## Abstract

Direct detection is traditionally regarded as a detection method that recovers only the optical intensity. Compared with coherent detection, it owns a natural advantage–the simplicity–but lacks a crucial capability of field recovery that enables not only the multi-dimensional modulation, but also the digital compensation of the fiber impairments linear with the optical field. Full-field detection is crucial to increase the capacity-distance product of optical transmission systems. A variety of methods have been investigated to directly detect the optical field of the single polarization mode, which normally sends a carrier traveling with the signal for self-coherent detection. The crux, however, is that any optical transmission medium supports at least two propagating modes (e.g. single mode fiber supports two polarization modes), and until now there is no direct detection that can recover the complete set of optical fields beyond one polarization, due to the well-known carrier fading issue after mode demultiplexing induced by the random mode coupling. To avoid the fading, direct detection receivers should recover the signal in an intensity space isomorphic to the optical field without loss of any degrees of freedom, and a bridge should be built between the field and its isomorphic space for the multi-mode field recovery. Based on this thinking, we propose, for the first time, the direct detection of dual polarization modes by a novel receiver concept, the Stokes-space field receiver (SSFR) and its extension, the generalized SSFR for multiple spatial modes. The idea is verified by a dual-polarization field recovery of a polarization-multiplexed complex signal over an 80-km single mode fiber transmission. SSFR can be applied to a much wider range of fields beyond optical communications such as coherent sensing and imaging, where simple field recovery without an extra local laser is desired for enhanced system performance.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Coherent detection has reshaped optical fiber communications during the past decade. Its capability of recovering the optical field benefits the transmission systems twofold: (i) it enables multi-dimensional signal modulations beyond the optical amplitude that significantly increases the capacity per fiber, exploiting phase [1-2], polarizations [3-4] and spatial modes of various types of fiber [5–8]; (ii) with the full access to the signal field, coherent receivers become versatile aided by the powerful digital signal processing (DSP) to compensate various types of fiber impairments, like chromatic dispersion (CD) [9-10], polarization mode dispersion (PMD) [10-11] and nonlinear interference noise [12], which greatly enhance the long-haul transmission performance. As the counterpart, direct detection still dominates short-reach transmissions, due to its natural advantage – the simplicity. In particular, the detection without the local oscillator not only saves the expense of one laser, but more importantly, realizes a completely colorless receiver; namely, there is no need to match the two lasers between transmitter and receiver. This avoids the sophisticated wavelength alignment (i.e. the uncooled transceiver) as well as carrier phase recovery [1-2] at receiver; and in consequence, relaxes the requirements of both laser linewidth and frequency stability at transmitter. Such characteristics are critical for short-reach networks with asymmetric nodes, where the remote unit is usually equipped with low-cost uncooled transceivers. However, one fundamental drawback of direct detection, which is distinguished from the coherent counterpart, is the intensity-only (instead of the field) signal recovery, which restrains not only the modulation degree of freedom, but also the receiver capability of compensating fiber impairments [13], limiting the achievable distance to tens of kilometers without optical dispersion management.

It is desirable to realize the direct detection with signal field recovery that combines the advantages between coherent and direct detection. In single-mode fiber (SMF), direct detection of the optical field of single-polarization mode has been realized by a variety of approaches [14–17]. This endows receivers the capability of compensating the intra-mode propagation impairments, such as fiber CD that is linear to the optical field. Nevertheless, within any optical waveguide, there are at least two propagating modes, and till now, there is no effective approach of recovering the entire set of optical fields beyond one polarization mode by direct detection, because it lacks the capability to rewind any linear inter-mode interaction such as polarization rotation and spatial mode coupling. We term any mode number beyond one polarization mode as multi-mode (e.g. SMF is regarded as two-mode waveguide), and for the first time, we propose a universal solution for the direct detection of multi-mode optical field, verified by a 256-Gb/s dual-polarization-mode complex-modulated transmission experiment over 80-km SMF. The concept can be extended to space division multiplexing (SDM) fibers (e.g. few-mode fiber [5–7] and coupled-core fiber [8]) beyond one spatial modes, and even the free-space optical transmission with orbital angular momentum (OAM) modes [18]. Furthermore, it can be implemented to much wider application ranges beyond optical communications, where the light source is distributed while the remote receiver requests the optical field without the local reference from an extra laser, for example, the coherent fiber sensing [19] and imaging [20] where field information can be applied to enhance the sensitivity.

