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Using adaptive optics (AO) and a directional coupler, we demonstrate adaptive control of linearly polarized (LP) modes in a two mode fiber. The AO feedback is provided by the coupling ratio of the directional coupler, and does not depend on the spatial profiles of optical field distributions. As a proof of concept demonstration, this work confirms the feasibility of using AO and all fiber devices to control the waveguide modes in a multimode network in a quasi-distributed manner.
© 2014 Optical Society of America
1. Introduction
Recent development of mode division multiplexing (MDM) has demonstrated that waveguide modes can serve as an independent multiplexing parameter for optical communications [1–3]. Used in conjunction with other multiplexing parameters such as wavelength and polarization, the application of MDM techniques can significantly increase the bandwidth of a communication network. To characterize the cross-talk between distinct mode channels, one typically describes the impact of intermodal coupling of the entire multimode fiber (MMF) network as an impulse response matrix. This is sufficient for communication related applications, where one is only concerned with the relationship of optical fields at the input and the output ports. However, the concept of MDM can also be extended to other branches of optics such as optical sensing. Figure 1(a) illustrates an example of a quasi-distributed sensor network with multiple MMF-based sensors. Since the response of a MMF-based fiber sensor can be highly mode dependent, one needs to be able to control the form of optical field within the MMF network, which requires us to overcome the challenge that the intermodal coupling within the MMF network can depend on both location and time.
Fig. 1 (a) Illustration of a quasi-distributed sensor network based on MMF. Each sensor should contain a mode control element to distinguish sensor responses due to different LP modes. (b) Using the coupling ratio of a directional coupler to achieve mode control. Schematic of the setup: λ/2, half-wave plate; Expander, 1:5 beam expander comprise of two lenses; M, mirror; SLM, phase-only spatial light modulator; P1 and P2, polarizers; L1, 20 × objective lens, NA = 0.40; Fiber, Thorlabs 980HP, ~1 m; Fiber Coupler, Thorlabs FC980-50B; L2, 100 × objective lens, NA = 0.70; CCD, CCD camera.
The main purpose of the paper is to experimentally demonstrate that one can use adaptive optics (AO) and all fiber devices to control the guided modes within a MMF fiber network, as long as the fiber devices can generate appropriate feedback signals. In the present paper, we use the coupling ratio of a directional coupler constructed using a two-mode fiber (TMF) for AO feedback. For the two linearly polarized (LP) modes, the higher order LP11 mode extends much further into fiber cladding, which leads to a much larger coupling coefficient for the LP11. Therefore, by maximizing or minimizing the coupling ratio, we should be able to adaptively control the LP modes in the TMF.
AO has previously been used in MDM. In particular, AO has been used to reduce intermodal dispersion [4], demonstrate mode multiplexing and demultiplexing [5–7], and achieve mode control [8, 9]. Additionally, by applying techniques of scattering medium focusing [10–14], one can use AO to achieve optical focusing [15–17], imaging [18, 19], and particle manipulation through a MMF [20, 21]. The present paper and aforementioned work differ in several key aspects. First, our approach makes no assumption on the optical system and essentially treats the entire optical system as a “black box” with unknown intermodal coupling. In contrast, the adaptive annealing method in [7] assumes a specific relation between the spatial light modulator (SLM) and the input port of the MMF. Second, our AO approach utilizes coupling ratio as the feedback signal, which, unlike [7], does not require any knowledge of the spatial profiles of the guided fields. Third, in [7], the multimode coupler is used essentially as a mode demultiplexer for MDM, whereas in our work, the fiber coupler generates feedback signals for mode control. Our ultimate goal is to show that as long as we incorporate appropriate fiber devices within the MMF network for optical feedbacks, we can control the form of optical field within the entire network in a quasi-distributed manner, which is essential for the development of a MMF-based sensor network. The work presented here serves as a proof-of-concept study to show the feasibility of this goal.
2. Experiment setup
The experimental system is shown in Fig. 1(b). Light from a linearly polarized He-Ne laser (632.8nm) is collimated and expanded. The expanded beam is reflected by a phase-only SLM (Holoeye, Pluto) and focused into a silica fiber (Thorlabs 980HP, length approximately 1m). According to the specification of the fiber, the fiber V-number is V = 3.57 at the operation wavelength, thus the fiber is a TMF that supports the LP01 and LP11 modes. The fiber is spliced to one of the two input ports of a fiber optic coupler (Thorlabs, FC980-50B), which has a specified V-number of V = 3.71 at the operation wavelength, and therefore supports both the LP01 and the LP11 modes.
