Ultra-wideband optical diffractive network for mode multiplexing across the entire telecommunication range

Domain: physics
Published: 2025-12-05
Editorial outcome: major_revision

Abstract
Advancements in multi-band and space division multiplexing (SDM) technologies are rapidly progressing to support growing data traffic. Current research focuses on hybrid multiplexing in the C+L bands, pushing SDM systems to achieve capacities exceeding 10 Pb/s. Integrating ultra-wideband SDM systems with expanded spectrum leaps forward in transmission capacity. Here, we proposed a comprehensive model of an O+E+S+C+L+U ultra-wideband mode multiplexing transmission system to enhance spectrum utilization. A 4-linearly polarized mode multiplexer, designed utilizing ultra-wideband optical diffractive network with an end-to-end inverse design algorithm, achieves over 90% mode purity and mode crosstalk less than −17 dB, with below −8.87 dB in multiplexer-demultiplexer back-to-back systems, supporting a bandwidth exceeding 420 nm in experiments. Validations across O, E, S, C, L, and U bands demonstrate its efficacy in broadband wavelength-mode hybrid division multiplexing communications. The breakthrough in bandwidth and compatibility with existing application scenarios promise highly efficient ultra-high-speed communication systems for future applications.

Introduction
The escalating demand for higher capacity data traffic in commercial communications systems, driven by the global deployment and rapid expansion of Internet services, has prompted to the exploration of multi-band transmission systems and innovative multiplexing technologies.

Ultra-wideband wavelength division multiplexing (WDM) technology holds the potential to significantly enhance the throughput of fiber communication networks. Extending ultra-wideband WDM systems beyond the commonly utilized C (1530−1565 nm) or L (1565−1625 nm) bands to additional bands, such as O (1260−1360 nm), E (1360−1460 nm), S (1460−1530 nm), and U (1625−1675 nm) is currently under investigation to augment the channel capacity of single-mode fiber (SMF) communication systems1–4. The advancement of ultra-wideband modulators, transceivers, and amplifiers has paved the way for the realization of ultra-wideband WDM applications5,6. In recent years, significant milestones have been achieved in this domain. Demonstrations of fiber throughput exceeding 100 Tbps have been conducted in triple-band configurations encompassing the S, C, and L bands7,8. Furthermore, the transmission of five-band WDM signals (O, E, S, C, and L bands) employing high-order modulation formats has been successfully presented9. The U band shows promise as the next transmission band for multi-band networks, primarily due to its advantage of lower transmission loss. The National Institute of Information and Communications Technology constructed the world’s first O to U band system capable of dense WDM (DWDM) transmission in a commercially available standard optical fiber achieved with custom designed amplifier technology10. It is anticipated that spectrum expansion will emerge as a prominent trend in augmenting channel capacity in the future.

Space division multiplexing (SDM) extends the spatial dimensions of electromagnetic waves and is recognized as an next-generation fiber communication technology with immense potential11–13. Compared to the Shannon’s law for SMF optical communication systems, the capacity of SDM systems that can be reliably communicated incorporates a factor of the number of spatial channels. SDM systems achieve continuous breakthroughs in channel capacity by leveraging spatially orthogonal modes in few-mode fiber (FMF)/multi-mode fiber (MMF)14–16 or multi-core multiplexing based on spatially separated cores in multi-core fiber (MCF)17–19. Various modes of FMF/MMF have been reported, including 2-mode20, 4-mode21, 6-mode22, 9-mode23, 15-mode24, and even 55-mode25 configurations. Through a multi-dimensional hybrid multiplexing system integrating wavelength, space, polarization, and higher-order signal modulation formats, the transmission capacity per fiber for SDM can reach the order of Pb/s. Demonstrations of FMF/MMF signal transmission have showcased significant capacities, such as over 1 Pb/s for 15-mode and up to 1.53 Pb/s for 55-mode configurations, achieved by Georg Rademacher et al. in 202126 and 202227 respectively. These transmissions have also achieved notable bandwidths, reaching up to 82 nm26. With the development of few-mode MCF (FM-MCF), SDM communication systems, through a combination of mode channels and multi-core channels, exhibit considerable potential to increase total capacity28–30. Experimental SDM-WDM hybrid multiplexing communications have reached transmission capacities of up to 10.66 Pb/s in a 38-core-3-mode FM-MCF configuration31. While existing SDM studies have predominantly focused on traditional C+L bands or single-span bands (e.g., O band), the deployment of ultra-wideband strategies and SDM technologies may occur at different times due to cost considerations. However, they exhibit synergistic effects. Looking ahead, ultra-wideband SDM transmission represents a forefront research area in optical communication systems, promising further improvements in capacity and performance.

Over the past few years, a plethora of mode multiplexers has become integral to SDM systems, sparking notable research interest. Investigations have delved into mode multiplexers/demultiplexers employing various optical components, such as free-space combining28, phase diffraction gratings32, all-optical fibers33, planar lightwave circuits34, photonic integration35, optical geometrical transformations36, metasurfaces37, and multi-layer optical networks38–42. Among these devices, the maximum number of multiplexed modes up to 1025 Hermite-Gaussian modes in free space43 and 55 linearly polarized (LP) modes in fiber communication systems27, respectively. Multiple methods seamlessly integrate with WDM for wavelength-mode hybrid transmission across the C+L bands or O band26,44. Metasurfaces and multi-layer optical networks mark significant progress in multi-wavelength device technology due to their precise beam control, versatility in both application and manufacturing, and high efficiency. However, they are susceptible to considerable chromatic aberrations due to their operation as diffractive optical elements. Several pioneering studies have demonstrated that multi-layer diffraction structures capable of operating over a broad wavelength range can be achieved through dispersion control. Notable examples include achromatic multi-layer metasurface lenses45, wavelength-multiplexed diffractive neural networks46, broadband optical computing systems47 and ultra-wideband 90-degree optical hybrid48. As demands surges for higher bandwidth in SDM transmission applications, ultra-wideband multiplexer remains a major challenge. Wavelength dispersion plays an important role in designing ultra-wideband multiplexer. Designing ultra-wideband multiplexers with high performance and low complexity presents significant challenges, particularly in ensuring low wavelength- and mode-dependent characteristics, high mode purity, and low insertion loss (IL).

In this work, we utilize a wide-spectrum, dispersion-compensated, multi-layer diffractive network within an end-to-end inverse design framework to create ultra-wideband optical components. As a proof of concept, we demonstrate an ultra-wideband, 4-LP mode multiplexer that achieves high efficiency over a broad bandwidth ranging from 1260 to 1680 nm. We present a general model for an ultra-wideband mode multiplexing transmission system with a bandwidth exceeding 400 nm, significantly enhancing transmission capacity. To our knowledge, there are no reports of ultra-wideband mode division multiplexing (MDM) covering the entire O+E+S+C+L+U bands. Strategies to increase capacity per fiber mode involve combining data signals and using higher-order modulation formats across ultra-wideband wavelength and mode channels spanning the O+E+S+C+L+U bands, as shown in Fig. 1. By integrating our ultra-wideband mode multiplexing system with existing commercial WDM communication systems, we achieved a series of hybrid multiplexed communication transmissions. Our approach to ultra-wideband multi-layer diffractive network introduces significant innovations in full-color light manipulation, imaging, optical computing and high-capacity optical communications, advancing these designs toward practical applications.

