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Design and optimization of fundamental mode filters based on long-period fiber gratings
Optical Fiber Technology 30 (2016) 89–94
Contents lists available at ScienceDirect
Optical Fiber Technology www.elsevier.com/locate/yofte
Design and optimization of fundamental mode filters based on long-period fiber gratings Ming-Yang Chen a,b,⇑, Jin Wei a, Yong Sheng c, Nai-Fei Ren c a
Institute of Opt-Electronics and Communication Technologies, Jiangsu University, Zhenjiang 212013, Jiangsu Province, China Jiangsu Tian Xing Optoelectronics Technology Co. Ltd, Zhenjiang 212013, Jiangsu Province, China c School of Mechanical Engineering, Jiangsu University, Zhenjiang 212013, Jiangsu Province, China b
a r t i c l e
i n f o
Article history: Received 6 December 2015 Revised 24 March 2016 Accepted 26 March 2016
Keywords: Optical fiber devices Optical fiber filters Gratings
a b s t r a c t A segment of long-period fiber grating (LPFG) that can selectively filter the fundamental mode in the fewmode optical fiber is proposed. By applying an appropriate chosen surrounding material and an apodized configuration of LPFG, high fundamental mode loss and low high-order core mode loss can be achieved simultaneously. In addition, we propose a method of cascading LPFGs with different periods to expand the bandwidth of the mode filter. Numerical simulation shows that the operating bandwidth of the cascade structure can be as large as 23 nm even if the refractive index of the surrounding liquid varies with the environment temperature. Ó 2016 Elsevier Inc. All rights reserved.
1. Introduction The first kind of optical fibers used for optical communication is multi-mode optical fibers (MMFs), however, due to the limits of modal dispersion, MMFs are mainly used in short distance communication. Sing-mode fibers (SMFs) can avoid the modal dispersion, leading to the great expansion of the transmission capacities of optical fiber communication systems. The capacities of SMFs have been exploited by the wavelength-division multiplexing (WDM), polarization-division multiplexing (PDM), and time-division multiplexing (TDM) technologies. However, owing to the nonlinear effects caused by the increased transmission power and the limited mode area of SMFs, the transmission capacity is close to the limit [1]. One of the solutions is to use space-division multiplexing (SDM) technology [2,3], which is based on multi-core optical fibers (MCFs) or multi-mode optical fibers (MMFs). Mode multiplexing in MMFs is difficult to realize owing to the large number of modes in MMFs. Therefore, few-mode optical fibers (FMFs) become a preferred choice. FMFs have aroused a lot of interests recently. FMFs can be used for long-distance transmission without modal dispersion and insertion loss penalty [4,5]. Single-mode operation can be realized by selectively exciting the fundamental mode, leading to large mode area operation [4,5], or low-bending loss operation [6]. It is well known that two-mode optical fibers (TMFs) can be ⇑ Corresponding author at: Institute of Opt-Electronics and Communication Technologies, Jiangsu University, Zhenjiang 212013, Jiangsu Province, China. E-mail address: [email protected] (M.-Y. Chen). http://dx.doi.org/10.1016/j.yofte.2016.03.006 1068-5200/Ó 2016 Elsevier Inc. All rights reserved.
worked as interferometric sensors [7]. Higher-order mode operating with ultra-large effective-area has been demonstrated and proposed as a new strategy for high-power lasers [8]. FMFs can also be designed to possess high-order modes with a variety of desired dispersive properties [9]. Mode tailoring devices, such as mode multiplexers/demultiplexers [10–20], mode converters [21–26], are the basic devices for FMF based applications. For example, mode converters should be used to convert between the fundamental mode in SMFs and the high-order modes in FMFs. One kind of the most commonly used mode converters is the long-period fiber grating (LPFG) based mode converter [18,21,27], which can ensure high conversion efficiency between the fundamental mode and the converted high-order mode. However, owing to the fact that the two core modes are propagating in the same core, the rest of the fundamental mode will become a source of cross-talk to the converted high-order mode. Generally, for communication applications, a high conversion efficiency of 20 dB is needed in order to suppress the fundamental mode. The operating bandwidth of the mode converter can be expanded effectively if more moderate crosstalk criterion can be accepted. Meanwhile, a mode filter that can selectively filter the fundamental mode and preserve the high-order modes simultaneously, can be applied to suppress the induced cross-talk. Mode filters can be used to suppress the unwanted modes during the demultiplexing process. Just like wavelength filtering devices for WDW systems, mode filtering devices should also be a basic component for mode-division multiplexing (MDM) systems.
