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Abstract
Mode-group multiplexing (MGM) can increase the capacity of short-reach few-mode optical fiber communication links while avoiding complex digital signal processing. In this paper, we present the design and experimental demonstration of a novel mode-group demultiplexer (MG DeMux) using Fabry-Perot (FP) thin-film filters (TFFs). The MG DeMux supports low-crosstalk mode-group demultiplexing, with degeneracies commensurate with those of graded-index (GRIN) multimode fibers. We experimentally demonstrate this functionality by using a commercial six-cavity TFF that was intended for 100 GHz channel spaced wavelength-division multiplexing (WDM) system.
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1. Introduction
In the past few decades, the spectral efficiency of optical fiber communication links has been increased by multiplexing independent degrees of freedom of light, including wavelength, polarization, and quadratures [1]. Using all available degrees of freedom, the capacity of a single-mode fiber (SMF) has reached its nonlinear limit [2]. To overcome the fundamental capacity limit of SMFs based on wavelength-division multiplexing (WDM), space-division multiplexing (SDM) has attracted much attention in the last ten years [2,3]. Multimode fibers (MMF) can support mode-division multiplexing (MDM) but the complexity and cost of transmission link increases because of random modal crosstalk which necessitates multiple-input multiple-output (MIMO) digital signal processing (DSP) [4]. For short-distance applications such as interconnects for data centers and super-computers, it has been shown that mode-group multiplexing (MGM) can be used without DSP [5]. In MGM, the same data is carried through all modes within the same mode group, so crosstalk among degenerate modes does not affect the performance of MGM transmission. In addition, large effective index differences between mode groups can be maintained to ensure low crosstalk.
Mode-group multiplexers (MG Muxes) and demultiplexers (MG DeMuxes) are the key components of transmission systems using MGM [6]. In MGM, the single-mode output from the transmitter for each MGM tributary is converted into a desired higher-order mode of the MMF and combined using MG Mux while MG DeMux performs the inverse operation at the receiver [6,7]. The most straightforward MG DeMuxes demultiplex all spatial modes using a mode demultiplexer. Subsequently, a mode multiplexer combines degenerate modes using devices such as photonic lanterns (PL). As a result, the number of components and associated insertion losses tend to be high in such MG DeMuxes [8]. The degenerate-mode-selective coupler [5] is an elegant solution based on directional coupling, however, it only works for mode groups with two-fold degeneracies. Therefore, degenerate-mode-selective couplers tend to only work with specially designed fibers supporting mode groups with at most two-fold degeneracies. Another example is the specially designed ring-core fiber that supports orbital angular momentum modes [9,10]. In all aforementioned MG DeMuxes, the number of supported mode groups are limited, and the crosstalk between mode groups is relatively high.
Conventional MDM-WDM receivers use a broadband MG demultiplexer followed by wavelength demultiplexers on each output channel of the MG DeMux to demultiplex all MDM-WDM channels [11]. Using this approach, the crosstalk between different channels is limited to the MG DeMux used at the beginning. In this paper, we propose a specially designed MG DeMux which enables swapping the order of MDM-WDM demultiplexing. The new configuration uses a multimode WDM at the beginning of the demultiplexer followed by the proposed MG DeMuxes, which can improve the crosstalk performance of the MDM-WDM receivers.
In [12] and [13], we presented the simulation results of the proposed MG DeMux using bulk FP filter and FP-TFF, respectively. In this paper, we design and experimentally demonstrate an MG DeMux that supports mode-group demultiplexing with degeneracies commensurate with mode degeneracies in graded-index (GRIN) fibers. It should be noted that FP filters can be made (spatial) mode-insensitive if the incident beam has a large Rayleigh range or if all incident spatial modes are well outside the Rayleigh range thus the Gouy phase shifts for all spatial modes are nearly zero. Using such a configuration, FP filters essentially operate as multimode wavelength-division (de)multiplexer that can demultiplex all the spatial modes with the same wavelength into the same channel. When the multimode WDM DeMux is followed by the proposed MG DeMux, the tandem supports WDM-MDM demultiplexing with low crosstalk. In multimode WDM DeMux, as a Fabry-Perot filter, the only source of inter-modal crosstalk is propagation between the two mirrors [14]. As a result, the mode coupling is negligible because of its very small thickness compared with mirror curvature. Also, provided that the signal bandwidth is within the frequency response of the multimode WDM filter, there is no crosstalk between the modes in different frequency channels. The remainder of this paper is organized as follows: Section II describes the principles and the structure of the proposed MG DeMux; in Section III, numerical simulations and an experimental demonstration of the MG DeMux are presented; Section IV concludes the paper.
