtain the transfer functions of bi-modal fibers and formulate a simple analytical model to describe the behaviors of measured transfer functions. The model allows one to extract the modal delay based on one single transfer function measurement, regardless of the launch condition. The transfer function and hence modal bandwidth with arbitrary launch condition can be calculated, in particular, a minimum or worst-case bandwidth can be defined, which is solely related to the modal delay. The measured transfer function agrees very well with the analytical transfer function. The analytical model is further validated using time domain measurements both with the direct measurement of the modal delay between two split pulses from the two modes and the calculated transfer function. The measurement method can also be applied to multiple wavelengths, from which the modal bandwidth or modal delay as a function of wavelengths can be fitted using polynomial functions with good agreements to experiments. The fitting allows us to extract additional information, specifically, the peak wavelength, a gauge how well the fiber is made relative to the targeted laser wavelength. The analytical model is also generalized to take into account effects such as polarization mode dispersion that occur in longer fibers when they become more observable. The modal delay obtained from longer fiber is in excellent agreement with the delay value from shorter fiber.
6. Generalization of the analytical formalism The testing done in Section 3 was for fiber samples with relatively short length (1 km). In other testing, we have prepared the fiber samples with much longer lengths (over 6 km). We observed that the transfer function of long fibers behaves differently. While the overall transfer function still shows oscillating behavior, the overall profile decays with increasing frequency. We believe the decay is due to pulse broadening effects using time domain terminology, which can result from polarization mode dispersion or chromatic dispersion effect. For longer fibers, such broadening effects should be considered; for example, the blue curve in Fig. 7(a) shows the measured transfer function of a 7-km fiber at 850 nm, and the overall profile decays with increasing frequency, which is presumably due to pulse broadening effects. As-
References [1] M. Li, X. Chen, K. Li, J.E. Hurley, J. Stone, Optical fiber for 1310nm single-mode and 850nm few-mode transmission, Proc. SPIE 10945, Broadband Access Communication Technologies XIII, 1094503, (2019). [2] K. Li, X. Chen, J.E. Hurley, J.S. Stone, M. Li, High data rate few-mode transmission over graded-index single-mode fiber using 850 nm single-mode VCSEL, Opt. Express 27 (2019) 21395–21404. [3] FOTP-220 (TIA-455-220-A), Differential mode delay measurement of multimode fiber in the time domain (2003). [4] R. Ryf, S. Randel, A.H. Gnauck, C. Bolle, A. Sierra, S. Mumtaz, M. Esmaeelpour, E.C. Burrows, R.-J. Essiambre, P.J. Winzer, D.W. Peckham, A.H. McCurdy, R. Lingle Jr., Mode-division multiplexing over 96 km of few-mode fiber using coherent 6X6 MIMO processing, J. Lightwave Technology 30 (4) (2012) 521–531. [5] IEEE Standard 802TIA/EIA 455–203, Launched power distribution measurement procedure for graded-index multimode fibre transmitters. [6] X. Chen, S.R. Bickham, J.E. Hurley, H. Liu, O.I. Dosunmu, M. Li, 25 Gb/s transmission over 820 m of MMF using a multimode launch from an integrated silicon photonics transceiver, Opt. Express 22 (2) (2014) 2070–2077. [7] C.-H. Cheng, C.-C. Shen, H.-Y. Kao, D.-H. Hsieh, H.-Y. Wang, Y.-W. Yeh, Y.-T. Lu, S.W.H. Chen, C.-T. Tsai, Y.-C. Chi, T.-S. Kao, C.-H. Wu, H.-C. Kuo, P.-T. Lee, G.-R. Lin, 850/940-nm VCSEL for optical communication and 3D sensing, Opto-Electron. Adv. 1 (3) (2018) 180005. [8] X. Chen, J.E. Hurley, M. Li, R.S. Vodhanel, Effects of multi-path interference (MPI) on the performance of transmission systems using fabry-perot lasers and short bend insensitive jumper fibers in optical fiber, Communication Conference and National Fiber Optic Engineers Conference, Paper NWC5, (2009).
( ) t2
suming Gaussian input pulse Hin (t ) = exp − 2σ 2 , the generalized 0
transfer function can be written as:
S21 = 20·log10 ⎡ ⎢ ⎣ +d
|σ1·exp (−2π 2σ12 f 2 ) + c·σ2·exp (−2π 2σ22 f 2 )·exp(−i2πτf )| ⎤ ⎥ σ0·exp (−2π 2σ02 f 2 ) ⎦ (11)
where σ1 and σ2 are the output pulse widths of the two modes. The model agrees with the experimental data very well, as shown in Fig. 7 (a), yielding a modal delay of 0.0537 ns/km. Note that although we
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