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In this article, an LPFG configuration that can operate as a mode filter is numerical demonstrated. In addition, we also present a method of cascading LPFGs with different periods to expand the operating bandwidth. 2. Investigation of single-LPFG based mode filters According to the mode coupling theory of LPFGs, the fundamental mode in the fiber core will couple with a specific cladding mode when the phase matching condition is met. The relationship between the grating period K and the effective index of the core and the cladding modes is K ¼ k=ðno ðkÞ nc ðkÞÞ, where k is central wavelength of grating, no ðkÞ is the effective index of a core mode, and nc ðkÞ is the effective index of a cladding mode. Therefore, the fundamental mode will couple with the cladding mode if the LPFG is written in the core. The segment of the FMF written with LPFG is generally a naked optical fiber, and the two ends of which are accompanied by FMF segments with coating. Generally, the refractive index of the coating is higher than the cladding index, therefore, the converted cladding mode will be extended to the coating of the FMF and dispelled. It would be difficult to couple only the fundamental mode with the cladding mode in a FMF, owing to the fact that a naked optical fiber is surrounded by air, which leads to a wide range of effective indices of cladding modes. Our idea is to introduce a surrounding material with refractive index only slightly lower than the cladding index, therefore, the effective indices of the cladding modes are limited to a small range. By designing a phased-matched LPFG for the fundamental mode (LP01 mode) and a specific cladding mode, the fundamental mode will be coupled to the cladding mode, and finally dispelled in the segment of coated optical fiber. On the other hand, the high-order modes in the fiber core are not phased-matched with cladding modes, therefore, they can still reside in the fiber core. The configuration is shown in Fig. 1. The FMF parameters are set as core refractive index no ðkÞ ¼ 1:46, cladding refractive index nc ðkÞ ¼ 1:45, core diameter do ¼ 10 lm, and cladding diameter dc ¼ 125 lm. The refractive index of the surrounding material is nt ¼ 1:44. The default operating wavelength is set as 1550 nm, at which the optical fiber can support the propagation of the LP01 and LP11 modes. The LPFG is designed to couple the LP01 mode with the LP03 cladding mode. The effective indices of the modes are solved numerically by mode solution using a full vectorial finite-element method with perfectly matched layer boundary conditions [28]. The period of the LPFG is set as 200 lm based on the solution and the modulation depth of the LPFG is set as 0.001. The mode coupling characteristics of the LPFG are demonstrated by launching the LP01 and LP11 modes independently into the fiber core, and recording the mode power by using the beam propagation method [29,30]. It’s well known that the LP01 mode
Coating
Surrounding material
Cladding LP01 mode LP11 mode
LP11 mode
Core L
Fig. 1. The configuration of the mode filter based on a single LPFG.
will couple with the azimuthally symmetric cladding modes if the applied LPFG is also azimuthally symmetric. Partial coupling could also happen between the LP11 mode and the cladding mode with similar symmetry. For example, partial coupling could happen between the LP11 mode and the cladding LP1n (n > 1) mode when the grating period is set to be 200 lm. The variations of the mode distribution in the LPFG are shown in Fig. 2, where LH is coupling period of LP11 mode. We can see the LP11 mode can be converted to the cladding mode although the majority of the mode field still locate in the fiber core. It is obvious that such coupling should be avoided in order to reduce the transmission loss of the LP11 mode. As shown in Fig. 3(a), the normalized mode power varies with the propagation distance in the proposed LPFG. We can see the LP01 mode shows a strong coupling characteristic with the cladding mode, therefore, effective suppression of the LP01 mode can be achieved. On the other hand, the LP11 mode also shows a slightly power variation, which means that partial coupling between the LP11 mode and the cladding modes happens. It is not a surprising result. As there are still a large number of the cladding modes, even if the high-order mode and the cladding modes do not meet the phase matching condition, partial coupling could happen. Just as expected, partial coupling leads to shorter coupling period as compared to the phase-matched coupling [21,22,24–26]. The suppression of the coupling between the LP11 mode and the cladding modes can be achieved by increasing the effective index difference between the LP01 and LP11 modes, and meanwhile, reducing the span range of the effective indices of the cladding modes by narrowing the refractive index difference between the cladding and the surround material. In addition, in this article we will explore the suppression of the high-order mode coupling by using an apodized LPFG. As mentioned before, the coupling period of partial coupling is shorter than that of phase-matched coupling. Also the amplitude of coupling is associated with the modulation depth of the grating. Therefore, we can apply an apodized LPFG so that the LP01 mode can still be converted to the cladding mode, whereas the LP11 mode will have weak coupling with the cladding modes at the two ends of the LPFG, owing to the lower modulation depth at the two ends of the LPFG. The modulation amplitude of the LPFG along the propagation direction is set as mðZÞ ¼ cosðpZ=LÞ, with L=2 < Z < L=2 and L is length of the LPFG. The length of the LPFG is determined to be 18 mm, which ensures the LP01 mode be converted to the cladding LP03 mode completely. The length of the LPFG is longer than that of the uniform LPFG, the reason is that its overall modulation depth is lower than that of the uniform LPFG. So it would need longer length to realize the conversion. Fig. 3(b) shows the results of the apodized LPFG, the LP01 mode shows similar characteristics with the uniform LPFG, whereas power transferring of the LP11 mode is effectively suppressed. Even though there is still coupling at the middle of the grating, owing to the higher modulation depth at the position, but the power will transfer back, and lead to low loss at the output port. Mode filter should work with sufficiently large bandwidth. Since the LP01 mode to be filtered in a few-mode optical fiber is generally a residual mode, the mode filter do not need high enough suppression ratio. For example, a conversion efficiency of 10 dB for a mode converter means that the power loss of the converted mode is lower than 0.5 dB which is low enough for the converted mode but the cross-talk should be improved. Therefore, if a mode filter is applied, an insertion loss of 10 dB is large enough to suppress the LP01 mode to better than 20 dB. Based on such consideration, the operating wavelength range is defined by the requirement of the output power of the LP01 mode less than 10%, that is, the device loss of the LP01 mode is higher than 10 dB. Normalized output powers as functions of wavelength for the proposed optical fiber with the apodized LPFG are plotted in
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(a)
(b)
(c)
(d)
Fig. 2. LP11 mode distribution in the LPFG at propagation distances, (a) Z = 0, (b) Z = LH/3, (c) Z = 2LH/3, and (d) Z = LH.
