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Abstract
Monte-Carlo simulations of optical speckle noise are performed to predict the range of gain fluctuations for photonic devices which multiplex many single-moded inputs into single multimode waveguides. Here, two waveguides are simulated which bound the cases of interest: a few mode fiber and a standard multimode fiber. When fully-excited and after spatial-filtering by a 10µm photodiode, the former’s gain variations can range up to the mean value of the gain itself, ΔG ≈ 〈G〉, whereas for the latter, ΔG ≈ 3.4〈G〉. In certain cases, ΔG can be reduced by offsetting the photodiode relative to the waveguide, results which cannot be predicted using standard analytical speckle noise theories.
1. Introduction
The coherent summation of multiple radio-frequency (RF) signals is a key signal processing functionality for many applications, from filtering [1,2] to phased array antennas (PAAs) in receive mode [3,4]. Coherent radio-frequency summation methods in the optical domain, where the microwave/RF signals are modulated onto one (or many) optical carriers, have many significant advantages compared to purely RF/electrical domain methods. In particular, the often cited advantages include ultra-wide bandwidths, low weights, and small footprints [5]. One of the least complex, smallest footprint, and arguably lowest cost optical domain coherent RF summation method is to couple the many separate single-mode inputs into a single multi-mode (MM) waveguide, which is then coupled to a solitary photodetector (PD). Typically, this form of mode-division multiplexing is accomplished using free-space optics into a multimode fiber [6,7] or by using adiabatically tapered photonic lantern-style devices deemed ‘optical concentrators’ [8,9]. Of course, the relative simplicity of this method comes at a cost. With a single laser source in the coherent regime, optical phase-to-intensity noise conversion can become a non-negligible source of RF gain instability if mode-selective losses in the multiplexers and, critically, losses due to overfilling of the PD are not minimized. These coherent interference effects can induce drastic power fluctuations or ‘interferometric fading’ effects, destroying the RF gain stability of a photonic link. The optical phase noise arises from, for example, the separate optical fiber delay lines of a transversal filter or from the individual receiving elements of a PAA.
In order to efficiently couple and guide all of the RF-modulated optical signals, the terminal MM waveguides in an RF summation link must, at the very least, support the same number of modes as the number of inputs [10]. Hence, the waveguides must have high V numbers (i.e. high numerical apertures (NAs) and/or large core diameters). On the other hand, surface-normal PD’s with ≥40 GHz 3 dB bandwidths (BWs) generally have small active areas with radii ≤5µm. Hence, coupling highly MM, high NA outputs onto small area, large bandwidth PDs without loss is a non-trivial design and engineering challenge. If the MM outputs are spatially-filtered, optical phase-to-intensity noise conversion can occur and the resulting optical power fluctuations could easily become a major and even dominant source of noise. While standard analytical modal noise and speckle noise theories are sufficient for predicting the statistics of overfilled (i.e. fully-excited) waveguides with multitudes of modes (usually >500) [11–13], they are generally insufficient to describe waveguides that are under-filled and/or that guide less than a few hundred modes [14], the typical operating regime for coherent RF summation devices at telecom wavelengths. Incidentally, it is also the operating regime for mode-division multiplexed few-mode and multimode fiber communications systems [15].
In this work, Monte-Carlo (MC) simulations are used to predict the maximum range of possible RF gain fluctuations arising from optical speckle noise in coherent RF summation links which terminate in MM waveguides. The MC simulations are performed for two specific example cases where N-separate single mode inputs are multiplexed into 1) a six mode Few-Mode Fiber (FMF) and 2) a standard Graded-Index Multi-Mode Fiber (GI-MMF). The predicted range of RF gain fluctuations display non-negligible dependence on the input-to-mode distributions and the geometry of the PD relative to the output of the waveguides, results which cannot be predicted using the limiting statistics of standard analytical theories [14]. In addition to the two examples shown in this work, these simulations could also be applied to any multimode waveguide where speckle noise is an issue and the output modes are known.