## 2. Single-polarization mode field recovery

For short-reach optical transmission, there is an incentive to retrieve the optical field by the simple direct detection, because it enables the digital CD compensation to elongate the achievable distance cost-efficiently. As a result, the direct detection of single-polarization (SP) field has been extensively investigated. In general, this field recovery (FR) exploits self-coherent configurations [14–17], where a reference carrier is sent together with the signal at transmitter, and they mix with each other at receiver for phase diversity. In this section, we retrospect two self-coherent categories by the receiver structure: (i) the intensity receiver; (ii) the coherent receiver.

#### 2.1 Intensity receiver

To recover the optical field from the intensity-only detection, the signal should be modulated with single sideband (SSB), where the optical carrier is out of the signal spectrum. Assuming the baseband complex signal has the bandwidth of $B$, the corresponding minimum sampling rate at receiver should be at least $2B$ according to the Nyquist sampling theorem. The doubling bandwidth is consistent with the fact that the 2-D field signal at transmitter is transferred by square-law detection into a 1-D intensity signal without information loss. Given an SSB signal $S(t)$, there is a determinant relation between its in-phase (I) and quadrature (Q) parts:

where $HT$ stands for Hilbert transform. Under polar coordinates, Eq. (1) can be converted to the relation between the intensity $I(t)={\left|S(t)\right|}^{2}$ and the phase $\phi (t)$:Using Eq. (1) for FR, the major limitation is that the square-law detection cannot recover the pure in-phase part, contaminated by the signal-to-signal beat noise (SSBN) (i.e. ${\left|S+C\right|}^{2}={\left|C\right|}^{2}+{\left|S\right|}^{2}+2Re(S{C}^{\text{*}})$), where $C$ stands for carrier. SSBN can be separated from the signal by a frequency gap between signal and carrier (i.e. offset-SSB [14]), which sacrifices half spectral efficiency, as shown by Fig. 1(a-1). Alternatively, the gapless methods [15] suppress SSBN by high carrier-signal power ratio (CSPR) and digital SSBN cancellation algorithms. Via the nonlinear logarithm, Eq. (2) can reconstruct signal phase from the photo-current directly [17] without distinguishing SSBN, but a CSPR threshold, also interpreted as the minimum-phase condition [17] is still required to guarantee the positive intensity envelope.#### 2.2 Coherent receiver

If the signal and carrier are separated before detection, they can be fed into a coherent receiver to recover the linear field. To purely separate the carrier without interference from the signal, the carrier normally occupies orthogonal modulation dimensions at transmitter. As an example, Fig. 1(b) realizes the signal-carrier separation by time-domain interleaving [16]. The coherent receiver takes the incoming light and its one-symbol-period delay as two inputs to guarantee the signal only beats with the carrier for every sampling point. Alternatively, the signal and carrier can be separated at frequency domain by a sharp optical filter, which normally requests a frequency gap beyond 10 GHz to cope with the filter roll-off [21].

Compared with the intensity receiver in Section 2.1, the coherent receiver is compatible with double-sideband (DSB) detection that owns higher receiver electrical spectral efficiency. However, it wastes the optical spectral efficiency in general, to guarantee a strict signal-carrier separation. In contrast, the intensity receiver can achieve the full optical spectral efficiency (using the gapless SSB methods) by sacrificing half receiver spectral efficiency.

