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We experimentally demonstrate an adaptive-optics-based approach that allows selective excitation of waveguide modes and their mixtures in a two-mode fiber (TMF). A phase-only spatial light modulator is used for wavefront control, using feedback signals provided by the correlation between the experimentally measured field distribution and the desired mode profiles. Experimental results show the optical field within the TMF can be shaped to be pure linearly polarized (LP) modes or their combinations. Analysis shows selective mode excitation can be achieved using only 5 × 5 independent phase blocks. With proper feedback signals, this method should enable one to precisely control the optical field within any multimode fiber or other types of waveguides in real time.
© 2014 Optical Society of America
1. Introduction
Recently, spatial division-multiplexing (SDM) has attracted much attention in optical communications [1–3]. A major goal of SDM is to use space as a multiplexing parameter to overcome the capacity limit of single mode fibers [4, 5]. One approach of SDM is mode division multiplexing (MDM), which utilizes individual waveguide modes within multimode fiber (MMF) as communication channels for signal multiplexing and demultiplexing. A key component of MDM is the excitation of specific waveguide mode in a MMF. In existing literature, this task can be accomplished using multiple techniques, such as stress fiber [6], spot-based coupler or spatial beam sampler [7, 8], photonic lanterns [9–11], phase masks [12, 13] or spatial light modulators (SLM) [14–18].
Despite impressive progress so far, most of the existing methods for mode control have certain drawbacks. For example, the phase mask approaches described in [12, 13] are static in nature and cannot be easily adjusted in real time. Other research groups have utilized SLM to achieve selective mode excitation [14–18]. However, the method described here is different from existing approaches in [14–18]. Specifically, our adaptive optics (AO) based approach requires no prior knowledge of the optical system or the intermodal coupling within the fiber network. In fact, our method treats the entire optical system, including both the input coupling system and the few-mode fiber (FMF) itself, as a “black box”. In contrast, the method reported in [15–17] requires knowledge of the optical system and is based on linking optical near field to far field through a Fourier transform [15]. Furthermore, the method in [15–17] is implemented using computer generated holography, which is both computationally intensive and time consuming. As an additional example, the method reported in [18] requires careful optical alignment and pre-calibration of the SLM to achieve pre-determined phase and amplitude modulation for selective mode excitation.
For many applications in optical communications and sensing, one needs to ensure that the optical field at a given location within a MMF network becomes either a specific linearly polarized (LP) mode, or a mixture of LP modes with proper amplitudes and phases for individual LP components. For a MMF network with strong and time-dependent intermodal coupling, we may not even know how optical waves propagate within such a network. Therefore, there is an urgent need to develop a technique that can achieve selective mode excitation within a fiber network that support multiple LP modes, without any prior knowledge of the underling optical systems. The main purpose of the present paper is to demonstrate the feasibility of use AO techniques to achieve this goal. Additionally, the method presented here is generic, easy to implement, requires a relatively few number of independent phase control elements, and can deliver mode control in real time.
The AO-based approach, as illustrated in Fig. 1(a), requires a feedback signal that is directly associated to the desired mode properties. For example, in present paper, the desired outcome is that the optical field at the fiber output becomes specific LP modes or their combinations. For the convenience of implementation, the feedback signal in present work is simply the correlation between the desired mode profile and the optical field distribution captured by a CCD camera. Potential candidates for feedback signals, however, are by no means restricted to this particular choice. For future applications, the feedback signal could be the reflection peak of a fiber Bragg grating, the coupling ratio of a directional coupler, or even nonlinear signals generated by processes such as Brillouin scattering, Raman scattering, and four wave mixing. After choosing the proper feedback, we use the SLM to modify the wavefront and measure the feedback signal produced by the new wavefront. The modified wavefront is retained (or rejected) depending on whether the feedback signal suggests better (or worse) match between the measured outcome and the desired target. This optimization cycle is repeated until the desired target is reached.