2. Device structure
There are two approaches for demultiplexing a WDM-MGM signal, as shown in Fig. 1. The input signal to the demultiplexer consists of N wavelength channels with M mode groups at each wavelength. Figure 1(a) is the schematic diagram of existing mode-group demultiplexers, in which mode-group demultiplexing precedes wavelength demultiplexing. The input signal first passes through a mode demultiplexer. Subsequently, a mode multiplexer combines all degenerate modes in each mode group into a few-mode fiber, followed by a multimode WDM demultiplexer. Figure 1(b) is the schematic diagram of our proposed MGM-WDM demultiplexer, in which wavelength demultiplexing precedes mode-group demultiplexing. In this scheme, a multimode WDM filter is used to demultiplex all mode groups with the same wavelength into the same channel followed by the proposed MG DeMux which lowers the crosstalk and reduces the number of components compared with the previous approach. For other methods with the MG DeMux before the WDM filter, the performance of the device will be limited by the MG DeMux even if only the fundamental mode exists in some wavelengths.
Fig. 1. Two approaches for demultiplexing a WDM-MGM signal: (a) using MG DeMux first followed by wavelength demultiplexers, (b) proposed method. In this scheme, a multimode WDM filter is used to demultiplex all mode groups with the same wavelength in the same channel followed by the proposed MG DeMux.
The concept of the proposed MG DeMux is schematically shown in Fig. 2(a) which resembles a TFF-based WDM filter. When we have successive FP filters as shown in Fig. 2(a), for the input MGM signal with the same carrier frequency ${\nu _0}$, only one mode group is transmitted through each FP filter and the remaining are reflected and directed toward the next FP filter. This process continues until all the mode groups are demultiplexed by their corresponding FP filter. Each FP filter has a specific cavity length, designed to transmit only one mode group. So, at the end all the mode groups are demultiplexed for the MGM signal carried in a single frequency. To clearly visualize this process, the spectrum of the input MGM signal and the spectrum transmitted through the first FP filter (FP1), as an example, is shown in Fig. 2(b) which, by design, only passes the fundamental mode at the design frequency ${\nu _0}$. Each of the subsequent FP filters will pass one of the remaining mode groups at the design frequency ${\nu _0}$. Even though curved mirrors representing focusing lenses are shown in Fig. 2(a), flat diffractive optical elements can be used instead in the actual devices.
Fig. 2. Schematic diagram of the proposed MG DeMux for the input MGM signal with the same carrier frequency ν0. (a) MG DeMux for an input beam with five LG mode groups (M = 1 to M = 5) at the design frequency ν0. The spatial mode profiles of the LG beams are commensurate with linearly polarized (LP) modes in GRIN fibers. (b) Schematic performance of FP1 for an input beam with five mode groups at frequency ν0. FP1 is designed to transmit mode group M = 1 at ν= ν0 and reflects all other mode groups (Same FP filter transmits other mode groups at frequency ${\nu _0} + (M - 1)\Delta f$). Reflected beam moves toward the next FP filters to be demultiplexed. (c) LG beam propagation and the positions of the mirrors relative to the beam waist position. The origin of the z axis is at the beam waist.