## 3. Isomorphic-space detection of multi-modes

Distinguished from the SP-FR above, the unique obstacle of the multi-mode FR (MM-FR) is the random inter-mode coupling during optical transmissions. To simplify the analysis, we start from the least multi-mode number, namely, the two polarization modes in SMF.

#### 3.1 Direct detection of the dual-polarization optical field: the fundamental barrier

Polarization states change randomly during propagation due to the time-varying birefringence distributed along the fiber. The receiver splits the light into two local polarization modes. Ignoring polarization dependent loss (PDL), the SMF channel can be characterized by a 2 × 2 unitary Jones matrix. The receiver should first recover the field of the two local modes, and then perform 2 × 2 MIMO equalization [9] in Jones space to retrieve the field of the two transmitter polarization modes. This requires an orthogonal pair of local carriers in Jones space with the same intensity. For conventional coherent detection with local oscillator (LO), the LO polarization can be fine controlled and then split to generate the orthogonal carriers, as shown by Fig. 2(a). Nevertheless, for self-coherent detection, the polarization state of the carrier that travels with the signal is not known at receiver. As a consequence, the receiver can no longer generate the proper pair of carriers. At the local Jones space, the two carriers would have different intensity during most time, which breaks the unitary condition of the 2 × 2 channel matrix, equivalent to PDL. One extreme case is the complete carrier fading of one local axis, as shown by Fig. 2(b), which results in channel singularity.

Carrier fading has been the fundamental barrier that prevents the direct detection of a polarization-multiplexed (POL-MUX) signal, since the first demonstration of SP-FR by direct detection [14] more than 10 years ago. For the offset-SSB, there emerges one approach that sends two orthogonal carriers at different frequencies to avoid the destructive interference [22], as shown by Fig. 2(c), followed by the 4 × 4 (instead of 2 × 2 owing to two carriers) MIMO to overcome the polarization-mode interaction. The time-domain interleaving [16] may utilize such method, but it sacrifices the spectral efficiency as well. The crucial drawback for such method is that the CSPRs of the four MIMO dimensions are not identical, which cannot guarantee a stable performance for arbitrary polarizations; moreover, for the FR that requires a CSPR threshold (like the gapless SSB), the insufficient CSPR of a MIMO dimension would result in the FR failure of that dimension. An alternative attempt, instead of splitting the incoming carrier, is to generate its orthogonal counterpart through a polarization rotator (e.g. Faraday rotator mirror) to form a complete polarization basis at receiver as shown by Fig. 2(d), namely, the self-polarization diversity. However, it is generally not possible to transform any arbitrary polarization into its orthogonal state by a static rotator [23]. Below, we briefly prove this claim by an arbitrary polarization in Jones space $E$ and a rotator with the fixed Jones matrix of $F$. The self-polarization-diversity should satisfy

where “H” stands for Hermitian conjugate. As a unitary matrix, F can be can be diagonalized as $F={P}^{H}DP$, where P is a diagonalization unitary matrix and D is a diagonal matrix with the diagonal elements ${\eta}_{1}$ and ${\eta}_{2}$ (i.e. $D=[{\eta}_{1}0;0{\eta}_{2}]$). Substituting $F$ to Eq. (3), we reach#### 3.2 Isomorphic-space direct detection

In general, it is not possible to regenerate an orthogonal basis in Jones space by only the transmitted carrier without active optical polarization control. As a result, the dual-polarization (DP) optical field cannot be recovered directly in Jones space by direct detection. Considering direct detection recovers the optical intensity (the 2^{nd}-order term of optical field), the DP field should be recovered in a 2^{nd}-order isomorphic space of Jones space to avoid any information loss from channel fading or even singularity. In terms of the complex-valued 2-D Jones space, its isomorphic pair is the real-valued 3-D Stokes space [24]. Via the Pauli spin vector $\stackrel{\rightharpoonup}{\sigma}$ in Stokes space, the Stokes vector $\widehat{s}$ can be related to the Jones vector $|s\u3009$ [24] as

*U*, can be directly retrieved from the Stokes space [24] aswhere

*R*is the real-valued 3 × 3 rotation matrix in Stokes space.