Fig. 1 (a) Principle of AO-based mode control. (b) 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, P2 and P3, polarizers; L1, 20 × objective lens, NA = 0.40; Fiber, Thorlabs 980HP, ~1.5m; L2, 100 × objective lens, NA = 0.70; CCD, CCD camera. (c) Optimization sequence of SLM blocks (N = 13).
Conceptually, the aforementioned AO-based approach is similar to those used in scattering medium focusing [19–27], imaging [28–31], or particle manipulation [32]. The main difference is that we are primarily interested in achieving highly selective excitation of LP modes for communication and sensing purposes, i.e., optical “focusing” in modal domain. As mentioned earlier, such ability is important for MDM-based fiber communications, and may find a wide range of applications in optical sensing and nonlinear fiber optics.
In this work, we demonstrate AO-based selective excitation of LP modes in a two-mode fiber (TMF). In Section 2, we describe the experiment setup. In Section 3, we introduce our adaptive algorithm for wavefront shaping. In Section 4, we present different experimental results of selective mode excitation. Examples include selective excitation of only the LP01 mode, only the LP11 mode, or a specific mixture of the LP01 and the LP11 modes. In Section 5, we describe several important features of the AO-based selective mode excitation processes. Additionally, we show that it is possible to utilize only 5 × 5 phase blocks to achieve highly selective mode excitations. Finally, we summarize our work in Section 6.
2. Experiment setup
Our 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 1.5m). 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 the LP11 modes.
The light polarization is controlled by a half-wave plate and three polarizers in the setup. The first two polarizers (P1 and P2) select optical waves with polarization direction parallel to the optical table, which happens to be the same as the phase modulation axis of the SLM. The polarizer in front of the CCD (P3) ensures that we only monitor a single polarization component of the optical output. The polarization direction of P3 is randomly chosen, excluding the obvious case where P2 and P3 are orthogonal to each other. The expanded He-Ne laser beam (FWHM ~8 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 6.2 mm by 6.2 mm.) This area is evenly divided into 13 × 13 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, a second objective lens (100 × , NA = 0.70) and a CCD camera are used to measure the output intensity profile. From the measured intensity profile, we calculate its correlation with the desired target mode profile as the feedback signal for SLM control.
3. AO-based mode control
AO-based mode control is carried out as follows. From the fiber specification, we can theoretically calculate the desired target profile at the fiber output. More specifically, a linear combination of the LP01 and the LP11 mode with a specific polarization can be expressed as:
Here E01(r, φ) and E11(r, φ) denote electric fields of the LP01 and LP11 modes with a specific polarization, respectively. A01 and A11 are the amplitudes of two modes, and β represents their relative phase difference. To simplify the procedure for mode decomposition in later sections, we assume both E01(r, φ) and E11(r, φ) to be real.With the theoretical mode profile known, we can calculate the correlation between the target intensity profile and the measured intensity profile at the fiber output, and use this correlation function as the feedback for mode control. Specifically, we define an objective function based on the CCD intensity profile Ik and the target intensity profile I0:
where and are the average of I0 and Ik, respectively. According to Eq. (2), it is clear that the optimization function f(k) quantitatively described the difference between the actual CCD intensity profile and the desired target, with smaller f(k) indicating better match. Ideally, if the CCD intensity profile forms a perfect match with the target profile, f(k) should become 0.The optimization procedure basically follows the stepwise sequential algorithm in [20]. In our case, mode control is achieved by adjusting the SLM to minimize the objective function f(k) defined in Eq. (2). The SLM is grouped into 13 × 13 equally sized blocks. Upon reflection, every block can produce 11 different phase shifts between 0 to 2π for wavefront control. Every block is optimized individually using the sequence shown in Fig. 1(c). During the optimization process, 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 optical intensity at the fiber output changes accordingly. In particular, at each phase level, we use the intensity profile captured by the CCD camera to calculate the optimization function f(k), which should depend on the value of the phase shift. 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.
The time required for one optimization cycle with 13 × 13 phase blocks is roughly 192 s. For a proof-of-concept demonstration, there is no need for using faster but more complex optimization algorithms. In future, we expect the faster algorithms and methods described in [21, 22, 24–27] should significantly improve the speed of selective mode excitation.