To understand how the FP filters discriminate between different mode groups, we first present the modal dependence of the transmission properties of FP filters with Laguerre–Gaussian (LG) input beams. The complex amplitude of an LG beam of order ($l$, $m$) is given by [15]
2.1 Bulk FP filters
For a bulk FP filter with two mirrors separated by a distance $d$ shown in Fig. 2(c), its transmittance is [15]:
where $F = {{4{r^2}} / {{{(1 - {r^2})}^2}}}$ is the coefficient of finesse, r is the reflection coefficient of the mirrors, and $\Delta \varphi $ is the roundtrip phase shift which depends on $d$ and the wavelength/frequency of light. As can be seen from Eq. (1), the phase of the $\textrm{L}{\textrm{G}_{l,m}}$ mode is given byIf the radius of curvature of the wavefront of the LG beam is matched to the radius of the FP filter mirrors, all the points on each mirror share the same phase and the reflected beam will retrace the incident beam [16]. Therefore, all LG modes are eigen modes of the spherical FP resonators [15]. From Eq. (3), degenerate modes within the same mode group of transverse order $M = 2m + l + 1$, but with different combinations of ($l,m$), share the same phase. Thus, as shown in Fig. 2(c), if the first mirror is located at z = z1, the phase shift for one round-trip is
whereFor a fixed mirror spacing d, $\Delta \xi ({z_1})$ is maximum for a symmetric non-confocal resonator with $|{{z_1}} |= {z_2} \ll {z_0}$, therefore, the transverse mode spacing $\Delta f$ is maximized. If the signal in each mode group has a bandwidth ($B$) smaller than the transverse mode spacing $\Delta f$, a particular FP filter will only pass one mode group and reflect the remaining at the designed frequency ${\nu _0}$. Therefore, a successive set of FP filters can be used to demultiplex all of the mode groups.
On the other hand, under two different conditions in Eq. (7), $\Delta f$ would vanish, leading to a multimode WDM filter. First, for nearly planar FP cavity with $d \ll |{{R_1}} |$ and ${R_2}$[17]
therefore, for a plane wave input (large z0), Δf is very small and all different mode groups have similar transmission resulting in a multimode WDM filter. Second, when all incident spatial modes are well outside the Rayleigh range (${z_1} \gg {z_0}$), $\Delta \xi ({z_1}) \to 0$ which results in a multimode WDM filter. Based on the design of single-mode WDM filters, multiple-cavity FP-TFFs can achieve flat-top transmission and reduce the skirt of transmission [18].In what follows, we show that this is the case for the proposed MG DeMuxes using thin-film filters.2.2 FP thin-film filters
FP-TFF cavity is an etalon where multiple reflective dielectric thin-film layers are used to form the mirrors. Figure 3(a) shows a schematic of a triple-cavity FP-TFF and its transmission characteristics are compared to a single-cavity FP-TFF in Fig. 3(b). It’s worthwhile to mention that the transmission response of a multi-cavity FP-TFF is derived from coupling between all cavities together and not simply cascading of several independent cavities. Therefore, the entire structure should be analyzed collectively. As shown in Fig. 3(b), the 3-dB bandwidth remains the same for both structures, but the transmittance spectrum is sharper for triple-cavity FP-TFF. An additional advantage of FP-TFFs over bulk FP resonators is that accumulative crosstalk is removed for demultiplexing applications for large-channel count systems due to their short cavities, which leads to a large free-spectral range [18].
Fig. 3. (a). The structure of a high-index narrow-bandpass triple-cavity FP-TFF and the positions of the layers in relation to the beam waist. Three single-cavity FP-TFF are cascaded using two coupling layers. (b) The transmission performance of a triple-cavity (solid) and a single-cavity (dashed) FP-TFF. (c) The structure of QWS used as a dielectric mirror in each cavity. (d-f) The LG beam waist at the center of the (d) first, (e) middle and (f) last cavity of the triple-cavity FP-TFF.
A narrow-bandpass N-cavity FP-TFF has N FP cavities cascaded through N-1 coupling layers. Each cavity can be described as ${({HL} )^q} - 2pH - {({LH} )^q}$[19], representing two quarter-wave stack (QWS) dielectric mirrors (${({HL} )^q}$ and ${({LH} )^q}$), where $q$ is the number of QWS pairs and $2pH$ is a spacer with optical thickness of a multiple of half-wavelength. As shown in Fig. 3(c), the refractive indices of the layers in the QWS alternate between high (${n_H}$) and low (${n_L}$) and each layer has a quarter-wavelength optical thickness (${d_H}$ and ${d_L}$).