The unique advantages of Stokes space detection are two-fold: (i) its components are the real-valued 2^{nd}-order terms of the optical field, which can be recovered by the intensity-only direct detection in nature; (ii) its isomorphism to Jones space endows the capability of polarization recovery (PR) in its own space via the rotation matrix *R* of Eq. (6) that gets rid of the FR ahead of PR.

Direct detection in Stokes space is a powerful tool for multi-dimensional intensity modulations [23,25–27] as shown by Fig. 3(a), because it can rewind the polarization-mode coupling linearly by 3 × 3 MIMO. However, there is no linear operation that converts the 2^{nd}-order intensities to the linear terms of optical field. As a consequence, the received signals in Stokes space are vulnerable to the intra-mode impairment, like the fiber CD, limiting the transmission distance up to tens of kilometers.

#### 3.3 Stokes-space field receiver

To achieve the direct detection of DP optical fields, a proper connection should be built between the Stokes-space optical intensities and the Jones-space optical field to linearize the optical channel. In this section, we propose a universal solution, named Stoke-space field receiver (SSFR), with the conceptual diagram shown by Fig. 3(b), using gapless-SSB modulation as an example. At the 1st stage, the POL-MUX signal is detected in Stokes space. The inter-polarization-mode coupling should be rewound at this stage by the Stokes-space MIMO. At the 2nd stage, parts of the Stokes components are picked up for parallel SP-FRs, taking a suitable FR approach introduced in Section 2. The intra-mode impairment that is linear to the optical field can be compensated at this step. The final recovered optical field can thus be free from both inter and intra mode impairments. By the isomorphic-space detection at the 1st stage, the MM-FR is decomposed to parallel SP-FRs at the 2nd stage that can be properly handled by existing schemes.

The isomorphism of Stokes space can be revealed by retrospecting the carrier fading issue. At receiver, when one polarization suffers from a complete carrier fading like Fig. 2(b), the other polarization would have the largest carrier due to the lossless power transfer of the unitary channel. Although the information within ${\left|X\right|}^{2}$ is lost, it can be retrieved from the extra two detections in Stokes space, because the strong carrier of *Y*-POL beats with the signal of *X*-POL in ${s}_{2}$ and ${s}_{3}$.

There are a variety of realizations of SSFR taking into account both optical hardware and DSP algorithms. In terms of receiver hardware, Stokes vector receiver (SVR) is a generalized architecture of 3 or 4 intensity detections to recover the signal in 3-D Stokes space, or a 3-D equivalent space which can be linearly transformed to Stokes space [23]. DSP algorithms are critical to perform DP-FR; while particularly in this section, we are interested in the scheme that realizes full optical spectral efficiency, namely, the gapless SSB in Section 2.1. In Fig. 3(b), a critical point is the Stokes-space MIMO for PR. While the DP optical field are 2-D complex-valued (or 4-D real-valued) Jones vector, the Stokes-space MIMO is a 3-D real-valued operation. The question is whether the two polarizations should lock their phases at transmitter to guarantee a correct PR. The phase locking condition shrinks the modulation dimension to 3-D [23,27], leading to an incomplete DP-FR. In this section, we reveal a unique feature for Stokes space to address the above concern: the non-coherent PR condition. To generalize the question, we consider two orthogonal polarizations in Jones space with an arbitrary frequency offset $\Delta f$ as well as phase mismatch: $|s={[\left|X\right|\mathrm{exp}\left(i{\phi}_{x}+i2\pi \Delta ft\right)\left|Y\right|\mathrm{exp}\left(i{\phi}_{y}\right)]}^{T}$, where the intensity $\left|X\right|$ and $\left|Y\right|$, the phase ${\phi}_{x}$ and ${\phi}_{y}$ are freely modulated, and superscript “T” stands for transpose. We convert this Jones vector to Stokes space using Eq. (5):