4. Experimental results
By using the experimental setup and adaptive algorithm described above, we have adaptively controlled optical mode content at the fiber output. During the experiment, only the optical field within or near the fiber core (approximately 5.4 µm × 5.4 µm in size) is used in the evaluation of the optimization function. (It corresponds to a 61 × 61 pixel block in the CCD image. The fiber core, with a radius of 1.8 µm, is completely contained within this region.) In our studies, we considered three different cases. In case 1 and 2, we adaptively covert the optical mode at the fiber output to be either purely LP01 (in case 1) or purely LP11 (in case 2). Then, in case 3, we excite a mixture of LP01 and LP11 with pre-determined mode coefficients.
The results for case 1, i.e., exciting only the LP01 mode, are shown in Fig. 2. Figures 2(a) and 2(b) show the “before” and the “after” optimization images for four experiments with randomly selected initial intensity profiles. In Figs. 2(a) and 2(b), the “before” and “after” images that belong to the same optimization process are denoted using the same sequence number. The difference between the optimized intensity distribution and the theoretical target is less than 0.19% for all four cases. The cross-sections of the optimized field intensity (as captured by the CCD camera, represented as dots) and the theoretical target (represented as solid lines) are shown in Figs. 2(c) and 2(d). After optimization, the agreement between the actual mode profile and the theoretical target is excellent.
Fig. 2 Selective excitation of the LP01 mode. Images in (a) are the initial field distributions. Results in (b) are the images captured after the optimization process. Figures with the same sequence number belong to the same optimization process; (c) and (d) show the horizontal and the vertical cross-section of the intensity profiles (along the dashed blue line in the inset) of both the theoretical target (solid line) and the experiment results (dots), respectively.
Figure 3 shows experimental results for case 2, where we excite only the LP11 mode at the fiber output. Again, Figs. 3(a) and 3(b) show the “before” and “after” optimization images for four experimental studies with different initial intensity profiles. After optimization, the difference between the actual CCD camera image and the theoretical target is less than 0.75% for all four cases. As previously, the cross-sections of theoretical and actual “after” optimization mode profiles are shown in Figs. 3(c) and 3(d).
Fig. 3 Selective excitation of the LP11 mode. Intensity profiles in (a) and (b) denotes the “before” and “after” optimization of four different optimization processes; (c) and (d) show the horizontal and the vertical cross-section (along the dashed blue line in the inset) of the intensity profiles for both the theoretical target (solid line) and the experiment results (dots), respectively.
Results in Figs. 2 and 3 demonstrate that the AO-based approach can produce either the LP01 or the LP11 mode with very high selectivity, regardless of the initial mode profiles. Next, we consider, in some sense, the worst case scenario, where we deliberately choose the LP01 mode as the initial profile and select the LP11 mode as our target. The “before” and “after” optimization images, as captured by the CCD camera, are shown in Figs. 4(a) and 4(b). Again, it is clear that we can achieve complete mode conversion from the LP01 to the LP11 mode.
Fig. 4 Converting the LP01 mode (shown as the initial distributions in (a)) to the LP11 mode (shown as the optimized intensity profiles in (b)). Figures labeled as the same sequence number belong to the same optimization process.
Finally, we consider case 3, where we use the adaptive algorithm to excite a superposition of LP01 and LP11 modes. The results are shown in Fig. 5. In Figs. 5(a)–5(d), the target intensity profiles are generated using the theoretical LP01 and LP11 mode profile with different amplitude ratios and phase contrasts. The difference between theoretical targets and actual CCD camera images are less than 1%. In Figs. 5(e)–5(h), we use some arbitrarily chosen CCD camera images as target profiles. The differences between the target profile and the “optimized” results are less than 0.7% for all four cases.
Fig. 5 Excitation of a specific mixture of the LP01 and the LP11 modes. Targets in (a) to (d) are obtained through theoretical calculations, with specific amplitude ratios (AR = A01/A11) and phase differences β, as defined in Eq. (1). (a) AR = 0.5, β = π/2. (b) AR = 0.5, β = π/4 (c) AR = 0.2, β = π/2. (d) AR = 0.2, β = π/4. In (e)-(f), we choose four representative CCD camera images, captured experimentally, as the targets used in the optimization process.