To find the transmission through the FP-TFF, one can use the characteristic matrix of the stack which is the product of the characteristic matrices of all the layers [20]:
For an FP-TFF placed near the waist of an LG beam, with its thickness much smaller than the Rayleigh range ${z_0} \gg \lambda$, the wavefront of the beam is nearly flat throughout the FP-TFF i.e. $R(z )> > W(z )$ [15]. Therefore, in each layer, all the points are in-phase and from Eq. (3), the mode-dependent phase thickness becomes,
where ${z_j}$ is the position of layer $j$ from the beam waist. It should be noted that the complex parameter $q(z)$, $\frac{1}{{{q_j}(z)}} =\frac{1}{{{R_j}(z)}} - i\frac{\lambda }{{{n_j}\pi w_j^2(z)}}$, of a Gaussian beam propagating through a planar interface between two dielectric layers of indices ${n_j}$ and nj+1 satisfies [15]As a result, Eq. (5) becomes [21],
Due to the mode-dependent phase thickness given by Eq. (10), the characteristic matrix of the filter of Eq. (9) and, consequently, the transmission of the filter into the exit medium depends on the mode order M [20]:
where $Y(M) = {H_a}/{E_a}$ is the equivalent optical admittance and, based on Eq. (10) is mode dependent. Accordingly, transmission performance of the $L{G_{l,m}}$ beams with the same mode order $M = 2m + l + 1$ is the same.It is worth noting that when the LG beam waist is at the center of the first, middle and last cavity of the triple-cavity FP-TFF shown in Fig. 3(d-f), the transmission performance of the MG DeMux is similar and the transverse mode spacing $\Delta f$ remains the same, illustrating that the MG DeMux can tolerate misalignment in the longitudinal direction. However, the flat-mirror assumption, i.e. W(z)/R(z) ratio is minimum, is most valid when the waist of the LG beams is placed at the center of the middle cavity of the FP-TFF as shown in Fig. 3(e).
Similar to bulk FP filters, maximum transmission of every mode group through FP-TFFs happens at a different frequency. Therefore, provided that for each mode group, the bandwidth B of the input signal is smaller than the transverse mode spacing Δf, FP-TFFs can be used for MG DeMux applications. On the other hand, if mode spacing Δf <<B (close to plane wave or outside Rayleigh range input beam), the FP-TFF can work as a multimode WDM. In the following, we present simulation and experimental results of the FP-TFF-based MG DeMux.
3. Device performance
In [12], we reported the simulation results of an MG DeMux for five mode groups using bulk FP filters, each bulk FP filter was designed to transmit a particular mode group and reflect all other mode groups at the design wavelength $1550nm$. Simulation results show that for $B = 100GHz$ and $FSR = 5THz$, crosstalk of lower than -20dB is obtained. Here, the ratio of the intensity summation of all undesired mode groups to the wanted one at the design frequency is defined as crosstalk. Also, the simulation results show that For the FPs with the same FSR, there is an optimized Rayleigh range ${z_0}$ to achieve the minimum crosstalk in demultiplexing a fixed number of mode-groups. Furthermore, to avoid crosstalk from neighboring free-spectral ranges, the number of mode groups is limited to ${{FSR} / {\Delta f}} \approx n{{\pi {z_0}} / d}$. Utilizing FP-TFFs can solve this problem due to intrinsically large FSR.
Here we report simulation results of MG DeMuxes using single-cavity FP-TFFs and triple-cavity FP-TFFs. We also report experimental results of an MG DeMux using a commercial six-cavity FP-TFFs for 100 GHz channel spacing WDM, which is the only FP-TFFs that we have access to. Having more cavities in WDM TFFs allows independent control of filter ripple, bandwidth, and edge steepness while in MG DeMux application, increasing the number of layers violates the flat-mirror assumption for outer layers.