where the intensity $\left|X\right|$ and $\left|Y\right|$, the phase ${\phi}_{x}$ and ${\phi}_{y}$ are freely modulated, and $\Delta f$ stands for the frequency offset. A remarkable characteristic of the Stokes vector in Eq. (7) is that ${s}_{0}$ and ${s}_{1}$ are independent from the angle behaviors of both polarizations (which are confined inside ${s}_{2}$ and ${s}_{3}$). This results in the following conclusion.

**Non-coherent PR condition**: Stokes-space polarization recovery would not be affected by the angle behaviors of both polarizations, if the modulated information can be fully retrieved from ${s}_{0}$ and ${s}_{1}$.

This condition enables PR as a transparent module of the subsequent DP-FR. After Stokes-space PR, DP-FR is decomposed to two completely independent SP-FRs, while each mode owns a complete 2-D degree of freedom, namely, any arbitrary complex modulation format and carrier wavelength. This condition is also the foundation that supports a 3-D SVR to recover the polarization of the 4-D (DP I/Q) optical field. The DP-FR principle now is fully revealed: (i) SVR detects the DP signal in Stokes space; (ii) Stokes-space MIMO performs intensity-based PR; (iii) ${s}_{0}$ and ${s}_{1}$ are picked up for the parallel SP-FRs of the two transmitted polarization modes. SSFR, therefore, realizes the direct detection of the DP optical field, without loss of optical spectral efficiency.

#### 3.4 Generalized Stokes-space field receiver

We now extend the discussion of MM-FR beyond SMF, namely, the SDM with more than one spatial mode. Because each spatial mode supports two orthogonal polarization modes, an *N*-spatial-mode SDM supports totally *2N* modes, represented by a *2N*-dimensional vector. The linear multi-mode interactions can be characterized by a *2N × 2N* channel matrix in the generalized Jones space. Its isomorphic real-valued space is the generalized Stokes space with *4N ^{2}-1* dimensions [28-29]. The generalized Jones and Stokes space can still be connected by Eq. (5) and (6), while the Pauli spin vector $\stackrel{\rightharpoonup}{\sigma}$ is extended to

*4N*dimensions as well. The generalized SSFR now performs

^{2}-1*4N*intensity detections, followed by the (

^{2}-1*4N*) × (

^{2}-1*4N*) MIMO to rewind the linear mode superposition. The conceptual flow is similar to Fig. 3.

^{2}-1To explain the generalized SSFR from an implementing perspective, we use two spatial modes as an example, namely, 4 modes in total.

For the gapless-SSB modulation, it is straightforward to extend the non-coherent PR condition to the first four components of $\tilde{v}$ (i.e. ${\left|{E}_{i}\right|}^{2}$) and pick them up after the 15 × 15 MIMO for the parallel FR of the four modes. Moreover, an attractive attempt for the multi-modes is to adopt a similar self-coherent modulation concept in Section 2.2, namely, assigning the carrier on one orthogonal mode. It is noted, however, the receiver self-coherent detection is totally different from Section 2.2, because SSFR need no signal-carrier separation and the signal-carrier beating is realized in the generalized Stokes space naturally. In terms of SMF, keeping one polarization mode as the constant carrier (e.g. *Y*-POL) linearizes the channel of *X*-POL, whose field can be directly recovered by ${s}_{2}+j{s}_{3}$ [25]. Obviously, it sacrifices half of the optical spectral efficiency. In contrast, for two spatial modes, occupying one polarization mode as carrier decreases the sacrifice of spectral efficiency to one-fourth. In return, the rewards are: (i) the double receiver electrical spectral efficiency due to the DSB detection; (ii) the completely linear FR operations by combining corresponding components of the vector $\tilde{v}$ after the Stoke-space MIMO, without the nonlinear SSBN equalization requested by Eq. (1) or the logarithm phase reconstruction by Eq. (2); (iii) higher system OSNR sensitivity due to the natural 3-dB advantage of DSB over SSB detection and the much lower CSPR requirement. With the increase of mode number, the sacrifice of optical spectral efficiency further decreases, leading to an even more attractive MM-FR.