2. Monte-Carlo simulations of multimode speckle noise
In order to numerically simulate multimode speckle noise, the eigenmodes of the FMF and MMF were first determined using Finite-Element Method (FEM) simulations using their known 2-D refractive index profiles. The normalized mode electric-fields, ${E_m}$, from the FEM simulations were then utilized in a model where the optical phase noise from each of the N-separate inputs were represented as N-individual stochastic processes. The N-inputs each excite a subset of adjacent modes in the fibers, resulting in a total electric field comprised of the double summation:
Here, ${\phi _n}$ is the randomized optical phase and ${a_n}$ is the amplitude of the nth input. in and jn are the indices of the adjacent modes excited by the nth single-mode input, and bm is the normalized weighting coefficients of the mth modes. Figure 1 displays the general summation architecture as well as a few example output speckle patterns, ${|{{E_T}} |^2}$, for a commercially-available six mode FMF at λ=1531 nm.
Fig. 1. Left- Simulated RF summation link architecture. Right- Example speckle patterns ${|{{E_T}} |^2}$ for three different iterations labeled k. The yellow circle corresponds to the geometric core of the FMF.
Assuming a very short propagation distance to the PD via the FMF or MMF, propagation effects, including mode coupling, mode-selective losses, dispersion, and modal noise, were considered negligible. The fibers were considered directly butt-coupled to a PD with zero standoff. Hence, for each k, the normalized optical power received by a PD was calculated using:
In speckle/modal noise theory the statistics are generally characterized by the signal-to-noise ratio (SNR), defined as the ratio of the ensemble average optical power to the standard deviation, $SNR = \left\langle P \right\rangle /{\sigma _P}$ [11,12], or by the speckle contrast, defined as the normalized variance, ${\gamma ^2} = SN{R^{ - 2}}$ [13,16]. However, from a purely practical standpoint, any instantaneous deviations in a photonic link’s RF gain could be disastrous in operation, regardless of the number of samples or probability of occurrence. Hence, this work will focus on quantifying the maximum possible range or, equivalently, the ‘worst-case’ RF gain fluctuations arising from variations in the output speckle pattern. By focusing on the worst case scenario, specific a priori knowledge of the optical phase noise distributions for the N-separate inputs are not required. To this end, the ${\phi _n}$s in Eq. (1) were generated from uniform random distributions ranging between ±π, ensuring the simulated speckle patterns assumed a majority of its possible configurations with a sufficiently large number of iterations (typically k=8000 was used to obtain relatively smooth probability distributions). Moreover, the N input polarizations were perfectly aligned, maximizing the possible optical interference. Since the RF gain is proportional the square to of the optical power ($G \propto P_{}^2$) [5], the variation can be defined as $\Delta G = 2P\Delta P,\; $ which can be interpreted as
where $\left\langle P \right\rangle $ is the ensemble average optical power and ${P_{max}}$ and ${P_{min}}$ are the maximum and minimum excursions of P over all ks. Finally, Eq. (4) is normalized to the ensemble average RF gain so3. Simulation results
3.1 Six mode few-mode fiber
For the first example case, N=6 inputs excite all six modes of a commercially-available FMF (excluding polarization degeneracies) at λ=1531nm, with a one-to-one input to mode distribution (bm=n=1) and with all ans equal to 1. As a reminder, the input polarizations are perfectly aligned; the worst case for optical interference.
Plotted in Fig. 2(a) are the calculated probability distributions of P for the six mode FMF as the PD radius, rPD, is varied. Here, the geometric centers of the PD and FMF are aligned with no offset. When rPD≥10µm, the ensemble averages of P, $\left\langle P \right\rangle $, of each distribution are nearly unity and their widths are extremely narrow (but still finite). In other words, practically the entire speckle patterns are captured by the PD and, consequently, there is minimal phase-to-intensity noise conversion due to spatial filtering. Notably, some of the higher-order modes of the FMF have Ems which extend into its cladding. Hence, the detrimental effects of spatial filtering begin to appear around rPD≈9µm rather than rPD=a=7.5µm, where a is the geometric radius of the FMF’s core. As rPD decreases below 9µm, the $\left\langle P \right\rangle $s of the distributions decrease and their respective widths increase, i.e. more light is lost off the PD and, as a result, the range of possible optical power fluctuations increases. The inset of Fig. 2(a) displays the probability distribution of P for rPD=5µm, the typical radii of commercially-available PDs with ≥40GHz BWs. Figure 2(b) plots the normalized RF gain variations (Eq. (4)), Δ${G_N} = \Delta G/\left\langle G \right\rangle $, which arises from the variations in P. Here, the effects of spatial filtering on ΔGN are relatively straightforward; as rPD decreases and hence spatial-filtering increases, ΔGN increases exponentially. For example, at rPD=a=7.5µm, the RF gain variations due to optical speckle noise could range up to ΔG≈ 0.15$\left\langle G \right\rangle$. However, with a reduction of only 2.5µm to rPD=5µm, ΔGN becomes approximately 1 and hence ΔG≈$\left\langle G \right\rangle $, representing potentially disastrous RF gain variations in operation.