Fig. 4. The operating wavelength range is 1548–1553 nm, at which the loss of the LP11 mode is less than 0.004 dB. As a result, the operating bandwidth of the mode filter is 5 nm. The main reason of the narrow bandwidth is that the variation of the effective index of the core mode with the wavelength is larger owing to the small size of the core, whereas the variation of the effective index of the cladding mode with the wavelength is smaller owing to the large size of the cladding, which leads to a strong wavelength dependent characteristic for the phase-matched grating period. 3. Investigation of mode filters based on cascading LPFGs Since we can filter the LP01 mode and at the same time maintain the LP11 mode at low loss, it is possible to increase the bandwidth of mode filter by cascading LPFGs. The structure of the cascading LPFGs is shown in Fig. 5. The mode filter is composed of three segments of LPFGs with different periods. Each LPFG can convert the LP01 mode to the LP03 cladding mode, and operates at different central wavelengths. As previously mentioned, the cladding mode cannot be dispelled directly. Therefore, each LPFG is followed by a segment of coated FMF. The refractive index of coating is higher than the cladding index, therefore, the converted cladding mode will experience high loss. Therefore, the cladding mode generated by each LPFG will then be extended to the coating and dispelled. Sufficiently long fiber length can ensure the full suppression of the cladding mode. The LP01 mode is therefore suppressed by the three segments of LFPGs with different operating wavelength ranges, and as a result, the operating bandwidth can be expanded. The central wavelengths of the three segments of LPFGs k1 , k2 , k3 are set as 1540, 1550, 1560 nm, respectively. The corresponding
periods of the LPFGs K1 , K2 , K3 are 198.7, 200, and 201.7 lm, respectively. The lengths of the LPFGs L1 ; L2 , L3 are determined to be 17.49, 18, and 18.15 mm, respectively. The length of the FMF between the gratings is set as LF ¼ 30 mm, and the refractive index of the coating is assumed to be nj ¼ 1:455: The mode filtering characteristics of the individual LPFGs are plotted in Fig. 6. The operating bandwidths of the three LPFGs are almost identical and the operating wavelength ranges have no overlapping with each other. We will then show the mode filtering characteristics of the cascading LPFGs by launching the LP01 and LP11 modes independently into the fiber core. The operating wavelength is chosen to be 1555 nm, which is out of the operating wavelength ranges of the individual LPFGs. As shown in Fig. 7(a), the power of the LP01 mode will transfer partly in each segment of LPFG, which means the power of LP01 mode at those wavelengths out of the operating wavelength ranges can be suppressed three times with different efficiencies. It should be noted that the total power in the fiber keeps unchanged when light transmits in each segment of the LPFGs, which means that the LP01 mode is only converted to the cladding mode and does not experience excess loss. The total power decreases rapidly in the FMF segment with high-index coating and will coincide with the power of the LP01 mode ultimately, which means that the power of the cladding mode can be dispelled completely. That is, the cladding mode will experience high loss owing to the high-index coating in the FMF segment. For the aim of preventing the cladding mode being converted back to the LP01 mode in the next segment of LPFG, the length of each segment of FMF must be enough long. Eventually, the output power of the LP01 mode is only 4.3% of the input power. Fig. 7(b) shows the result of the LP11 mode, the power of the LP11 mode can also trans-
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Coating
Surrounding material
LF
Cladding LP01 mode LP11 mode
LF LP11 mode
Core L1
L2
L3
Fig. 5. The configuration of the mode filter based on cascading LPFGs.
Fig. 6. Normalized output powers as functions of wavelength for the LP01 mode for the LPFGs with different central wavelengths.