The light polarization is controlled by a half-wave plate and two polarizers (P1 and P2), which select optical waves with polarization direction parallel to the optical table. The incident wave polarization also coincides with the phase modulation axis of the SLM. The expanded He-Ne laser beam (FWHM ~6.5 mm) is projected onto the SLM and forms an optical beam of similar size. (The total area of the SLM pixels that are phase modulated is approximately 5.3 mm by 5.3 mm.) This area is evenly divided into 11 × 11 phase blocks to control the wavefront of the incident beam. Within each phase block, the phase shifts produced by the SLM are identical. An objective lens (20 × , NA = 0.40) is used to focus the beam into the fiber. At the output end of the fiber coupler, two pieces of fused silica fiber (Thorlabs 980HP, length approximately 1m) are spliced to the two output ports (designated the throughput port and crossover port) of the fiber coupler. Their output facets are carefully aligned to ensure that the end facets are located in the same object plane for imaging. A second objective lens (100 × , NA = 0.70) and a CCD camera are used to capture the intensity profiles of both coupler ports simultaneously. The coupling ratio is extracted from the CCD camera images, where optical power within the throughput or the crossover port is calculated by summing the values of CCD pixels within the given coupler port. In practice, it would be much easier to measure the coupling ratio using two photodetectors. The main reason that we use a CCD camera to measure coupling ratio is to ensure that we can easily obtain the spatial profiles of the guided modes for validation only. During the experiments, only the optical fields within two square windows (~5.6 µm × 5.6 µm in size, one for crossover port and one for throughput port) are used in coupling ratio calculation and data analysis. This area corresponds to a 49 × 49 pixel block in the CCD images. The fiber core, with a radius of 1.7 µm, is completely contained within this region.
3. AO-based mode control
AO-based mode control is carried out as follows. First, we define an objective function:
where Icross and Ithrough are the total optical intensities carried by the crossover and throughput ports respectively, and K is the target coupling ratio. (The coupling ratio always refers to the percentage of optical power within the crossover port.) The optimization procedure basically follows the stepwise sequential algorithm described in [9]. In our case, mode control is achieved by adjusting the SLM to minimize the objective function f(k) defined in Eq. (1). In this process, each one of the 11 × 11 SLM blocks produces 11 different phase shifts between 0 to 2π for the reflected wave. During optimization, we change the phase shift of one, and only one phase block. As we cycle the phase shift of each block from 0 to 2π, the coupling ratio changes accordingly. After obtaining the 11 objective functions produced by the 11 different phase shifts, we fix the phase shift of the current block to the one that minimizes the objective function. Then, we move to the next phase block and repeat this optimization procedure. This optimization process is sequentially carried out for all phase blocks, and is referred to as one optimization cycle.Using the algorithm described above, we adaptively set the coupling ratio of the directional coupler to 7 different values. The target coupling ratio ranges from 0 to 0.27, with 0.27 being the highest coupling ratio that can be achieved in our experiments. For all 7 cases, the initial field distributions and the final optimized intensity profiles for the crossover and the throughput ports are shown in Fig. 2. The initial coupling conditions are arbitrarily chosen, which typically leads to some small initial coupling into the crossover port. As a result, the initial field distribution in the crossover port tends to weak and contains substantial noises, as can be seen in Fig. 2.
Fig. 2 Optimization results with different crossover coupling ratios K. The optimized images of both ports are labelled accordingly. All images are normalized to their maximum intensities.
In Fig. 2(a), the coupling ratio of the crossover port is optimized towards zero. After adaptive optimization, the field distribution within the throughput port is almost purely LP01. This is to be expected, since within the coupler, the LP11 mode extends much further into the fiber cladding. As a result, the coupling ratio for the LP11 mode should be much higher than that of the LP01 mode. Hence the condition of zero crossover coupling should be equivalent to the requirement that the mode within the throughput port should be LP01 only. For all other cases, after optimization, the intensity profile of the crossover port is dominated by the LP11 mode. Again, this is consistent with the observation that only LP11 mode exhibit significant crossover coupling. For the throughput port, its intensity profile gradually changes from the LP01 mode to the LP11 mode, as we increase the target coupling ratio from 0 in Fig. 2(a) to 0.27 in Fig. 2(g).
Figure 3(a) shows the variations of coupling ratios during 7 optimization cycles for K = 0, 0.10, and 0.27, with individual cycles indicated by the dashed lines. Within each cycle, the coupling exhibits significant variations as we adjust the phase shift for each of the 11 × 11 SLM blocks. This is to be expected, since for any SLM block, a “wrong” phase shift can certainly increase the difference between the actual and the target crossover coupling. The results in Fig. 3(a) suggest that it only takes 3 to 4 cycles to reach the target coupling ratios.