Results
Typically, light waves of different wavelengths exhibit varying diffraction effects, impacting phase control. The phase distribution resulting from an N-layer diffraction network is as follows:

\phi_N(\lambda) = \phi(\lambda_r) + \delta\phi(\lambda, Multilayers) (1)

where \phi(\lambda_r) represents the basic phase profile related to the reference wavelength \lambda_r, and \delta\phi(\lambda, Multilayers) denotes the additional phase shift between \lambda and \lambda_r. This phase shift, crucial for system sensitivity and accuracy, is influenced by the phase response of each unit element in the multi-layer network. It is vital for achieving stable, high modulation efficiency over an ultra-wideband range. Designing the multi-layer network to compensate for phase shifts is therefore crucial.

Here, a wide-spectrum dispersion-compensated multi-layer optical diffraction network is depicted in Fig. 2a. The N cascaded diffractive layers, positioned between the input and output apertures, handle the manipulation and dispersion compensation of complex light across different wavelengths. Each layer is coded with spatial features and a trainable thickness that allows full phase modulation from 0 to 2\pi. It is worth noting that all layers in the described framework are transmissive, although reflective layers can also be employed in practical experiments. The phase profiles of all the layers can be printed on a single reflective phase plate. In this work, the subsequently fabricated device is experimentally realized by utilizing a multi-pass reflective cavity formed by the phase plate and a mirror. The input, diffractive layers and output apertures are connected to each other through free space. The free space propagation can be modeled using the scalable angular spectrum method or the Fresnel diffraction approach. The phase shift introduced by the wavelength can be compensated by modulation, the effect of which can be evaluated and optimized by the modulation efficiency of the system. In our design, we propose a synergistic modeling mechanism with multi-objective and multi-physics coordination through a wide-spectrum modulation phase function \psi(\lambda) for compensation of ultra-wideband chromatic dispersion and spatial phase mismatch:

\psi(\lambda) = \frac{\lambda}{\lambda_r} \psi_r \left(\frac{B}{W}\right) (2)

where W denotes the spectral width of training and B is the working bandwidth. The reference wavelength \lambda_r is used to convert the pixel matrix of the optical diffraction layer to the physical structure. \psi_r is the basic phase profile at \lambda_r. The phase function \psi(\lambda) establishes a non-linear mapping from the coupled space into the global modulation efficiency \eta, which jointly evaluates mode purity, IL, and intermodal crosstalk:

\eta \propto f(K \cdot \psi(\lambda)) (3)

This mapping defines a dual-parameter dispersion balancing strategy based on reference wavelength (\lambda_r) and a wide spectral window (W), which balances spectral performance across short and long wavelengths across the entire target bandwidth B. A normalization factor (\lambda/\lambda_r) is applied to align the compensation phase across different wavelengths, effectively eliminating absolute phase offsets. Additionally, the coefficient K, associated with the number of diffractive layers, governs overall efficiency. Excluding free space propagation effects, the mean value \mu of \psi(\lambda) across B determines \lambda_r. Setting \lambda_r where \mu = \psi(\lambda_r) equalizes efficiency at the bandwidth boundaries:

\lambda_r = \frac{2 \cdot \lambda_{min}^0 \cdot \lambda_{max}^0}{\lambda_{min}^0 + \lambda_{max}^0} (4)

Where \lambda_{min}^0 and \lambda_{max}^0 are the maximum and minimum wavelengths of the working bandwidth, respectively. Due to free space propagation, shorter wavelengths experience stronger diffraction. Therefore, as the number of diffracting layers or the propagation distance increases, \lambda_r should be adjusted to a shorter wavelength to compensate for this effect. It is noted that the standard deviation of efficiency across the working bandwidth B is positively correlated with the dispersion compensation effect. Thus, we use this parameter to characterize the wavelength dependence of the system.

Based on the wide-spectrum dispersion-compensated design framework, a four LP-mode multiplexer consists of six-layer optical diffraction network with a bandwidth 420 nm was developed. When the training spectral width W is set within the range of 600 to 800 nm, the mode purity within the bandwidth ranges of 1260 nm−1680 nm remains above 90% with low standard deviations (see Supplementary Fig. S3). Figure 2b shows that the design of W = 700 nm (1100−1800 nm). Six diffraction layers exhibit smooth phase distributions. Figure 2c illustrates the simulated wavelength-dependent average mode purity, maximum mode crosstalk, average IL and mode dependent loss (MDL) for all mode channels of the multiplexer. The mode purity exceeds 90% across the entire 420 nm bandwidth, with the highest mode purity exceeding 94%. The crosstalk remains below −15 dB, while the IL and MDL are below 0.53 dB and 0.22 dB, respectively. Additionally, the IL and mode-dependent loss exhibit low wavelength dependence, with fluctuations of less than 0.21 dB and 0.12 dB, respectively. With respect to its ultra-wideband characteristics, the device demonstrates a certain tolerance for alignment and propagation distance errors, remaining within the practically acceptable range (see Supplementary Note 2). Notably, the performance of the device varies significantly in the short wavelength range, which is an effect of the enhanced short-wavelength attenuation due to free space propagation. To ensure balanced performance across the band, the reference wavelength was set to the shorter wavelength (1380 nm) rather than 1440 nm as calculated from Eq. 4 (see Supplementary Fig. S4). This design simultaneously achieves performance balancing and comprehensive performance enhancement across the entire telecommunication range, advancing MDM technology from fixed-wavelength/band or narrowband dedicated devices toward an ultra-wideband universal platform (see Supplementary Fig. S5).

In the experiments, a reflective ultra-wideband mode multiplexing system was implemented, as depicted in Fig. 3a. The experimental setup comprises a 2×2 fiber-microlens array with a standard 250 \mu m pitch and a reflective cavity, which includes a silicon-based gold-plated phase plate and a reflector. Six phase planes were printed on the same reflective phase plate and fabricated using a combination of ultraviolet exposure and dry etch etching in a four-cycle process (see Supplementary Fig. S6). A deep-to-shallow etching sequence was adopted, resulting in etching depths of 345, 173, 86, and 43 nm in each cycle. The spatial resolution was 8 \mu m. The distance between the fiber-microlens array and the reflective cavity was approximately 15 mm. Ultra-wideband fundamental mode Gaussian beams were incident at an angle of approximately 9.8° and reflected back and forth 11 times within the reflective cavity before being output as ultra-wideband LP01, LP11a, LP11b and LP21 mode light.