Fig. 3. Normalized mode powers as functions of propagation distance in the proposed optical fiber with (a) a uniform LPFG, and (b) an apodized LPFG.
reach 99% of the input power, which ensures low transmission loss for the LP11 mode. Normalized output powers as functions of wavelength for the cascading LPFGs are plotted in Fig. 8. The operating wavelength range is 1537–1563 nm when the LP01 mode loss is determined to be higher than 10 dB, and the LP11 mode loss is lower than 0.066 dB at the wavelength range. The operating bandwidth is expanded to 26 nm, which is 4 times larger than that of the single LPFG. Obviously, elimination of the LP01 mode at wider wavelength range can be achieved by the cascading of more LPFGs, the numbers of the cascaded LPFGs is mainly limited by the excess loss induced by the higher-order modes in the fabrication process of the LPFGs. 4. Characteristics of the cascading LPFGs with the same period In theory, the LP01 mode can also be suppressed by cascading LPFGs with the same period. Fig. 9 shows normalized output powers as functions of wavelength for the structure of three segments of cascading LPFGs with the same period of 200 lm (the central wavelength of the LPFGs is 1550 nm). The operating wavelength range is 1544–1556 nm, and the LP11 mode loss is lower than 0.044 dB at the wavelength range. The operating bandwidth is only 12 nm, which is smaller than that of the structure of cascading LPFGs with different periods.
Fig. 4. Normalized output powers as functions of wavelength for the proposed optical fiber with the apodized LPFG.
fer in each segment of LPFG, but its major power transfers back and its tiny power transfers to LP1n (n > 1) mode at the end of the LPFGs. As a result, the output power of the LP11 mode can still
5. Characteristics of the cascading LPFGs at different environmental temperatures As stated before, the FMF should be surrounded by the material with refractive index lower than the cladding index. This can be
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Fig. 9. Normalized output powers as functions of wavelength for the structure of three segments of cascading LPFGs with the same period of 200 lm and the central wavelength of the LPFGs is 1550 nm.
Fig. 7. Normalized mode powers as functions of propagation distance in the cascading LPFGs with (a) the LP01 mode at the wavelength of 1555 nm, and (b) the LP11 mode at the wavelength of 1555 nm.
Fig. 8. Normalized output powers as functions of wavelength for the cascading LPFGs.
Fig. 10. Normalized output powers as functions of wavelength for different refractive indices of the liquid with (a) nl ¼ 1:444, and (b) nl ¼ 1:432.
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achieved by immersing the FMF into a refractive index matching liquid (Cargille Labs). The temperature coefficient of the liquid is almost always around 0.0004 refractive index units per degree centigrade, therefore, the refractive index will vary strongly with the surrounding environment temperature. Obviously, it is important that the mode filter can still work even though the environment temperature varies. For the results shown in Fig. 8, the refractive index of the surrounding material is assumed to be nl ¼ 1:44, which is assumed to work at 20 °C. Then the results of the mode filter works at 10 and 40 °C are simulated. Fig. 10(a) shows that the operating wavelength range is 1537–1562 nm when nl ¼ 1:444, which is corresponding to the environment temperature of 10 °C. Fig. 10(b) shows the operating wavelength range is 1539–1565 nm when nl ¼ 1:432, which is corresponding to the environment temperature of 40 °C. The above results show that the operating wavelength range could be at least 1539–1562 nm when the environment temperature range is 10–40 °C. Therefore, the operating bandwidth of the mode filter can still reach 23 nm and the LP11 mode loss is always lower than 0.088 dB under all these cases. 6. Conclusion This article puts forward the design of an apodized LPFG to couple the fundamental mode with the cladding mode, and at the same time keep the high-order modes in the fiber core at low loss, so as to achieve the purpose of selectively filtering the fundamental mode. We also present the technique of cascading LPFGs with different periods to increase the operating bandwidth effectively. The proposed mode filter can be applied to MDM systems and also the FMF based applications. The proposed cascading LPFGs can be fabricated in a single optical fiber. Firstly, the period of each grating should be determined. Then each LPFG will be written into the optical fiber and there should be long enough distance between the adjacent LPFGs to leak out the cladding modes. Finally the LPFGs will be immersed into the index-matching liquid. No addition connection loss will be introduced for the writing of multiple segments of LPFGs. However, the higher-order modes will experience excess loss in the LPFGs, which is generally caused by the process of fabricating LPFGs. The excess loss induced by the LPFGs will be the main factor that limits the numbers of apodized LPFGs. Acknowledgment This work is supported by the 2nd Technology Plan of Zhenjiang (Key R & D Program – Industry Prospective and Common Key Technologies (Grant No. GY2015033)). References [1] R.-J. Essiambre, G. Kramer, P.J. Winzer, G.J. Foschini, B. Goebel, Capacity limits of optical fiber networks, J. Lightwave Technol. 28 (4) (2010) 662–701. [2] G. Li, N. Bai, N. Zhao, C. Xia, Space-division multiplexing: the next frontier in optical communication, Adv. Opt. Photon. 6 (4) (2014) 413–487. [3] D.J. Richardson, J.M. Fini, L.E. Nelson, Space-division multiplexing in optical fibres, Nat. Photonics 7 (5) (2013) 354–362.