3.1 FP-TFF simulation
The simulation results of the proposed MG DeMux for five LG mode groups is shown in Fig. 4. Figure 4(a) shows the transmission spectra of an FP-TFF through a high-index single- and triple-cavity FP-TFFs with nH = 1.96, ${n_L} = 1.404$, $p = 6$, $q = 8$, the design wavelength of ${\lambda _0} = 1550nm$ and $B = 0.6nm$ using MATLAB simulation. As shown in Fig. 4(a), The M = 1 mode group is transmitted through this filter and all others are reflected at λ0. For the Rayleigh range ${z_0} = \textrm{ }145\mu m$, a transverse mode spacing $\Delta f = 100GHz$ is achieved which is one of the International Telecommunication Union (ITU) standard grid spacings [22]. As shown in Fig. 4(a), the addition of extra layers does not affect the transverse mode spacing Δf but increases the transmission sharpness within the bandwidth and reduces the skirt effect outside. As a result, the designed triple-cavity FP-TFF has crosstalk lower than -30dB which compared to previously reported MG DeMuxes shows a significant improvement (better than 11dB) [5,8]. At the same wavelength, demultiplexing other mode groups requires adjusting the thickness of each layer to:
Fig. 4. Simulation results of the proposed MG DeMux for five mode groups, for single- (dashed) and triple-cavity (solid) designs (a) Transmission of first five mode groups vs. wavelength for B = 0.6 nm. (b) Crosstalk vs. Rayleigh range z0 for $B = 0.6nm$ and $B = 0.3nm$.
It should be noted that, for ${z_0} = \textrm{ }145\mu m$, the flat-mirror assumption is valid since $W(z )/R(z )= 0.305\%< 1\%$. Crosstalk variations for different Rayleigh range z0 is illustrated in Fig. 4(b) for single- and triple-cavity FP-TFFs and for B = 0.6nm and B = 0.3nm. The same FP-TFF layer structure is used for both B = 0.6nm and B = 0.3nm, however, the refractive index ${n_H}$ was changed from 1.96 to 2.045 to account for the narrower bandwidth. As shown in Fig. 4(b), by changing the Rayleigh range z0 of the input beam, the crosstalk can be controlled. For B = 0.6nm and Rayleigh range z0= 100µm, the crosstalk lower than -40 dB can be achieved.
From Eqs. (5) and (7), it can be shown that the mode spacing is inversely proportional to the effective index of the FP-TFF, therefore, by choosing a cavity with a lower effective index, for the same Rayleigh range z0, the transverse mode spacing increases. Equivalently, the filter with a lower effective index meets the required crosstalk for a larger Rayleigh range z0 which better supports the flat mirror approximation.
3.2 FP-TFF experiment
Figure 5(a) shows the experimental setup to demonstrate the proposed MG DeMux by measuring the transmission performance of one FP-TFF. In this setup, an Erbium-doped fiber amplifier (EDFA) was used as a broadband source with a nearly flat spectrum over the entire C-band. An all-fiber PL mode multiplexer capable of selectively exciting the first six spatial modes of a multimode fiber ($L{P_{01}}$, $L{P_{11a}}$, $L{P_{11b}}$, $L{P_{21a}}$, $L{P_{21b}}$, and $L{P_{02}}$) was deployed to convert the fundamental mode of the EDFA to high-order spatial modes [23]. Each time, the output of the EDFA was connected to one of the input ports of the PL to excite one specific spatial mode of the few-mode fiber (FMF). The output of the PL was collimated using a collimator lens L1. Because the degeneracy in fiber LP modes is the same as those in free-space LG modes, the corresponding LG mode propagates in free space [24]. In this experiment, the output fiber of the PL is a step-index FMF with a core diameter of 15.2µm and mode field diameter (MFD) of 12.9µm±0.5µm. Using an objective lens L2, the collimated beam was focused on the FP-TFF which was placed normal to the propagation direction and at the beam waist ${w_0}$ equal to the mode field radius (MFD/2) of the FMF. Figure 5(b) is a micrograph of the commercial 100GHz six-cavity FP-TFF used in the experiment which is designed for the C20 channel from the ITU grid [22] having a 0.6nm measured bandwidth. The transmitted light was then coupled to an MMF and the spectrum of the transmitted light was monitored using an optical spectrum analyzer (OSA) with a resolution of 0.05nm for GRIN multimode input fiber [25].