The performance of the multiplexer was experimentally characterized in the O+E+S+C+L+U bands. To cover the entire bandwidth range, measurements were conducted separately using four tunable laser light sources, each operating in a different spectral band. Figure 3b illustrates the measured intensity distributions of the generated LP modes in these six bands. It can be observed that the mode fields exhibit concentrated intensity with minimal scattering. Figure 3c depicts the wavelength-dependent maximum mode crosstalk among the four modes, calculated from the reconstructed mode fields obtained through interferometric measurements captured by a digital holography experiment38. The maximum mode crosstalk measured in the 1260−1675 nm band is less than −17 dB, with the minimum crosstalk occurring near the center wavelength. As shown in Fig. 3d, the measured fiber-to-fiber losses in the 1260−1675 nm wavelength range were found to be less than 4.9 and 4.4 dB when coupling the LP modes generated in free space into a four-mode fiber (FMSI-4, YOFC) and a six-mode fiber (FMSI-6, YOFC), respectively. These fiber-to-fiber losses encompass the device loss of the optical field from the incident side before the SMF array to the free space side of the multiplexer, as well as the coupling losses of the mode field from free space to FMF. These results demonstrate that the designed ultra-wideband mode multiplexer is compatible with commercially available FMF standards, maintaining fiber-to-fiber losses at a low-loss level over the ultra-wideband range. This characteristic significantly reduces power consumption in the transmission system26,49–51. When employing a six-mode fiber, the fiber-to-fiber losses of the different modes are lower. Furthermore, it was observed in the experiments that the mode degradation of the four-mode fiber becomes more severe in the short wavelength range as the transmission distance increases. This finding is particularly crucial for subsequent communication transmission experiments.

Figure 4a illustrates a fundamental framework for ultra-wideband mode multiplexing communication systems, incorporating critical components, such as optical transceivers (Transmitter and Receiver), wavelength multiplexers (WDM MUXs), a mode multiplexer, a mode demultiplexer, and wavelength demultiplexers (WDM DEMUXs). Each optical transceiver consists of a transmit port and a receive port, independently generating and detecting individual wavelengths. A polarization controller (PC) is utilized to adjust the coupling of modes in fiber transmission. The measured maximum mode crosstalk of a typical back-to-back mode multiplexer-demultiplexer system with three modes (LP01, LP11a and LP21) across the O+E+S+C+L+U bands is below −8.87 dB (see Supplementary Fig. S7). Experimental validation of mode multiplexing and wavelength-mode hybrid multiplexing communication transmission in the O, E, S, C, L, and U bands was conducted under two typical optical transceiver mechanisms.

In communication systems operating within the O and E bands, a cost-effective communication mechanism based on direct detection necessitates low mode coupling and stable transmission characteristics. A coarse WDM (CWDM)-MDM transmission system over a 5 km six-mode fiber using direct detection technology was demonstrated. In this scenario, six-wavelength channels (1271, 1291, 1311, 1331, 1351, and 1371 nm) and three-mode channels (LP01, LP11b, and LP21) were utilized for signal transmission. Signals modulated with on-off keying (OOK) were transmitted at a data rate of 10 Gbps per channel. Due to the strong coupling of modes between the same mode groups, one of the LP11a and LP11b mode channels was selected, posing a challenge for detection without a multiple-input multiple-output algorithm. The measured bit error rate (BER) values of all 18 channels in the CWDM-MDM transmission scenario are depicted in Fig. 4b. All BER values are below the forward error correction (FEC) threshold of 3.8E-3, indicating successful data signal transmission. These results validate that the designed mode multiplexer exhibits excellent compatibility with standard CWDM systems.

Based on an intensity modulation direct detection (IMDD) system, a three-mode multiplexed transmission of OOK signals with a channel rate of 10 Gbps over a 5 km four-mode fiber was conducted in the U band. In the absence of optical amplifiers, a 3×3 multiple-input multiple-output (MIMO) architecture with digital signal processing (DSP) was implemented at the receiver. The measured BER values remained below the FEC threshold of 2.4E-2 across the 1630−1670 nm wavelength range, as demonstrated in Fig. 4c.

Using coherent detection technology, the C band DWDM-MDM system, S band and L band broadband MDM transmission systems were constructed over a span of 20 km of four-mode fiber. The S band employed dual-polarization quadrature phase shift keying (DP-QPSK) modulation, whereas DP 16-quadrature amplitude modulation (DP-16QAM) was utilized in the C and L bands, both operating at a symbol rate of 32 Gbaud. All three systems employed 8×8 MIMO-DSP configuration at the receiver. At the same optical signal-to-noise ratio (OSNR), measured BER values for all mode channels were similar with low wavelength dependence, demonstrating the system’s ability to transmit high-quality broadband signals within a given OSNR range (see Supplementary Fig. S8). Figure 4d illustrates the BER scatters of 80-wavelength 4-mode multiplexing transmission in the C band (1530.33−1561.43 nm) and 4-mode multiplexing transmission in the S band (1460−1510 nm) and L band (1570−1608 nm). It can be seen that the BER values of all measured channels were below the FEC threshold of 2.4E-2, affirming that the proposed ultra-wideband mode multiplexer effectively supports mode multiplexing transmission in the S+C+L bands. Deeper MIMO equalizer enhancements (e.g., increased taps or advanced nonlinear compensation) demonstrate effective BER reduction (see Supplementary Fig. S9), enabling our system to achieve longer-distance transmission potential while using the current multiplexer. Since the device was designed to excite only the LP21a mode, it inherently implemented selective demultiplexing of LP21a at the receiver for coherent detection. When the multiplexer is specifically designed to excite both LP21a and LP21b modes simultaneously, the system enables joint detection of degenerate LP21 modes at the receiver without additional PC for suppressing coupling (see Supplementary Fig. S10).

Methods
To model the diffractive network, its diffractive layers are treated as thin planar elements. For the i-th diffractive feature of the k-th layer at the spatial coordinates (x, y, z), the wavelength-dependent complex transmission coefficient t_k(x, y, z) at the c-th mode channel is represented as a function of the wide-spectrum modulation phase function \psi_k(\lambda) at the spatial coordinates (x, y, z). The relationship can be mathematically expressed as:

t_k(x, y, z) = a_k(x, y, z, \lambda) \exp(j\psi_k(x, y, z)) (8)

here, a and \psi represent the amplitude and phase coefficients, respectively. The diffractive layers are connected by free-space propagation, which is modeled through the band-limited angular approach. Consequently, the transfer function for free-space propagation over a distance d between two successive layers is given by:

H_k(f_x, f_y, d, \lambda) = \exp(j2\pi d \sqrt{\frac{1}{\lambda^2} - f_x^2 - f_y^2}), f_x^2 + f_y^2 < \frac{1}{\lambda^2} (9)

where f_x and f_y represent the spatial frequencies along the x and y directions, respectively.

Assuming u_k(x, y, z) represents the input complex light field of wavelength \lambda at the c-th mode channel on the k-th optical layer. The input complex light field of the k-th optical layer is obtained through the convolution operation between the scattered field generated by the (k-1)-th optical layer and the diffraction kernel. For the k-th layer, the forward optical field u_k(x, y, z) can be expressed as:

U_k(f_x, f_y, d) = \mathcal{F} \{ u_{k-1}(x, y, z_{k-1}) \cdot \exp(j \cdot Arg(t_{k-1}(x, y, z_{k-1})) \cdot \frac{\lambda}{\lambda_r}) \}

u_k(x, y, z_k) = \mathcal{F}^{-1} \{ U_{k-1}(f_x, f_y, d) \cdot H_{k-1}(f_x, f_y, d) \} (10)

where \mathcal{F}{·} and \mathcal{F}^{-1}{·} denote the Fourier transform and inverse Fourier transform operations, respectively. \lambda_r represents the reference wavelength.