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Contents lists available at ScienceDirect
Optical Fiber Technology www.elsevier.com/locate/yofte
Design and optimization of fundamental mode filters based on long-period fiber gratings Ming-Yang Chen a,b,⇑, Jin Wei a, Yong Sheng c, Nai-Fei Ren c a
Institute of Opt-Electronics and Communication Technologies, Jiangsu University, Zhenjiang 212013, Jiangsu Province, China Jiangsu Tian Xing Optoelectronics Technology Co. Ltd, Zhenjiang 212013, Jiangsu Province, China c School of Mechanical Engineering, Jiangsu University, Zhenjiang 212013, Jiangsu Province, China b
a r t i c l e
i n f o
Article history: Received 6 December 2015 Revised 24 March 2016 Accepted 26 March 2016
Keywords: Optical fiber devices Optical fiber filters Gratings
a b s t r a c t A segment of long-period fiber grating (LPFG) that can selectively filter the fundamental mode in the fewmode optical fiber is proposed. By applying an appropriate chosen surrounding material and an apodized configuration of LPFG, high fundamental mode loss and low high-order core mode loss can be achieved simultaneously. In addition, we propose a method of cascading LPFGs with different periods to expand the bandwidth of the mode filter. Numerical simulation shows that the operating bandwidth of the cascade structure can be as large as 23 nm even if the refractive index of the surrounding liquid varies with the environment temperature. Ó 2016 Elsevier Inc. All rights reserved.
1. Introduction The first kind of optical fibers used for optical communication is multi-mode optical fibers (MMFs), however, due to the limits of modal dispersion, MMFs are mainly used in short distance communication. Sing-mode fibers (SMFs) can avoid the modal dispersion, leading to the great expansion of the transmission capacities of optical fiber communication systems. The capacities of SMFs have been exploited by the wavelength-division multiplexing (WDM), polarization-division multiplexing (PDM), and time-division multiplexing (TDM) technologies. However, owing to the nonlinear effects caused by the increased transmission power and the limited mode area of SMFs, the transmission capacity is close to the limit [1]. One of the solutions is to use space-division multiplexing (SDM) technology [2,3], which is based on multi-core optical fibers (MCFs) or multi-mode optical fibers (MMFs). Mode multiplexing in MMFs is difficult to realize owing to the large number of modes in MMFs. Therefore, few-mode optical fibers (FMFs) become a preferred choice. FMFs have aroused a lot of interests recently. FMFs can be used for long-distance transmission without modal dispersion and insertion loss penalty [4,5]. Single-mode operation can be realized by selectively exciting the fundamental mode, leading to large mode area operation [4,5], or low-bending loss operation [6]. It is well known that two-mode optical fibers (TMFs) can be ⇑ Corresponding author at: Institute of Opt-Electronics and Communication Technologies, Jiangsu University, Zhenjiang 212013, Jiangsu Province, China. E-mail address: [email protected] (M.-Y. Chen). http://dx.doi.org/10.1016/j.yofte.2016.03.006 1068-5200/Ó 2016 Elsevier Inc. All rights reserved.
worked as interferometric sensors [7]. Higher-order mode operating with ultra-large effective-area has been demonstrated and proposed as a new strategy for high-power lasers [8]. FMFs can also be designed to possess high-order modes with a variety of desired dispersive properties [9]. Mode tailoring devices, such as mode multiplexers/demultiplexers [10–20], mode converters [21–26], are the basic devices for FMF based applications. For example, mode converters should be used to convert between the fundamental mode in SMFs and the high-order modes in FMFs. One kind of the most commonly used mode converters is the long-period fiber grating (LPFG) based mode converter [18,21,27], which can ensure high conversion efficiency between the fundamental mode and the converted high-order mode. However, owing to the fact that the two core modes are propagating in the same core, the rest of the fundamental mode will become a source of cross-talk to the converted high-order mode. Generally, for communication applications, a high conversion efficiency of 20 dB is needed in order to suppress the fundamental mode. The operating bandwidth of the mode converter can be expanded effectively if more moderate crosstalk criterion can be accepted. Meanwhile, a mode filter that can selectively filter the fundamental mode and preserve the high-order modes simultaneously, can be applied to suppress the induced cross-talk. Mode filters can be used to suppress the unwanted modes during the demultiplexing process. Just like wavelength filtering devices for WDW systems, mode filtering devices should also be a basic component for mode-division multiplexing (MDM) systems.