## 4. Field recovery of dual-polarization modes: an experimental demonstration

In this section, we demonstrate, for the first time, the direct detection of optical field beyond single polarization mode. Without loss of generality, we select SMF as the light propagation medium that supports two polarization-modes, and perform a high-speed optical transmission experiment to verify the SSFR concept.

#### 4.1 Experiment setup

To maximize the optical spectral efficiency, we adopt the gapless-SSB modulation for FR. The transmitter contains two parts: SSB generator and POL-MUX emulator. The SSB generator contains two lasers for signal and carrier, respectively. Their frequency offset is around 17 GHz for SSB modulation. It is noted that the SSB can also be generated by splitting one laser output with an optical frequency shifter for the carrier, to match the phase noise of the two paths. The signal path contains an I/Q modulator driven by an 80-GSa/s arbitrary waveform generator. The baseband DSP is shown by inset (ii). The signal is OFDM-modulated whose DFT size is 800 (to match the fiber delay in the subsequent POL-MUX emulator; in practice, the size should be ${2}^{n}$) filled with 320 subcarriers. The redundant OFDM guard interval is avoided by the dispersion post-compensation at receiver. The baseband baud-rate is 32 Gbaud, and the raw data-rate is 256 Gb/s using POL-MUX 16-QAM. The carrier path contains a low-speed IQ modulator to generate MHz-level polarization training for the Stokes-space MIMO. When the carrier path sends the training, the signal path stays at null-point, as shown by inset (i). We adopt the basis vector training for PR [25], which contains the three basis polarizations in Stokes space. Considering the subsequent one-symbol delay of Y-POL in the POL-MUX emulator, we send the Jones space training as inset (iii). In practice, for a DP signal, the basis vectors can be generated by only 1 low-speed modulator [25]. The carrier may even be simplified to pure continuous-wave light if the PR is designed to be adaptive in the future. The two paths are combined with 9-dB CSPR, whose polarizations are aligned to the axis of a polarizer by polarization controllers to guarantee a single polarization signal before POL-MUX emulator. The POL-MUX emulator splits the SSB signal into two paths, with 2-m fiber delay (corresponding to one OFDM symbol length) on one path to de-correlate the two polarizations. These two paths are polarization combined to form a DP signal. The signal is launched into 80-km standard single-mode fiber (SSMF) with 6-dBm optical power.

In terms of the receiver architecture, we use a simplified SVR based on 3 × 3 coupler and four intensity detections [23], which can be transformed to the Stokes vector definition in Eq. (5) by a fixed 4 × 4 matrix linearly. The PDs have 40-GHz bandwidth and the oscilloscope has 33 GHz sampling at 80 GSa/s. Inset (iv) shows receiver DSP. The 3 × 3 MIMO is performed in Stokes space for PR. After this, ${\left|X\right|}^{2}$ and ${\left|Y\right|}^{2}$ (i.e. ${s}_{0}$ and ${s}_{1}$) are picked up for parallel SP-FRs via Eq. (2). The subsequent DSP processes the optical field of the two polarizations independently, following the traditional steps in coherent receiver including CD compensation, matched filtering, channel equalization and carrier recovery. We evaluate the FR performance by SNR or BER of the complex 16-QAM constellation for each polarization.