It is assumed that g_k(x, y, z) represents the output complex light field of a wavelength of \lambda at the c-th mode channel on the k-th optical layer. The output complex light field of the k-th optical layer is obtained through the convolution between the scattered field generated by the (k+1)-th optical layer and the diffraction kernel. The backward optical field g_k(x, y, z) of the k-th layer can be written as follows:

G_k(f_x, f_y, d) = \mathcal{F} \{ g_{k+1}(x, y, z_{k+1}) \cdot \exp(-j \cdot Arg(t_{k+1}(x, y, z_{k+1})) \cdot \frac{\lambda}{\lambda_r}) \}

g_k(x, y, z_k) = \mathcal{F}^{-1} \{ G_{k+1}(f_x, f_y, d) \cdot H_{k+1}(f_x, f_y, d) \} (11)

The correlation function is described as follows:

\rho_{\lambda_c} = \frac{\sum_x \sum_y (O_{\lambda_c} \cdot O_{\lambda_c}^*) \times \sum_x \sum_y (T_{\lambda_c} \cdot T_{\lambda_c}^*)}{\sum_x \sum_y (O_{\lambda_c} \cdot T_{\lambda_c}^*)^2} (12)

The function is defined as mode purity, which is used to evaluate modulation efficiency. Here, O_{\lambda_c} and T_{\lambda_c} denote the generated and target modes light fields of wavelength \lambda at the c-th mode channel on the output plane, respectively.

In the implementation of the CWDM-MDM transmission, 18 optical transceivers are all driven by an optical switch (HUAWEI CE6865-48S8CQ-EI). 10 Gbit/s fundamental mode light signals using OOK at wavelengths of 1271, 1291, 1311, 1331, 1351, and 1371 nm were generated independently from the transmit port of different optical transceivers and coupled into six-wavelength CWDM broadband signals after passing through a commercial CWDM multiplexing module. The three channels of fundamental mode broadband signals were then modulated by the wideband mode multiplexer to generate LP01, LP11b and LP21 three spatial modes channels of broadband signals and transmitted over a 5 km six-mode fiber. At the receiving side, the broadband LP modes signals were split into three channels of broadband fundamental mode signal lights by a wideband mode demultiplexer. After passing through three CWDM demultiplexing modules, three channels of broadband signals channels were separated into 18-channel single wavelength signals and detected by the receive port of the corresponding transceiver.

In the broadband MDM transmission experimental setup, an external cavity laser (ECL) was utilized to generate five optical carriers in the range of 1630−1670 nm with a 10 nm wavelength spacing. The OOK signals modulated by the DSP were loaded onto the arbitrary waveform generator (AWG) to generate the electrical waveform, which was amplified by the electrical amplifier (EA) to drive a Mach-Zehnder modulator (MZM) biased at the quadrature point to implement intensity modulation. Then, the signals were divided into three channels by a 1×4 optical coupler (OC) to be incident on the input ports of a mode multiplexer. The three channels of fundamental mode signals were further modulated by the wideband mode multiplexer to generate LP01, LP11, and LP21 spatial modes of broadband signals, which were transmitted over a 5 km four-mode fiber. At the receiver, followed by the fiber was a PC to adjust the coupling of modes in fiber transmission. Then, the LP modes signals were split into three channels by the wideband mode demultiplexer. Photoelectric signal conversion detection was performed by a photodiode (PD) and the waveform was displayed on the mixed signal oscilloscope (MSO) (see Supplementary Fig. S12). Given the use of a single polarization state per spatial mode and a total of three spatial channels, a 3×3 MIMO equalization structure was implemented.

In the implementation of C band DWDM-MDM transmission, ECLs were utilized to generate 80 optical carriers with a 50 GHz frequency spacing at the transmitter. These carriers spanned the wavelength range of 1530.33−1561.43 nm, following the international telecommunication union (ITU) standard. The 80 channels were divided into even and odd channels and underwent modulation in two IQ-modulators after passing through polarization-maintaining array waveguide gratings (PM-AWGs). The modulators were driven by a 4-channel arbitrary waveform generator (AWG) operating at a 64 GSa/s sampling rate. The modulation format of the generated optical signals was DP-16QAM with a symbol rate of 32 Gbaud. Post electro-optical modulation, the two sets of 40-channel optical carriers were combined into 80-channel DWDM signals via a 1×2 optical coupler. Subsequently, the DWDM signals traversed an analog polarization multiplexer (PM) containing a 1 m length of polarization-maintaining (PM) fiber serving as a delay line for decorrelation between the two polarization signals. They were then amplified in an erbium-doped fiber amplifier (EDFA) to equalize the power spectrum and divided into four channels by a 1×4 optical coupler to be incident on the input ports of a mode multiplexer. The four channels of fundamental mode signals were further modulated by the wideband mode multiplexer to generate LP01, LP11a, LP11b, and LP21 four spatial modes of broadband signals, which were transmitted over a 20 km four-mode fiber. SMF delay lines, with lengths of 10, 6, and 3 m, were introduced to three of the input optical signal channels of the mode multiplexer to facilitate decorrelation between signals of different modes. Meanwhile, a variable optical attenuator (VOA) was placed after each branch of the 1×4 optical coupler to equalize the optical power of the four modes. At the receiver, the broadband LP modes signals were split into four channels by the wideband mode demultiplexer. A time-division multiplexing (TDM) receiver was employed, receiving a single channel of four-mode broadband signals output from the mode demultiplexer for each time slot. The signals channel with single wavelength and single mode were separated after passing through a wavelength selector switch (WSS). Finally, the signals were received and demodulated by an integrated coherent receiver (ICR), and the waveform was displayed on an oscilloscope (OSC). Given the total eight spatial channels of four modes and two polarization states, an 8×8 MIMO-DSP configuration was employed (see Supplementary Fig. S13).

In the implementation of broadband MDM transmission in the S band, ECL generated 26 optical carriers with varying wavelengths intended for transmission into the IQ modulator. These carriers spanned the wavelength range of 1460−1510 nm with a wavelength spacing of 2 nm. Unlike the facilities used in C band transmission, a thulium-doped fiber amplifier (TDFA) is adopted to amplify the signals, which can be better adapted to the S band transmission. The sample rate of AWG is set at 64 GSa/s. The modulation format of the generated optical signals was DP-QPSK with a symbol rate of 32 Gbaud (see Supplementary Fig. S14).

Simultaneously, in the implementation of broadband MDM transmission in the L band, the ECL generated 20 optical carriers with varying wavelengths intended for transmission into the IQ modulator. These carriers spanned the wavelength range of 1570−1608 nm with a wavelength spacing of 2 nm. Parameters, such as the AWG sample rate, modulation format, and symbol rate remained consistent with those of the C band system.