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In this article, an LPFG configuration that can operate as a mode filter is numerical demonstrated. In addition, we also present a method of cascading LPFGs with different periods to expand the operating bandwidth. 2. Investigation of single-LPFG based mode filters According to the mode coupling theory of LPFGs, the fundamental mode in the fiber core will couple with a specific cladding mode when the phase matching condition is met. The relationship between the grating period K and the effective index of the core and the cladding modes is K ¼ k=ðno ðkÞ nc ðkÞÞ, where k is central wavelength of grating, no ðkÞ is the effective index of a core mode, and nc ðkÞ is the effective index of a cladding mode. Therefore, the fundamental mode will couple with the cladding mode if the LPFG is written in the core. The segment of the FMF written with LPFG is generally a naked optical fiber, and the two ends of which are accompanied by FMF segments with coating. Generally, the refractive index of the coating is higher than the cladding index, therefore, the converted cladding mode will be extended to the coating of the FMF and dispelled. It would be difficult to couple only the fundamental mode with the cladding mode in a FMF, owing to the fact that a naked optical fiber is surrounded by air, which leads to a wide range of effective indices of cladding modes. Our idea is to introduce a surrounding material with refractive index only slightly lower than the cladding index, therefore, the effective indices of the cladding modes are limited to a small range. By designing a phased-matched LPFG for the fundamental mode (LP01 mode) and a specific cladding mode, the fundamental mode will be coupled to the cladding mode, and finally dispelled in the segment of coated optical fiber. On the other hand, the high-order modes in the fiber core are not phased-matched with cladding modes, therefore, they can still reside in the fiber core. The configuration is shown in Fig. 1. The FMF parameters are set as core refractive index no ðkÞ ¼ 1:46, cladding refractive index nc ðkÞ ¼ 1:45, core diameter do ¼ 10 lm, and cladding diameter dc ¼ 125 lm. The refractive index of the surrounding material is nt ¼ 1:44. The default operating wavelength is set as 1550 nm, at which the optical fiber can support the propagation of the LP01 and LP11 modes. The LPFG is designed to couple the LP01 mode with the LP03 cladding mode. The effective indices of the modes are solved numerically by mode solution using a full vectorial finite-element method with perfectly matched layer boundary conditions [28]. The period of the LPFG is set as 200 lm based on the solution and the modulation depth of the LPFG is set as 0.001. The mode coupling characteristics of the LPFG are demonstrated by launching the LP01 and LP11 modes independently into the fiber core, and recording the mode power by using the beam propagation method [29,30]. It’s well known that the LP01 mode
Coating
Surrounding material
Cladding LP01 mode LP11 mode
LP11 mode
Core L
Fig. 1. The configuration of the mode filter based on a single LPFG.
will couple with the azimuthally symmetric cladding modes if the applied LPFG is also azimuthally symmetric. Partial coupling could also happen between the LP11 mode and the cladding mode with similar symmetry. For example, partial coupling could happen between the LP11 mode and the cladding LP1n (n > 1) mode when the grating period is set to be 200 lm. The variations of the mode distribution in the LPFG are shown in Fig. 2, where LH is coupling period of LP11 mode. We can see the LP11 mode can be converted to the cladding mode although the majority of the mode field still locate in the fiber core. It is obvious that such coupling should be avoided in order to reduce the transmission loss of the LP11 mode. As shown in Fig. 3(a), the normalized mode power varies with the propagation distance in the proposed LPFG. We can see the LP01 mode shows a strong coupling characteristic with the cladding mode, therefore, effective suppression of the LP01 mode can be achieved. On the other hand, the LP11 mode also shows a slightly power variation, which means that partial coupling between the LP11 mode and the cladding modes happens. It is not a surprising result. As there are still a large number of the cladding modes, even if the high-order mode and the cladding modes do not meet the phase matching condition, partial coupling could happen. Just as expected, partial coupling leads to shorter coupling period as compared to the phase-matched coupling [21,22,24–26]. The suppression of the coupling between the LP11 mode and the cladding modes can be achieved by increasing the effective index difference between the LP01 and LP11 modes, and meanwhile, reducing the span range of the effective indices of the cladding modes by narrowing the refractive index difference between the cladding and the surround material. In addition, in this article we will explore the suppression of the high-order mode coupling by using an apodized LPFG. As mentioned before, the coupling period of partial coupling is shorter than that of phase-matched coupling. Also the amplitude of coupling is associated with the modulation depth of the grating. Therefore, we can apply an apodized LPFG so that the LP01 mode can still be converted to the cladding mode, whereas the LP11 mode will have weak coupling with the cladding modes at the two ends of the LPFG, owing to the lower modulation depth at the two ends of the LPFG. The modulation amplitude of the LPFG along the propagation direction is set as mðZÞ ¼ cosðpZ=LÞ, with L=2 < Z < L=2 and L is length of the LPFG. The length of the LPFG is determined to be 18 mm, which ensures the LP01 mode be converted to the cladding LP03 mode completely. The length of the LPFG is longer than that of the uniform LPFG, the reason is that its overall modulation depth is lower than that of the uniform LPFG. So it would need longer length to realize the conversion. Fig. 3(b) shows the results of the apodized LPFG, the LP01 mode shows similar characteristics with the uniform LPFG, whereas power transferring of the LP11 mode is effectively suppressed. Even though there is still coupling at the middle of the grating, owing to the higher modulation depth at the position, but the power will transfer back, and lead to low loss at the output port. Mode filter should work with sufficiently large bandwidth. Since the LP01 mode to be filtered in a few-mode optical fiber is generally a residual mode, the mode filter do not need high enough suppression ratio. For example, a conversion efficiency of 10 dB for a mode converter means that the power loss of the converted mode is lower than 0.5 dB which is low enough for the converted mode but the cross-talk should be improved. Therefore, if a mode filter is applied, an insertion loss of 10 dB is large enough to suppress the LP01 mode to better than 20 dB. Based on such consideration, the operating wavelength range is defined by the requirement of the output power of the LP01 mode less than 10%, that is, the device loss of the LP01 mode is higher than 10 dB. Normalized output powers as functions of wavelength for the proposed optical fiber with the apodized LPFG are plotted in
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(a)
(b)
(c)
(d)
Fig. 2. LP11 mode distribution in the LPFG at propagation distances, (a) Z = 0, (b) Z = LH/3, (c) Z = 2LH/3, and (d) Z = LH.