Fig. 2. a) Probability distributions of P plotted for various values of rPD for N=6 inputs exciting all six modes of the FMF. Inset— Probability distribution of P for an rPD of 5µm. b) rPD vs. ΔGN determined from Eq. (5). The orange line demarks the defined radius of the FMF’s core.
Figure 3(a) plots the probability distributions of P for a fixed rPD of 5µm as the FMF-to-PD offset is varied, illustrating the detrimental effect even small offsets could have on the $\left\langle P \right\rangle $s and the distribution widths. Notably, their widths increase and then narrow as the offset increases since for the lower order modes Em→0 at larger radial offsets, i.e. the lower order modes have been effectively filtered out. Finally, Fig. 3(b) plots ΔGN vs. the FMF-to-PD offset. Here, with spatial-filtering, misalignments of only a few micrometers could drastically increase ΔGN, which increases linearly with offset with a slope of ≈0.3 $\left\langle G \right\rangle $/µm.
Fig. 3. a) Probability distributions of P vs. FMF-to-PD offset for the six mode FMF impinging on an rPD=5µm PD. b) ΔGN for the rPD=5µm vs. FMF-to-PD offset.
3.2 Graded-index multimode fiber
For the second example case, N inputs excite a selectable subset of modes in a standard a=25µm radius core GI-MMF at 1531nm with a one-to-one input to mode distribution (bm=n=1), all ans equal to 1, and with the input polarizations again perfectly aligned.
Figure 4(a) and (b) display the probability distributions of P as rPD is varied for two example input-to-mode distributions in the GI-MMF, with the geometric centers of the PD and MMF axially-aligned with no offset. In Fig. 4(a), N=7 inputs excite the first seven lowest order modes of the MMF (a ‘restricted-launch’ or ‘under-filled’ case), with the mode distribution shown in the inset. Each bar in the inset is a separate input. Similar to the FMF, the effects of spatial filtering are relatively straightforward; a reduction in $\left\langle P \right\rangle $ and an increase in the widths of the distributions occur as the spatial-filtering and hence optical loss off the PD increases. Since the mode-distribution is weighted toward the lower order modes with smaller mode areas, noticeable (in Fig. 4(a)) changes in $\left\langle P \right\rangle $ do not begin to appear until rPD ≈18µm. In contrast, Fig. 4(b) plots the probability distributions of P for N=56 inputs exciting all 56 modes of the MMF (the theoretical maximum number of inputs for a single polarization [6]). Here, as rPD decreases below a, the detrimental effects of spatial-filtering begin to appear almost immediately due to the larger mode areas of the higher-order modes. Finally, Fig. 4(c) plots ΔGN vs. rPD for the mode distributions of a) and b). Since increasing N fundamentally requires exciting more modes [10], an increased ΔGN is incurred over all rPDs.
Fig. 4. a) Probability distributions of P vs. rPD for the GI-MMF where N=7 inputs excite modes 1 through 7 of the example GI-MMF. Inset: Mode distribution for a). Each red bar corresponds to a different input. b) Probability distributions of P vs. rPD where N=56 inputs excite modes 1 through 56. Inset: Mode distribution for b). c) ΔGN vs. rPD for the two example mode distributions of a) and b).