Peer review rounds
[{"round": 1, "reviewer_comments": {"reviewer_1": "Please see the attached file.", "reviewer_2": "This paper described a design methodology, a prototyped device and its characterizations, and proof-of-concept system demonstration of a ultra-wideband mode multiplexer across 420 nm spectrum range, covering almost all the telecom wavelengths of interest. The motivation of this work is valid and considered very timely, and the characteristics and performance of the device are quite good. The authors also performed careful review on the state-of-the-art technologies that are relevant to this study. The transmission experiments are carefully carried out with stringent configurations to de-correlate all the adjacent channels, which is highly appreciated. The design methodology is also considered valuable takeaway information from this paper. This work is indeed, to my knowledge, the first report on a close-to-feasible miniaturized mode mux for such a wide range of spectrum coverage, which can directly support multi-band transmission system upgrade. Therefore, I consider this work highly novel and quite significant in driving forward mode multiplexing on top of ultra-wide multiband optical communications beyond the C-band.\nBefore recommending accepting the manuscript for publication, I have a list of comments for the authors to address, detailed as follow.\n1. Figure 1a shows the end-to-end inverse design framework. The way showing such a design is intuitive and clear. However, considering each layer of the later fabricated device in practice is reflective rather than transmissive, it's suggested to clarify this early in the text when describing the design concept in order to avoid confusion.\n2. In the Subsection \"Experimental characterization for ultra-wideband mode multiplexing\", the authors described briefly the characterization setup. For a better understanding, please elaborate on the fiber-microlens array configuration. Does each input fiber carry the same ultra-wideband fundamental mode Gaussian light? How was this input light generated? Was it also a WDM muxed signal similar to the transmission experiment, or was it a single ultra-wideband light source? What's the distance between the array and the reflective cavity? Please clarify.\n3. In the same subsection, the authors tried to explain the performance variations in the short wavelength range, which is still not very clear to me. What does it exactly mean by enhanced short-wavelength attenuation due to free-space propagation? Could you please quantify or provide references?\n3. A follow up question on the previous comment, is it possible to include such performance variation aspect in the analytical expressions Eq.4？This can provide more precise guidelines on selecting the reference wavelength. Beside, could you please explain how exactly did you choose the reference wavelength of 1380 nm? Is it manually optimized?\n4. Figure 3c shows the crosstalk results through measurements using a digital holography setup, which is not described in the manuscript. Can the authors provide the descriptions of this experimental configuration or provide a reference?\n5. In the same section, the authors stated that \"This characteristic significantly reduces power consumption in the transmission system.\" Please explain with which reference schemes is it compared to draw such a conclusion?\n6. Also the authors stated that the mode dispersion becomes more severe in the short wavelength region. Please show the characterized result to support this statement, or explain qualitatively how you made such an observation.\n7. The caption of Figure 4d states that BER curves are shown. These are not BER curves. Please rephrase.\n8. In the Subsection \"Ultra-wideband mode multiplexing communication transmission experiments\", I would suggest to include the data rates and modulation formats for both the O-band CWDM and C+L band DWDM transmissions already here in the text, besides the descriptions later in the Methods section. It helps to better understand the transmission results.", "reviewer_3": "Dear Authors,\nThank you submitting this very interesting paper. It presents very important research results on enhancing the capacity of fiber-optic telecom networks by means of mode multiplexing and addressing several optical transmission bands. The results are very readable and convincing, which pave the road for further development in the SDM subject area.\nI have nothing to suggest regarding methodology, analysis of the obtained results.\nI recommend this paper to be accepted provided:\n1. It is thoroughly checked in terms of English language use. There are some minor mistakes regarding punctuation or grammar e.g. 5-6 line of the abstract \"... system to enhancing...\";\n2. Some graphics need improvement to be better readable. Perhaps, it is enough to take advantage of the complete column width e.g. Fig.1, 2b (indexes are hardly visible)."}, "author_response": "Response to the reviewers’ comments (Nature Communications, Research Article, No. NCOMMS-24-56122)\nWe sincerely thank all reviewers for your valuable comments and suggestions on our manuscript titled \"Ultra-wideband optical diffractive network for mode multiplexing across the entire telecommunication range\" (Manuscript ID: [NCOMMS-24-56122]). We greatly appreciate the time and effort you have invested in reviewing our work, and your insightful feedback has significantly improved the quality of the manuscript. We highly cherish the opportunity to revise our work. All the reviewer’s suggestions have been taken into account.\nThe followings are the changes made in the manuscript to fully address the reviewers’ concerns:\nBlue Italic: Original comments from reviewers.\nBlack: Our response including the action taken. The position of the changes made in the manuscript has been highlighted in this response.\nResponse to reviewer 1:\nReviewer #1: The author proposes a wide-spectrum, dispersion compensated, multi-layer diffractive network design method to create ultra-wideband optical components, based on which an ultra-wideband, 4-LP-mode MPLC mode multiplexer is designed and fabricated for proof-of-concept demonstration. The MPLC component can achieve high efficiency over a very broad bandwidth ranging from 1260 nm to 1680 nm. However, I do not recommend this paper to be published by Nature Communications for the following reasons: ultra-wideband MPLC-based devices and transmission experiments have been experimentally demonstrated in previous works. Its novelty, progress, and practical significance must be clarified and highlighted before it be considered for publication. Here are my comments:\n1. The authors claim a wide-spectrum device design method for multi-layer diffraction network by compensating the spatial dispersion. Please clarify the innovations of your approach by comparing it with previous work in reference [1], in which an ultra-wideband MPLC with a bandwidth over 900 nm is achieved by optimizing phase planes at various wavelengths. Mathematical and simulation descriptions must be added to highlight the practical significance of this work.\nWe sincerely appreciate the reviewer’s thoughtful comments and the opportunity to clarify and further elaborate on the innovative aspects of our work. In comparison with the study in reference [1], our research exhibits innovation across multiple dimensions. It not only extends and enhances existing technologies but also introduces novel contributions in several key areas.\nBelow, we address the specific points raised in the reviewer’s comments, offering a detailed comparison with the work presented in reference [1] from several critical perspectives.\n1. Application in combined WDM+MDM multiplexing\nIn optical communications, traditional wavelength division multiplexing (WDM) technology has made significant advancements in enhancing transmission capacity and bandwidth utilization. However, with the continuous growth in communication demands, the single WDM technology is increasingly inadequate to meet the requirements for high capacity and efficiency. Mode division multiplexing (MDM), as an emerging optical communication technology, further extends the transmission capacity and bandwidth by incorporating spatial dimension multiplexing.\nOur research utilizes multi-layer diffraction network technology to advance combined WDM+MDM multiplexing technology, enabling the parallel transmission of multiple wavelengths and spatial modes within a single optical fiber. This approach is applicable across the entire telecommunication range (O+E+S+C+L+U bands), overcoming the limitations of traditional WDM technology. It also addresses the challenge of dispersion compensation in integrating spatial and wavelength domain multiplexing, thereby overcoming the transmission bandwidth limitations typically encountered in conventional MDM systems. This technology holds significant practical value for real-world communication networks, particularly in high-capacity optical communication systems.