Fig. 4. The operating wavelength range is 1548–1553 nm, at which the loss of the LP11 mode is less than 0.004 dB. As a result, the operating bandwidth of the mode filter is 5 nm. The main reason of the narrow bandwidth is that the variation of the effective index of the core mode with the wavelength is larger owing to the small size of the core, whereas the variation of the effective index of the cladding mode with the wavelength is smaller owing to the large size of the cladding, which leads to a strong wavelength dependent characteristic for the phase-matched grating period. 3. Investigation of mode filters based on cascading LPFGs Since we can filter the LP01 mode and at the same time maintain the LP11 mode at low loss, it is possible to increase the bandwidth of mode filter by cascading LPFGs. The structure of the cascading LPFGs is shown in Fig. 5. The mode filter is composed of three segments of LPFGs with different periods. Each LPFG can convert the LP01 mode to the LP03 cladding mode, and operates at different central wavelengths. As previously mentioned, the cladding mode cannot be dispelled directly. Therefore, each LPFG is followed by a segment of coated FMF. The refractive index of coating is higher than the cladding index, therefore, the converted cladding mode will experience high loss. Therefore, the cladding mode generated by each LPFG will then be extended to the coating and dispelled. Sufficiently long fiber length can ensure the full suppression of the cladding mode. The LP01 mode is therefore suppressed by the three segments of LFPGs with different operating wavelength ranges, and as a result, the operating bandwidth can be expanded. The central wavelengths of the three segments of LPFGs k1 , k2 , k3 are set as 1540, 1550, 1560 nm, respectively. The corresponding
periods of the LPFGs K1 , K2 , K3 are 198.7, 200, and 201.7 lm, respectively. The lengths of the LPFGs L1 ; L2 , L3 are determined to be 17.49, 18, and 18.15 mm, respectively. The length of the FMF between the gratings is set as LF ¼ 30 mm, and the refractive index of the coating is assumed to be nj ¼ 1:455: The mode filtering characteristics of the individual LPFGs are plotted in Fig. 6. The operating bandwidths of the three LPFGs are almost identical and the operating wavelength ranges have no overlapping with each other. We will then show the mode filtering characteristics of the cascading LPFGs by launching the LP01 and LP11 modes independently into the fiber core. The operating wavelength is chosen to be 1555 nm, which is out of the operating wavelength ranges of the individual LPFGs. As shown in Fig. 7(a), the power of the LP01 mode will transfer partly in each segment of LPFG, which means the power of LP01 mode at those wavelengths out of the operating wavelength ranges can be suppressed three times with different efficiencies. It should be noted that the total power in the fiber keeps unchanged when light transmits in each segment of the LPFGs, which means that the LP01 mode is only converted to the cladding mode and does not experience excess loss. The total power decreases rapidly in the FMF segment with high-index coating and will coincide with the power of the LP01 mode ultimately, which means that the power of the cladding mode can be dispelled completely. That is, the cladding mode will experience high loss owing to the high-index coating in the FMF segment. For the aim of preventing the cladding mode being converted back to the LP01 mode in the next segment of LPFG, the length of each segment of FMF must be enough long. Eventually, the output power of the LP01 mode is only 4.3% of the input power. Fig. 7(b) shows the result of the LP11 mode, the power of the LP11 mode can also trans-
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Coating
Surrounding material
LF
Cladding LP01 mode LP11 mode
LF LP11 mode
Core L1
L2
L3
Fig. 5. The configuration of the mode filter based on cascading LPFGs.
Fig. 6. Normalized output powers as functions of wavelength for the LP01 mode for the LPFGs with different central wavelengths.