5. Optimization process analysis
In this section, we discuss several important aspects of the optimization processes. Figure 6 shows the variations of objective functions during 7 optimization cycles. We consider three different cases, with the LP01 mode being the target in Fig. 6(a), the LP11 mode as the target in Fig. 6(b), and a mixed mode as the target in Fig. 6(c). All results in Fig. 6 are divided into individual cycles, as indicated by the dashed lines. Within each cycle, the objective function exhibits significant variations as we adjust the phase shift for each of the 13 × 13 SLM blocks. This is to be expected, since for any SLM block, a “wrong” phase shift can certainly increase the difference between the captured CCD image and the target profile. Furthermore, recall that for each optimization cycle, we use the minimum value of the objective function to determine the optimal phase shift for the desired incident wavefront. Consequently, the results in Fig. 6 suggest that it only takes 3 to 4 cycles to reach the optimal wavefront for selective mode excitation. And after 7 optimization cycles, the deviation between the actual CCD image and the target profile are less than −27 dB for the LP01 case and fewer than −20 dB case for the LP11 case. Several factors may account for the differences between the theoretical targets and the actual optimized field distributions. For example, the actual index distribution of the fiber may not be the simple step-index profile assumed in our theoretical calculations. Additionally, the fiber output facet may not be perfectly flat, which can also cause deviations between the theoretical target profiles and the CCD camera images.
Fig. 6 The variation of the objective function obtained during the optimization process: (a) Optimize the mode to LP01 mode; (b) Optimize the mode to LP11 mode; (c) Optimize towards a mixed LP01/LP11 mode.
We also analyze the mode profiles captured by the CCD camera after each optimization cycle. Specifically, we decomposed the CCD image into a linear superposition of the LP01 and the LP11 mode components. The square of the amplitude ratios of LP01 mode to LP11 mode are shown in Fig. 7. Figures 7(a) and 7(b) confirms that after optimization, the optical mode at the fiber output is clearly dominated by the desired target, i.e., LP01 for Fig. 7(a) and LP11 for Fig. 7(b). Figure 7(c) shows the mode decomposition results of the four cases shown in Figs. 5(a)–5(d), where we use a mixed mode as the optimization target. In Figs. 5(a) and 5(b), the target mode amplitude ratio between the LP01 and the LP11 mode is A01/A11 = 0.5. The corresponding values of A012/A112 measured during the optimization process, are shown as the red and green lines in Fig. 7(c). In Figs. 5(c) and 5(d) the target mode amplitude ratio is A01/A11 = 0.2, and the corresponding values of A012/A112 measured during the optimization processes, are represented as the blue and magenta lines in Fig. 7(c). Results in Fig. 7 also suggest that we can reach the LP modes or their mixtures after only 3-4 optimization cycles.
Fig. 7 The variations of mode ratio A012/A112 during the optimization cycles. Mode is shaping to (a) LP01 mode, (b) LP11 mode, and (c) mixed modes. The legend in (c) shows the mode ratios A012/A112 of the targets used in optimization.
Finally, for certain applications in optical communications and sensing, it is often desirable to reduce the amount of independent SLM phase blocks used in selective mode excitation. (For example, fewer independent blocks should increase optimization speed and reduce system cost.) Here, we show that it is possible to use only the central 5 × 5 SLM blocks (out of the 13 × 13 phase blocks) to achieve highly selective mode excitation. Three different sets of “before” and “after” optimization images, as well as their corresponding target intensity distributions, are shown in Figs. 8(a)–8(c). For all three cases, the deviation between the target and the optimized image is less than 0.6%. The variations of objective functions during 7 optimization cycles are also shown in Figs. 8(a)–8(c). Comparing the results shown in Fig. 8 (5 × 5 blocks) with the results obtained using 13 × 13 blocks in Figs. 6(a)–6(c), there is no significant degradation in the performance of SLM-based optimization.