Fig. 5. (a) The Schematic of the experimental setup. Each time, the EDFA output is connected to one of the inputs of the all-fiber multiplexer input port to excite the corresponding LP mode. PL: Photonic Lantern, EDFA: Erbium Doped fiber Amplifier, TFF: Thin-Film Filter, OSA: Optical Spectrum Analyzer. (b) The microscope photograph of one of the tested FP-TFF. dmeasured varied between 47.9µm and 53.3µm for different FP-TFFs that were tested. (c) Experimental results for different mode groups. The normalized transmission spectrum of each mode group shifts by $\Delta \lambda = 0.55nm$ relative to that of its neighboring mode group. (d) and (e) The normalized transmission spectra of the modes in the same mode group: (d) for M = 2 and (e) for M = 3 mode group.
Figures 5(c-e) show the normalized transmission spectra of the six-cavity FP-TFF for the first 6 LP modes. The measured 3-dB bandwidth for all the modes was similar, about 0.6nm and the minimum transmission was below -30dB for all three mode groups at the edges of spectrum window. The transverse mode spacing between two consecutive mode groups is $\Delta f = 68GHz(\Delta \lambda = 0.55nm)$. Due to the relatively large thickness (${d_{measured}} \in [47.9\textrm{ }53.3]\mu m\; $) of the FP-TFF compared to the Rayleigh range ${z_0}$, W(z)/R(z) is ∼1.3%, which violates the flat mirror approximation. As a result, LG modes do not completely match the spatial modes of the FP cavity; hence, there was some unwanted coupling to other modes, especially for higher-order modes [26]. Therefore, the sharpness of the transmission spectrum reduces for each higher order mode group.
Since the FP-TFF in this experiment is a commercial WDM-TFF, the exact filter configuration such as the refractive index of each layer is not available. We have calculated the transverse mode spacing $\Delta f$ of the FP-TFF for two extreme cases of maximum and minimum possible refractive indices while maintaining the known bandwidth of 0.6nm and the total FP-TFF thickness of ${d_{measured}}$. The refractive index combinations were chosen to yield zero reflectance for each dielectric mirror at one wavelength [27,28]. The simulation results predict a transverse mode spacing $\Delta f$ between $\Delta {f_{\min }} = 39GHz$ (for ${n_{Hmax}} = 3.71$[29]) and $\Delta {f_{\max }} = 147GHz$(${n_L}_{min} = 1.39$[20]). The measured mode spacing $\Delta f = 68GHz$ indeed falls within this range.
Furthermore, for the LP modes in the same mode group, as shown in Figs. 5(d) and (e), similar transmission performance was achieved which confirms that the proposed MG DeMux is well-suited for mode-group demultiplexing applications. The structure of the proposed MG DeMux is similar to a WDM filter. Therefore, the expected insertion loss is the same as the reported WDM filters and could go to lower than 0.2 dB [30]. Also, the temperature stabilization techniques that have been used for WDM TFFs can be applied to our MG DeMux [31].
4. Conclusion
In this paper, deploying FP-TFFs, we introduced a novel mode group DeMux with a similar structure to TFF-based WDM filters. The proposed MG DeMux can support low-crosstalk mode-group demultiplexing with degeneracies commensurate with mode degeneracies in GRIN multimode fibers. The experimental tests utilized off the shelf FP-TFFs, which shows the manufacturability of this design. Preceded by a (spatial) mode-insensitive TFF-based WDM filter, the tandem can support WDM-MGM demultiplexing.
Funding
Office of Naval Research (N00014-20-1-2441); Army Research Office (W911NF1910385); National Science Foundation (ECCS-1932858).
Acknowledgment
The authors would like to thank Dr. Sethumadhavan Chandrasekhar for his helpful advice and comments on various technical issues examined in this paper.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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