\nThis aspect is not addressed in reference [1], which focuses on the ultra-wideband 90-degree optical hybrid, primarily used for signal mixing and phase modulation. Although this optical hybrid achieves high bandwidth, it does not tackle spatial mode multiplexing or the combined multiplexing of WDM and MDM, nor does it overcome the limitations associated with the integration of wavelengths and spatial modes.\n2. Experimental and practical application validation\nWe have not only demonstrated the effectiveness of our technology through simulations but also conducted real-world optical characterization and communication tests across the entire communication bands (O + E + S + C + L + U bands). The experimental results indicate that our research has significantly overcome the bandwidth limitations of traditional MDM systems. The joint WDM and MDM multiplexing technology can substantially enhance network capacity in optical communication systems. This experimental validation proves the feasibility of our design method and showcases its practical application value, particularly in high-capacity communication systems.\nHowever, the study in reference [1] was merely tested at selected wavelengths within the O, S, C, and L bands and primarily focused on signal mixing in the broadband range, lacking a comprehensive approach to spatial mode multiplexing and wavelength multiplexing. As a result, it has not contributed to the expansion of optical communication system capacity.\n3. Application scenarios and functional expansions\nAlthough reference [1] and our work utilize similar technologies (multi-layer diffraction networks and MPLC), their application scenarios and functional expansions are fundamentally different.\nReference [1] focuses on achieving an ultra-wideband 90-degree optical hybrid. The primary function of the 90-degree optical hybrid is to mix two input fundamental mode optical signals. The input optical signals and the local oscillator (LO) laser light can be expressed as:\nE = P *exp[j(2*pi*f *t+phi)]， (R1)\ns s s s\nE = P *exp[j ( 2*pi*f *t+phi )]， (R2)\nLO LO LO LO\nwhere P and P represent the optical powers of the input signals. f and f denote the frequencies of the input signal and LO, respectively. phi and phi are the phases of the input signal and the LO.\nThe output signal of the 90-degree optical hybrid consists of the sum and difference between the two input fields (E and E ):\ns LO\n[E E ]\ns + LO\n[ 2 2 ]\n[ ]\n[E 1 ] [1 1 ] [E E ]\n[ ] [ ] s + j LO\nE = [ E 2] = 1 *[ 1 j ]* [ ]= [ 2 2 ] (R3)\nout [E ] 2 [1 -1] E E E\n[ ] [ ] LO s - LO\n[ 3] [ ] [ 2 2 ]\n[E ] [ ] [ ]\n[ 4 ] [1 -j] [ E E ]\n[ s - j LO]\n[ 2 2 ]\nThe output generates sum and difference frequencies, focusing on frequency mixing and phase adjustment. It depends on the relative phase between the two input signals. Reference [1] primarily addresses signal mixing and phase control, which are crucial for signal processing and coherent optical communication.\nIn contrast to the focus of reference [1], our work focuses on ultra-wideband mode multiplexing. The objective of mode multiplexing is to convert and coaxial transmit multiple orthogonal modes simultaneously. Given M fundamental modes denoted as u(x, y, z) at different positions, the input signal light for each mode can be expressed as follows:\nE = A *u (x,y,z)*exp(j*phi (lambda))， (R4)\ni i i i\nwhere A represents the amplitude of each mode, u(x, y, z) stands for the spatial distribution of the fundamental mode. phi (lambda) indicates the phase of each mode at wavelength lambda. The total output signal is the superposition of all modes, which is given by:\nM\nE = sum A *F (x,y,z)*exp(j*(phi (lambda)) (R5)\nout i i i\ni=1\nHere, F (x, y, z) refers to the spatial distribution of the high-order mode. phi (lambda) represents the phase of each output mode at the wavelength lambda. The objective of mode multiplexing is to guarantee that each fundamental mode is transmitted and converted into different spatial modes simultaneously while minimizing interference. These generated modes are coaxial and spatially orthogonal to each other. Our work applies multi-layer diffraction network technology to handle multi-mode conversion and solve dispersion compensation issues within the ultra-wideband range. This allows our technology to support more complex multi-wavelength and multi-mode transmission, thereby achieving higher-capacity optical communication systems.\n4. Technological advancements in ultra-wideband mode multiplexing\nOur research advances the realization of ultra-wideband mode multiplexing, particularly in the parallel transmission across multiple modes and wavelengths. This requires technology capable of supporting large-scale signal multiplexing and efficiently converting modes across a wideband wavelength range. Reference [1] does not address similar challenges or offer corresponding solutions. A simplified working schematic of the ultra-wideband 90-degree optical hybrid is presented in Fig. R1. In this schematic, the input local oscillator (LO) and signal light are spatially separated Gaussian beams. The target output modes are composed of four equally spaced Gaussian beams with identical power but different phases. The phase differences between the beams generated by the LO and those generated by the signal are 0, pi, pi/2 and -pi/2 degrees, respectively. In reference [1], the MPLC technology is predominantly employed for signal mixing, with an emphasis on phase control across a broad range.\nIn our research, multi-layer diffraction network technique is not only used for mode multiplexing but also addresses the issue of dispersion compensation for ultra-wideband transmission. Our study identifies several key factors that impact the scalability of these models for broadband targets, including the spatial transverse structure of the beam, the training wavelength, and the training bandwidth. These factors play a crucial role in achieving mode multiplexing with low mode crosstalk, high mode purity and low wavelength-dependent across the ultra-wideband range.\nFig. R1. Schematic of the ultra-wideband 90-degree optical hybrid using MPLC.\nAs illustrated in Fig. R2, arrays of Gaussian spots, spanning multiple wavelengths of the ultra-wideband spectrum, simultaneously enter the cascaded optical layers. The final output is coaxial orthogonal multi-mode light with multispectral characteristics. The N diffractive layers address ultra-wideband complex optical fields, focusing, converting modes and compensating for dispersion. During model training, each layer is constructed through end-to-end design and updated by establishing the relationship between the objective function and various parameters. These parameters include the wavelength, mode, coordinates, working bandwidth, spectral width of training, free space transfer function, and the sequence of diffraction layers. For a more detailed algorithm descriptions and mathematical formulas, please refer to our responses to questions 3b and 3d.\nFig. R2. Schematic of the ultra-wideband mode multiplexing using multi-layer diffractive network.\nIn summary, although reference [1] and our research utilize similar technologies (such as MPLC and multi-layer diffraction networks), our work introduces significant innovations in application scenarios, functional implementation, technological scalability, and practical applications. By applying these technologies to more complex scenarios such as ultra-wideband mode multiplexing and WDM+MDM combined multiplexing, we have advanced optical communication systems, particularly in improving network capacity and bandwidth utilization, showcasing clear technological innovation.\nWe have added and highlighted the above-mentioned main discussions in section [Result] (Free-space architecture design of cascaded optical layers constructed by the multi-focal optical system) in the revised manuscript as follows:\nParagraph 2: “Through the establishment of the relationship between the objective function and various parameters, including wavelength, mode, coordinates, working bandwidth, spectral width of training, free space transfer function and the sequence of diffraction layers, each layer is then updated accordingly.”\n2. The ultra-wideband transmission covering OESCLU bands has been demonstrated in reference [2]. Please clarify the novelty of your experimental transmission work.