Figure 5(a), (b) and (c) illustrates the effects of a MMF-to-PD offset on P for a fixed rPD of 10µm for three different input-to-mode distributions. Specifically, Fig. 5(a) displays the probability distribution of Ps for the ‘low’ input-to-mode distribution where the seven lowest order modes are excited by N=7 inputs (inset). Here, the probability distributions of P behave similar to those of the FMF when the offset is varied, with a straightforward reduction in $\left\langle P \right\rangle $ and increase in distributions widths as the offset increases. In contrast, Fig. 5(b) displays the Ps vs. offset for the ‘mid’ input-to-mode distribution, where modes 24 through 30 are excited. Compared to a), the $\left\langle P \right\rangle $s of the ‘mid’ distribution decrease significantly to ≈0.2 due to their larger mode areas. Interestingly, however, as the offset increases the widths of the distributions decrease to a minimum near ≈5µm. Similarly, the high input-to-mode distributions’ ΔPs decrease to a minimum near an 8µm offset. Finally, Fig. 5(c) displays the ΔGN vs. MMF-to-PD offset for the three example mode distributions, showing the resulting minimums in RF gain variations due to the minimums shown in Fig. 5(b) and (c). These spatially-anisotropic statistics in P and therefore ΔGN may be related to, for example, offsetting of the fiber cores in double-clad fiber amplifiers to improve the pump absorption/efficiencies [17,18] and will be the subject of further investigation.
Fig. 5. a) P vs. MMF-to-PD offset for the GI-MMF with N=7 inputs exciting modes 1 through 7 (inset). b) P vs. offset for N=7 inputs exciting modes 24 through 30 (inset). c) P vs. offset for N=7 inputs exciting modes 49 through 56 (inset). d) ΔGN vs. offset for a fixed rPD of 10µm showing the minimums in RF gain variations for the higher-order mode distributions.
4. Discussion and conclusion
MC simulations of optical speckle noise have been performed which predict the maximum possible range of RF gain fluctuations for two specific example cases where many single-moded inputs are multiplexed into a FMF or a GI-MMF for coherent RF summation onto a single PD. For the fully-filled FMF and MMF cases, spatially-filtering their outputs could incur drastic penalties in their RF gain stabilities. For these specific cases, ΔGN could be reduced by utilizing high NA micro-optics in order to more tightly focus their outputs onto a PD. However, extremely tight fabrication and alignment tolerances would be required. As another mitigation measure, the relative polarizations of adjacent inputs could be rotated by $n\pi /2$ in order to minimize optical interference effects. Unfortunately, these options would significantly increase the complexity of the links. In the end, for the fully-filled cases the simplest recourse towards minimizing the gain variation is to capture all of the light by using a larger area PD, at the cost of reduced RF BW. Intriguingly, however, the simulations show for the under-filled MMF the RF gain variations could be minimized by weighting the input-to-mode distributions towards the higher order modes and adding a deliberate spatial-offset relative to the detector. Of course, in practice it may be difficult to excite and maintain the input-to-mode distributions shown in Fig. 5. Regardless, the example MC simulations demonstrate the nuance and complexity of MM speckle noise in the under-filled regime, where the actual mode profiles of the specific waveguides must be taken into account to fully predict the effects of spatial-filtering.
Of course, here only the DC photocurrent has been considered. It would be interesting to incorporate non-zero RF-frequency components into the MC simulations in order to determine how, for example, the input-to-mode distributions and waveguide-to-PD offsets affect the shape (i.e. the filter coefficients) of an RF transversal filter, particularly at the gain variation minimums of Fig. 6. Additionally, the simulations could be expanded to include propagation before, within, and after the multiplexer using a transfer matrix approach in order to simulate a real-world device or system. For example, it could be used for predicting the noise statistics in a mode-division multiplexed few-mode or multimode fiber communications system. However, with additional degrees of freedom these expanded simulations will require significantly more computing resources than were available at the time of this work. Nevertheless, these MC simulations show great utility for predicting the statistics of speckle noise in the few-to-many mode regime where the standard analytical speckle theories fall short.
Disclosures
The author declares no conflict of interest.
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