Fig. 8 Results of selective mode excitation using 5 × 5 blocks with different initial and target intensity profiles. The initial intensity distributions, the target intensity distributions, and the optimized images are labelled accordingly in (a)-(c). The optical output field is shape to (a) the LP01 mode, (b) the LP11 mode and (c) a mixture of the LP01/LP11 modes (the target is an experimentally captured CCD camera image). The objective functions obtained during the optimization process are also shown in (a)-(c).
We further reduce the independent SLM phase blocks from 5 × 5 to 3 × 3 central blocks (out of the 13 × 13 phase blocks). With fewer phase blocks, the performance of SLM optimization degrades noticeably. Figure 9 shows two representative examples, where the target mode is the LP01 and the LP11 mode, respectively. The difference between the target mode and the optimized intensity distribution is ~7.5% and ~6.0% respectively, and is noticeable through visual comparison.
Fig. 9 Results of selective mode excitation using 3 × 3 blocks with different target intensity profiles. The initial intensity distributions, the target intensity distributions, and the optimized images are labelled accordingly in (a)-(b). The optical output field is shape to (a) the LP01 mode, (b) the LP11 mode. The objective functions obtained during the optimization process are also shown in (a) and (b).
6. Discussion and conclusion
Three aspects of our method deserve further comments. First, in present work, we do not consider the issue of coupling efficiency. Primarily, this is because for applications such as optical sensing, coupling efficiency is not as critical as in optical communications, where every dB counts. Additionally, it should not be too difficult to incorporate coupling efficiency into the optimization process. For example, we may first choose total output power as the optimization parameter and increase it to the largest possible extent. Then, we may apply the method described here for selective mode excitation. We may also alternate optimization parameters in different optimization cycles. For example, we may select optical output power as the to-be-optimized parameter for the odd optimization cycles, and choose the objective function in Eq. (2) as the optimization parameter for the even cycles. We may also define a different optimization parameter that depends on both the coupling efficiency and the correlation with the desired mode patterns. Given the fact that the algorithm described here has been widely used for optical focusing through diffusive media [19–27], where optical power is a critical factor, the issue of coupling efficiency should not become a major hurdle for our AO-based approach.
Additionally, we consider only a single polarization component for both the input and the output optical fields. For more general scenarios, we may replace P2 in Fig. 1(b) with a polarization beam splitter (PBS) and P3 with another PBS. In this case, the study carried out here corresponds to a specific combination (out of four possible choices) for the input / output polarization channels [17]. We can add more bulk optics components so that all four input / output polarization channels are monitored simultaneously using the same SLM and CCD camera. The added complexity, however, is unlikely to reveal anything new. This is because in our studies, the orientation of P3 is randomly chosen, excluding the obvious case where P2 and P3 are orthogonal to each other. We did not observe any dependence or correlation between the optimization results and P3 orientation.
Finally, the results presented here are obtained using a TMF. For a FMF that supports a limited number of modes (for example, 4 or 6), the results reported here should remain applicable. However, for a MMF that supports hundreds of modes, some of the conclusions may no longer apply. For example, we may need to use more phase control elements to distinguish different LP modes within the MMF. However, the overall framework of the AO-based approach should still apply. Again, this expectation is based on the successful demonstration of AO-based focusing [19–27], which involves hundreds, if not tens of thousands independent scattering channels.
In summary, we experimentally demonstrate the feasibility of using AO to achieve highly selective mode excitations in a TMF. In this proof-of-concept demonstration, we use the correlation between the CCD camera image and the theoretical target profile as the feedback for SLM control. The target profile can be purely LP01, purely LP11, or a mixture of the two modes. Furthermore, selective mode control can be accomplished using as few as 5 × 5 independent phase elements.
The method developed here can be easily generalized to cases where the feedback is provided by signals produced by mode-selective elements, including fiber Bragg gratings and mode selective couplers. The AO-based approach may find applications in MDM-based fiber communications and optical sensing.
Acknowledgments
The research is partially funded by NIH (NIBIB, HL098912, 1R21EB017819-01), whose support is gratefully acknowledged.
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