\nWe thank the reviewer for pointing out these important aspects that will contribute to enhancing the technical soundness of our paper. Reference [2] demonstrates ultra-wideband transmission covering the O, E, S, C, L, and U bands, leveraging the wavelength dimension of electromagnetic waves. This approach primarily focuses on using multiple wavelength channels through wavelength division multiplexing (WDM), without incorporating mode division multiplexing (MDM). In contrast, our research introduces an approach that emphasizes ultra-wideband MDM, which expands the mode dimension of electromagnetic waves. By combining with WDM, our system enables a hybrid wavelength-mode multiplexing scheme for optical transmission. This scheme facilitates a more efficient utilization of the available electromagnetic spectrum, presenting significantly enhanced capacity and improved flexibility compared to the traditional WDM-only approach. Moreover, our system is entirely compatible with the WDM-based system demonstrated in Reference [2], enabling wavelength-mode hybrid multiplexing for optical transmission.\n3. The descriptions of this algorithm should be clarified for it’s a main innovation of this work, such as:\na. Which phase plane is the spatial dispersion compensation algorithm applied to?\nWe thank the reviewer for raising this question. The dispersion compensation algorithm is implemented on each plane. During the optimization process of the algorithm, the end-to-end inverse design for each plane is employed. The iterative update of each phase plane is associated with both the forward and backward transmitted light fields, which are obtained through the phase modulation and free-space transmission of the preceding phase plane. To sum up, the algorithm functions iteratively throughout all the planes, and each plane collaborates with one another to accomplish the overall dispersion compensation.\nWe have added and highlighted the above main discussion in section [Results] (Free-space architecture design of cascaded optical layers constructed by the multi-focal optical system) of the revised manuscript in context as follows:\nParagraph 2: “All layers within the diffraction network collaborate to handle focusing, mode conversion and dispersion compensation for ultra-wideband complex optical fields.”\nb. How does the algorithm be applied? Is it just in the judgment step after calculating the correlation?\nWe thank the reviewer for pointing out this question. The algorithm is applied iteratively throughout the entire process. Calculating the correlation is only one of the steps used to determine when the algorithm terminates. In our work, the algorithm utilizes an end-to-end inverse design framework for a multi-layer diffractive network to address ultra-wideband targets.\nAs illustrated in Fig. R3(a), the N cascaded diffractive layers, which are positioned between the input and output apertures, play a crucial role in handling the manipulation of complex optical fields and compensating for dispersion across a wide spectrum of wavelengths. These layers are interconnected through free space, and the angular spectrum method is used to calculate the wave propagation between adjacent layers. Once the input and target light fields are defined, the algorithm iteratively optimizes the transmission coefficient of each diffractive layer. This optimization is based on an objective function that establishes a relationship with several critical parameters, including wavelength, mode, coordinates, working bandwidth, spectral width of the training, and the sequence of diffraction layers. In this process, the algorithm continuously refines the design of the diffractive layers to achieve low mode crosstalk, high mode purity, and effective dispersion compensation across the ultra-wideband range.\nAs illustrated in the flowchart of the algorithm in Fig. R3(b), the process begins with an initial phase profile, in which the transmission coefficient of each layer is set to 1. During each iteration, the input optical fields propagate forward across a wide spectral range, and subsequently, the desired output optical fields propagate backward across the identical range. The forward and backward diffracted optical fields, which are contingent upon factors such as wavelength, mode, coordinates, working bandwidth, and spectral width, are calculated both before and after each diffraction layer. By employing the end-to-end algorithm, the phase differences between the forward fields and the backward fields at each layer for every mode channel under multi-spectrum conditions are computed and summed following phase conversion at the reference wavelength. These phase differences function as objective functions, which are utilized to adjust the phase distribution of the corresponding diffraction layer, thereby facilitating the minimization of phase shifts across the ultra-wideband range.\nThe objective functions are updated iteratively in each step until the output of the f forward fields in the target plane meets the criterion for iteration termination defined by the correlation function.\nFigure R3 and above algorithm-related details are added and highlighted in section [Supplementary Notes] (Note 1) of the revised Supporting Information. We have added and highlighted the note in section [Results] (Free-space architecture design of cascaded optical layers constructed by the multi-focal optical system) of the revised manuscript as follows:\nParagraph 2: “(See Supplementary Note 1 for algorithm-related details).”\nFig. R3. (a) Schematic diagram of the training model and (b) the flow chart of algorithm.\nc. The significance of this algorithm in reducing the number of phase planes and decreasing computational complexity?\nWe thank the reviewer for these thoughtful comments. The primary goal of the algorithm is to exploit the properties of ultra-wideband systems. Compared to the four phase planes commonly used in four-mode multiplexers, expanding the bandwidth of the mode multiplexer to cover the entire communication band in our work results in a slight increase in the number of phase planes. However, this increase does not significantly impact computational complexity. Although reducing the number of phase planes and lowering computational complexity isn’t the main emphasis of this work, we consider it a promising direction for future research that could potentially drive further progress.\nWe have added and highlighted the above-mentioned conclusion in section [Conclusion and Discussion] (Paragraph 4) in our revised manuscript as follows:\n“Expanding the bandwidth of the mode multiplexer to cover the entire communication band in our work results in a slight increase in the number of phase planes, but this does not significantly affect computational complexity.”\nd. I suggest that the detailed mathematical algorithm should be described like that in the reference [2].\nWe thank the reviewer for insightful thoughts and professional comments.\nOptical forward model of the ultra-wideband diffractive network\nThe ultra-wideband optical diffractive network consists of cascaded diffractive layers that collectively manipulate and compensate dispersion of the incoming ultra-wideband complex optical fields. To model the diffractive network, its diffractive layers are treated as thin planar elements. For the i-th diffractive feature of the k-th layer at the spatial coordinates (x, y, z ), the wavelength-dependent complex transmission coefficient tk (x, y, z ) at the c-th mode channel can be represented as a function of the wide-spectrum modulation phase function Ψ(lambda) at the spatial coordinates (x , y , z ). The relationship can be mathematically expressed as:\ntk (x,y,z )=a *exp(j*psi(x,y,z ,lambda)), (R6)\nlambda_c i i k i i k\nhere, a and psi represent the amplitude and phase coefficients, respectively. The diffractive layers are connected by free-space propagation, which is modeled through the band-limited angular approach. Consequently, the transfer function for free-space propagation over a distance d between two successive layers is given by:\nHk(f , f ,d,lambda)=... (R7)\nwhere f and f represent the spatial frequencies along the x and y directions, respectively.\nAssuming uk (x, y, z ) represents the input complex light field of wavelength lambda at the c-th mode channel on the k-th optical layer. The input complex light field of the k-th optical layer is obtained through the convolution operation between the scattered field generated by the (k-1)-th optical layer and the diffraction kernel. For the k-th layer, the forward optical field uk (x, y, z ) can be expressed as:\nlambda_c i i k\nu_lambda_c^k(x_i,y_i,z_k)=... (R8)\nwhere Phi{·} and Phi^-1{·} denote the Fourier transform and inverse Fourier transform operations, respectively. lambda_r represents the reference wavelength.\nIt is assumed that gk (x, y, z ) represents the ouput complex light field of a wavelength of lambda at the c-th mode channel on the k-th optical layer. The ouput complex light field of the k-th optical layer is obtained through the convolution between the scattered field generated by the (k+1)-th optical layer and the diffraction kernel. The backward optical field gk (x, y, z ) of the k-th layer can be written as follows:\ng_lambda_c^k(x_i,y_i,z_k)=... (R9)\nChromatic dispersion compensation modeling and training\nTo achieve stable and high modulation efficiency ultra-wideband optical diffractive network across the entire telecommunication range, the manipulation of complex light fields and the compensation of dispersion are accomplished through the combined effect of all diffractive layers. By carrying out forward and backward propagation, the objective function used for updating the parameters at each layer can be expressed as:\nphi_k = Arg{...