#### 4.2 Recovery of the DP optical field

To reveal how the SSFR recovers the dual-polarization optical field, we compare the RF spectra of X-POL (defined by the receiver POL axes, i.e. port 1 of the SVR in Fig. 4) at various signal processing stages, corresponding to Fig. 3(b). In particular, Fig. 5(a) selects a polarization state when X-POL suffers from severe carrier fading. Without a strong carrier, the output of port 1 is dominant by the 2^{nd}-term of the signal field. As shown by the curve “before PR”, the power density keeps dropping at higher frequency, similar to the SSBN behavior depicted in Fig. 1(a). The power density at edge-frequency ranges has approached the receiver noise floor. A direct detection receiver in Fig. 2(b) would lose the entire information of X-POL in this case. In contrast, by SVR and Stokes-space MIMO, the faded signal at X-POL is retrieved, verified by the greatly-enhanced signal power density after PR in Fig. 5(a). Moreover, the power density becomes flat across the spectrum. Figure 5(a) also illustrates the field spectrum after FR, where the left sideband is completely suppressed, leading to a SSB spectrum that resembles a linear replica of the optical counterpart.

Figure 5(b) illustrates the recovered optical field of both polarizations, namely, the 16-QAM constellations, which are clear without nonlinear distortions. This proves the optical field has been linearly retrieved by the intensity-based direct detection. It also reveals the signal SNR across the spectrum. Both polarizations present flat SNR performance without severe frequency-selective fading, except the slight roll-off at spectrum edges of the due to the limited receiver bandwidth. More importantly, even if we vary the incoming polarization states of the receiver by the polarization controller arbitrary, there is no significant SNR difference between the two polarizations. This verifies the Stokes-space MIMO has successfully rewound the polarization variations, leading to two parallel FRs which are transparent from the inter-polarization interference.

#### 4.3 Transmission performance

We evaluate the OSNR sensitivity of this 256-Gb/s system for both back-to-back (B2B) and 80-km transmission. A 480-Gb/s signal reception via a time-domain multiplexed receiver is demonstrated in Ref [30]. In Fig. 6, the OSNR requirements at 20% FEC are about 32.5 dB for B2B and 33.5 dB for 80-km. These OSNR are higher than the theory, because the transceiver has been pushed to its bandwidth limit that induces a large performance penalty. Moreover, a sharp optical filter that matches the signal bandwidth can be added before optical detection to further increase the OSNR sensitivity by 3 dB, due to the SSB detection. A remarkable phenomenon is that the performance gap between B2B and 80-km transmission is narrow. Compared with B2B, the major transmission impairment of the 80-km SSMF is the fiber CD. This intra-mode impairment can be compensated by linear equalizers applied to the optical field. Therefore, the digital CD post-compensation equipped in the SSFR significantly shrinks the OSNR gap, which reveals a typical advantage of the field recovery over the intensity-only recovery for direct detection receivers.

## 5. Summary

We analyze the necessity as well as the sufficiency of using the (generalized) Stokes-space field receiver (SSFR) for the direct detection of multi-mode optical field. SSFR combines the isomorphic-space direct detection of optical field that guarantees the completeness of detection degree of freedom, the intensity-based MIMO that rewinds the linear mode coupling, and the parallel single-polarization field recovery that bridges the optical intensity and field. The concept is verified by a 256-Gb/s polarization-multiplexed complex signal transmission over 80-km single-mode fiber (SMF). For traditional SMF transmissions, SSFR supports the complete modulation degrees of freedom of SMF and is robust against both the inter-polarization coupling and the intra-mode chromatic dispersion. It is no doubt a powerful solution for future high-spectral-efficiency short-reach communications over single or a few spans fiber. Furthermore, SSFR reveals the first and perhaps the only universal approach for multi-mode field recovery by direct detection without active mode decoupling in optical domain (e.g. the adaptive polarization stabilizer). Its application can be extended to communication channels with spatial division multiplexing, and even scenarios beyond communications like coherent sensing and imaging where simple field recovery is desired to enhance system performance.

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