Fig. 3. Normalized mode powers as functions of propagation distance in the proposed optical fiber with (a) a uniform LPFG, and (b) an apodized LPFG.
reach 99% of the input power, which ensures low transmission loss for the LP11 mode. Normalized output powers as functions of wavelength for the cascading LPFGs are plotted in Fig. 8. The operating wavelength range is 1537–1563 nm when the LP01 mode loss is determined to be higher than 10 dB, and the LP11 mode loss is lower than 0.066 dB at the wavelength range. The operating bandwidth is expanded to 26 nm, which is 4 times larger than that of the single LPFG. Obviously, elimination of the LP01 mode at wider wavelength range can be achieved by the cascading of more LPFGs, the numbers of the cascaded LPFGs is mainly limited by the excess loss induced by the higher-order modes in the fabrication process of the LPFGs. 4. Characteristics of the cascading LPFGs with the same period In theory, the LP01 mode can also be suppressed by cascading LPFGs with the same period. Fig. 9 shows normalized output powers as functions of wavelength for the structure of three segments of cascading LPFGs with the same period of 200 lm (the central wavelength of the LPFGs is 1550 nm). The operating wavelength range is 1544–1556 nm, and the LP11 mode loss is lower than 0.044 dB at the wavelength range. The operating bandwidth is only 12 nm, which is smaller than that of the structure of cascading LPFGs with different periods.
Fig. 4. Normalized output powers as functions of wavelength for the proposed optical fiber with the apodized LPFG.
fer in each segment of LPFG, but its major power transfers back and its tiny power transfers to LP1n (n > 1) mode at the end of the LPFGs. As a result, the output power of the LP11 mode can still
5. Characteristics of the cascading LPFGs at different environmental temperatures As stated before, the FMF should be surrounded by the material with refractive index lower than the cladding index. This can be
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Fig. 9. Normalized output powers as functions of wavelength for the structure of three segments of cascading LPFGs with the same period of 200 lm and the central wavelength of the LPFGs is 1550 nm.
Fig. 7. Normalized mode powers as functions of propagation distance in the cascading LPFGs with (a) the LP01 mode at the wavelength of 1555 nm, and (b) the LP11 mode at the wavelength of 1555 nm.
Fig. 8. Normalized output powers as functions of wavelength for the cascading LPFGs.
Fig. 10. Normalized output powers as functions of wavelength for different refractive indices of the liquid with (a) nl ¼ 1:444, and (b) nl ¼ 1:432.
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achieved by immersing the FMF into a refractive index matching liquid (Cargille Labs). The temperature coefficient of the liquid is almost always around 0.0004 refractive index units per degree centigrade, therefore, the refractive index will vary strongly with the surrounding environment temperature. Obviously, it is important that the mode filter can still work even though the environment temperature varies. For the results shown in Fig. 8, the refractive index of the surrounding material is assumed to be nl ¼ 1:44, which is assumed to work at 20 °C. Then the results of the mode filter works at 10 and 40 °C are simulated. Fig. 10(a) shows that the operating wavelength range is 1537–1562 nm when nl ¼ 1:444, which is corresponding to the environment temperature of 10 °C. Fig. 10(b) shows the operating wavelength range is 1539–1565 nm when nl ¼ 1:432, which is corresponding to the environment temperature of 40 °C. The above results show that the operating wavelength range could be at least 1539–1562 nm when the environment temperature range is 10–40 °C. Therefore, the operating bandwidth of the mode filter can still reach 23 nm and the LP11 mode loss is always lower than 0.088 dB under all these cases. 6. Conclusion This article puts forward the design of an apodized LPFG to couple the fundamental mode with the cladding mode, and at the same time keep the high-order modes in the fiber core at low loss, so as to achieve the purpose of selectively filtering the fundamental mode. We also present the technique of cascading LPFGs with different periods to increase the operating bandwidth effectively. The proposed mode filter can be applied to MDM systems and also the FMF based applications. The proposed cascading LPFGs can be fabricated in a single optical fiber. Firstly, the period of each grating should be determined. Then each LPFG will be written into the optical fiber and there should be long enough distance between the adjacent LPFGs to leak out the cladding modes. Finally the LPFGs will be immersed into the index-matching liquid. No addition connection loss will be introduced for the writing of multiple segments of LPFGs. However, the higher-order modes will experience excess loss in the LPFGs, which is generally caused by the process of fabricating LPFGs. The excess loss induced by the LPFGs will be the main factor that limits the numbers of apodized LPFGs. Acknowledgment This work is supported by the 2nd Technology Plan of Zhenjiang (Key R & D Program – Industry Prospective and Common Key Technologies (Grant No. GY2015033)). References [1] R.-J. Essiambre, G. Kramer, P.J. Winzer, G.J. Foschini, B. Goebel, Capacity limits of optical fiber networks, J. Lightwave Technol. 28 (4) (2010) 662–701. [2] G. Li, N. Bai, N. Zhao, C. Xia, Space-division multiplexing: the next frontier in optical communication, Adv. Opt. Photon. 6 (4) (2014) 413–487. [3] D.J. Richardson, J.M. Fini, L.E. Nelson, Space-division multiplexing in optical fibres, Nat. Photonics 7 (5) (2013) 354–362.
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