Fig. 3 (a) The variation of the coupling ratios obtained during the optimization process. (b) Definition of the ratio MCR.
In order to quantitatively analyze the mode composition of the two coupler ports, we define a mode contrast ratio (MCR) as illustrated in Fig. 3(b). First, we define a small 5 × 5 pixel block that corresponds to the center of the fiber core. Then, we find the minimum intensity Imin within the central block as well as the maximum intensity Imax in the entire intensity profile. The MCR is defined as:
The MCR is quantitatively related to the percentage of LP01 mode power. To see this, we note that at the center of the fiber core, the LP11 mode intensity is zero. Therefore, a small MCR indicates that the majority of the optical power is carried by the LP11 mode, whereas a large MCR suggests that the field distribution is dominated by the LP01 component.We analyzed the MCR for both the crossover and the throughput ports for different target coupling ratios. The values of MCR during the entire optimization cycles are shown in Fig. 4 for several different target coupling ratios. From Fig. 4(a) we can see for the crossover port, the fluctuation of MCR is relatively small for three representative cases, and the ratio value is close to 0.06. This quantitatively confirms that the optical field within the crossover port is dominated by the LP11 mode. For the throughput port, it is clear that the MCR decreases as the target coupling ratio increases. This suggests that as the target K increases, the mode composition of the throughput port is increasingly dominated by the LP11 mode. Again, this result is consistent with the images in Fig. 2.
Fig. 4 The variation of the MCR obtained during the optimization process for (a) the crossover port; (b) the throughput port.
To further illustrate the relation between K and MCR, we plot MCR versus K for both the crossover and throughput port in Fig. 5, where each dot represents the values of K and MCR obtained using a specific SLM phase setting. Different dot color represents data obtained under different target K, with red for K = 0, green for K = 0.05, blue for K = 0.10, cyan for K = 0.15, yellow for K = 0.20, magenta for K = 0.25, and black for K = 0.27. In Fig. 5(a), we can see that excluding the obvious exception of K = 0, the MCR is always less than 0.1, regardless of K values. In Fig. 5(b), it is clear that the MCR of the throughput port decreases as the coupling ratio of the crossover port increases. Again, for the throughput port, the relationship between K and MCR clearly indicates that as we increase the target K values, we can gradually shift the composition of the throughput port mode from being dominated by the LP01 component to being dominated by the LP11 component.
Fig. 5 MCR versus the coupling ratio of the crossover port: (a) MCR of the crossover port; (b) MCR of the throughput port.
It is worth pointing out that the intensity distributions in Fig. 2 suggest that the guided mode in the crossover port may contain a small percentage of the LP01 mode. The deviation is likely caused by intermodal coupling in the TMF. First, we note that the fused directional coupler contains taper transition regions, which may induce some coupling between the LP01 and the LP11 mode. Second, the TMF is not completely straight and contains circular bends. Third, the TMF and the directional coupler are not based on the same type of TMF. Therefore, any offset at the splice point may cause intermodal coupling.
Finally, we point out that all experimental data are obtained without using any polarizer in front of the objective lens L2 or the CCD camera. In fact, the images in Fig. 2 are in general the sum of optical intensities associated with two independent polarization channels. It is interesting to note that using coupling ratio as the feedback, we can adaptively control the spatial profiles of the transmitted modes without considering their polarization.
4. Discussion
Several aspects of our work warrant additional discussion. Our experimental results suggest that for the directional coupler we use, only the LP11 mode can be coupled from the input port to the crossover port by a significant amount. The LP01 mode, on the other hand, remains mostly in the throughput port and experiences little crossover coupling. This phenomenon can be explained using the coupled mode theory [22]. Figure 6(a) depicts a directional coupler, where both coupler arms are constructed using the same TMF. To distinguish the two fibers, we denote the fiber associated with the input and the throughput port as fiber 1, and the fiber associated with the crossover port as fiber 2. Correspondingly, we use a superscript “1” to label themode in fiber 1 as , whereas denotes themode in fiber 2. Using this notation, the coupling ratio between the mode (in the input port) and themode (in the crossover port) can be expressed as [22]:
whereis the incident power (in the form of the mode), and is the coupled power into the crossover port (in the form of the mode), L is the coupling length, represents the difference in the propagation constant of the and modes (i.e., phase mismatch), and is the coupling constant between the and modes.Fig. 6 (a) Schematic of a two mode fiber 2 × 2 coupler and its refractive index profile; (b) Schematic of a 2 × 2 MSC that contains a MMF arm and a SMF arm. Its refractive index profile is shown in the bottom panel. Dashed lines in the refractive index profiles in (a) and (b) indicate the effective indices of various LP modes in fiber 1 and 2. (c) Schematic of a quasi-distributed sensor network that utilizes MSCs to provide feedback signals.
We can use Eq. (3) to estimate the coupling ratio between various LP modes in fiber 1 and fiber 2. Experimentally, we find a substantial percentage (~27%) of the LP11 mode in fiber 1 can be coupled into the LP11 mode in fiber 2. Since the two LP modes in the two fibers are identical, we have Δβ11;11 = 0. From Eq. (3), we can estimate the coupling coefficient between the two LP11 modes as κ11;11L = 0.55. Since the LP11 mode extends much further into fiber cladding, the coupling between the two LP11 modes should be much stronger than that of the two LP01 modes. Following [22], we can estimate κ01;01L to be 10 times smaller than κ11;11L . Assuming and Δβ00;00 = 0, we estimate the coupling ratio between the two LP01 modes in the fiber 1 and the fiber 2 is very small, i.e., . Similarly, we can also estimate the ratio for cross-mode coupling, i.e., from the LP01 mode in fiber 1 to the LP11 mode in fiber 2. For an order of magnitude calculation, we can assume the effective index difference between the LP01 and the LP11 mode to be. Using um, we find mm−1. According to Eq. (3) and assuming a reasonable coupler length, i.e., L = 2 mm [22], we find the cross-mode coupling should be less than 0.3%. In this estimate, we assume the sin2 term in Eq. (3) is one, and κ01;11L = κ11;11L, where both assumptions overestimate the value of . These estimates, based on standard coupled mode theory, suggest that only the LP11 mode can be coupled from the input port to the crossover port, which is consistent with our experimental results.
The method reported here can also be generalized to other types of mode selective couplers (MSCs) [22–24]. Figure 6(b) illustrates a MSC where fiber 1 (i.e., the input and the throughput ports) is a MMF that supports N LP modes (N ≥ 2), and fiber 2 (i.e., the crossover port) is a single mode fiber (SMF). Through proper fiber design or by applying techniques such as fiber pre-pulling, we can ensure that the effective index of the LP01 mode in fiber 2 to be very close to the effective index of the ith LP mode in fiber 1, i.e., , as illustrated in the bottom panel of Fig. 6(b). Since different LP modes possess different effective indices, it should be possible, according to Eq. (3), to ensure that only the ith LP mode in the MMF is coupled into the SMF. Such a MSC can be demonstrated using existing methods [22–24]. For mode monitoring in a sensor network, we can use the MSCs depicted in Fig. 6(b) and choose small coupling ratios (e.g., 1%). In this case, the MSCs essentially serve as mode selective tap couplers. Figure 6(c) shows the schematic of a quasi-distributed MMF sensor network with n fiber sensors, where each sensor is monitored by a MSC. Under this design, if we fix the total power of the incident signal, by maximizing or minimizing the optical power carried by the crossover port of a specific tap coupler, we should be able to select the form of the LP mode that enters into the corresponding fiber sensor. Our results in Fig. 5(b) suggest that it may also be possible to adaptively control the interrogation signal entering into any given fiber sensor to be a specific mixture of different LP modes. A detailed design of such a sensor network, however, is beyond the scope of this paper.
Finally, it is worth mentioning that in our experiments, the orientations of the polarizers (P1 and P2) are chosen to ensure proper SLM operation. Based on the coupled mode analysis given earlier, our coupler-based method for mode control should not depend on light polarization, as long as the coupler does not exhibit significant polarization dependent coupling or loss.
5. Conclusion
In conclusion, we demonstrate that it is possible to use the coupling ratio of a directional coupler as feedback signal for adaptive mode control. Specifically, by minimizing the coupling ratio, we can ensure that the optical field within the throughput port is dominated by the LP01 component. Conversely, by maximizing the coupling ratio, we can ensure that the optical fields in both the crossover and the throughput ports are dominated by the LP11 component. The work can serve as a proof-of-concept demonstration that by incorporating multiple feedback elements within a MMF network, it is possible to adaptively control the guided modes within the MMF network in a quasi-distributed manner.
Acknowledgment
The research is funded by NSF (CMMI 1400195), whose support is gratefully acknowledged.
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