} (R10)\nwhere uk and gk denote the forward and backward light fields on the k-th optical layer, respectively. Meanwhile, lambda_r refers to the reference wavelength, with W denoting the training spectral width and B representing the working bandwidth respectively.\nThe correlation function is described as follows:\nrho = ... (R11)\nThe function is defined as mode purity, which is used to evaluate modulation efficiency. Here, O and T denote the generated and target modes light fields of wavelength lambda at the c-th mode channel on the output plane, respectively. Additionally, m, q and n represent the position of each pixel and the number of pixels.\nWe have added and highlighted in section [Supplementary Notes] (Note 2) of the revised Supporting Information. Additionally, we have also noted and highlighted it in section [Results] (Free-space architecture design of cascaded optical layers constructed by the multi-focal optical system) of the revised manuscript as follows:\nParagraph 4: “(see Supplementary Notes 2).”\n4. With the same number of phase planes, the optimization of bandwidth will cause a deterioration in assembly tolerance such as the error planes or incident angle. Please add mathematical and simulation descriptions for them.\nWe thank the reviewer for these valuable comments. We fully agree with the point that it is essential to consider the effects of bandwidth optimization on assembly tolerance, including error planes or incident angles. Below, we provide a response that incorporates both mathematical descriptions and simulation results to analyze these effects.\nWhen optimizing the bandwidth of the ultra-wideband optical diffractive network with a fixed number of layers, the precision required for aligning each phase plane increases. This is because increasing the bandwidth necessitates more precise control over the diffraction characteristics across a wider range of wavelengths. As the system’s operating spectrum widens, slight errors in the misalignments of diffractive layers or variations in the incident angle can result in performance degradation, particularly in terms of mode purity and dispersion compensation.\n1. Analysis of errors\nMisalignments between successive diffractive layers can induce phase errors that accumulate over the propagation path, ultimately degrading the system’s performance. we need to consider the phase shifts introduced by these errors, particularly their effects on the forward field uk . The propagation of the light field is affected by alignment errors between phase planes, propagation distance errors and other factors. These errors will accumulate in the system, ultimately degrading system performance. Here, we divide the error analysis into three key aspects within the coordinate system: the (X, Y) coordinates in the phase plane and the propagation distance in free-space along the Z-direction.\nEffect of errors in X and Y coordinates: Assuming that the k-th phase plane has alignment errors Delta_x and Delta_y in the X- and Y-directions, respectively. These errors result in phase shifts in the wavelength-dependent complex transmission coefficient. The effect of these errors can be expressed as:\ntk (x+Delta_x,y+Delta_y,z )=tk (x,y,z )*exp(j*Delta_Psi(x,y,lambda)), (R12)\nlambda_c i i i i k lambda_c i i k i i\nwhere Delta_Psi (x_i, y_i, lambda) is the phase shift at wavelength lambda caused by positional alignment errors in the k-th phase plane, which affects the forward optical field uk as follows:\nlambda_c\nu_lambda_c^k(x_i,y_i,z_k)=... (R13)\nSince these phase errors accumulate layer by layer through the system layer by layer, they can cause changes in the overall light fields, ultimately affecting mode purity and modulation efficiency.\nEffect of propagation distance error in the Z-direction: In addition to alignment errors in the X- and Y-coordinates, misalignments in the propagation distance along the Z-axis (the propagation direction between layers) also contribute to phase shifts. The error in the Z-direction, denoted as Delta_z_k, alters the relative positioning of successive diffractive layers and affects the transfer function of free-space propagation between layers. For a given propagation distance d between two successive layers, the misalignment in the Z-direction results in the following effects on the transfer function:\nHk(f , f ,d+Delta_z_k,lambda)=Hk(f , f ,d,lambda)*exp(j*Delta_phi(d,lambda)), (R14)\nx y k x y\nwhere Delta_phi (d, lambda) is the propagation phase shift of wavelength lambda caused by errors in the propagation distance between two successive layers, which affects the forward optical field uk as follows:\nlambda_c\nu_lambda_c^k(x_i,y_i,z_k)=... (R15)\nThe propagation distance error alters the phase of the light field, thereby affecting the diffraction characteristics of the system. This is especially significant for ultra-wideband signals, as their diffraction properties change in accordance with the wavelength. This effect accumulates as the light propagates through successive layers.\nIncident angle error: In the reflective configuration, the incident angle is determined by the spacing of adjacent phase planes on the phase plate and the distance between the phase plate and the reflector. The former is associated with the coordinates (X, Y). The latter affects the free-space propagation distance. The effect of the incident angle error is related to both the alignment errors between phase planes and the propagation distance errors. Therefore, we will not analyze the incident angle in detail here. However, reviewers interested in further details can refer to the relevant reference [arXiv:2304.11323].\n2. Simulation of assembly tolerances\nTo quantify the effects of these errors, we simulated the system’s response to these tolerances. In our simulation, we introduced several types of errors: Delta_x and Delta_y errors in the X- and Y-directions for each phase plane, as well as propagation distance errors denoted as Delta_z. The mode purity of the corresponding forward output fields was calculated across the ultra-wideband range to assess the system’s performance.\nFigure. R4(a) illustrates that the effects of alignment errors in the X- and Y-directions on system performance are quite similar. When the alignment errors Delta_x or Delta_y of all planes amount to 8 um, the degradation in mode purity is less than 2% throughout the ultra-wideband range spanning from 1260 to 1680 nm. Additionally, the performance fluctuation among different wavelengths remains below 1%. As these errors grow larger, the degradation becomes more pronounced at shorter wavelengths.\nFig. R4. (a) Deviations in mode purity due to different alignment errors of Delta_x and Delta_y. (b) Deviations in mode purity due to different propagation distance errors of Delta_z.\nIt is worth noting that the system demonstrates greater tolerance to propagation errors Delta_z in the Z-direction. As depicted in Fig. R4(b), when the spacing between adjacent layers is increased, the performance at shorter wavelengths improves. However, the tolerance at long wavelengths is relatively lower. The tolerance for distance errors that result in a 2% degradation in mode purity across the entire wavelength range lies between -0.5 mm and +1 mm. It is evident that the system is less tolerant to errors associated with distance reduction, particularly at shorter wavelengths. It is clear that the system is less tolerant to errors associated with distance reduction, particularly at shorter wavelengths. Nevertheless, these errors remain within the tolerable range for practical operation.\nFigure R4 and above discussions are added and highlighted in section [Supplementary Notes] (Note 3) of the revised Supporting Information. We have added and highlighted the main conclusion in section [Results] (Free-space architecture design of cascaded optical layers constructed by the multi-focal optical system) in our revised manuscript as follows:\n“With respect to its ultra-wideband characteristics, the device demonstrates a certain tolerance for alignment and propagation distance errors, remaining within the practically acceptable range (see Supplementary Note 3).”\n5. Please provide some details for the transmission experiment:\na. Are both LP21a and LP21b modes demultiplexed at the receiver?\nWe thank for the reviewer’s insightful comments. In this work, only LP21a is utilized. To investigate the coherent reception characteristics of both LP21a and LP21b degenerate modes, we designed and fabricated two multiplexers: a five-mode multiplexer encompassing LP01, LP11a, LP11b, LP21a, and LP21b, as well as a four-mode multiplexer consisting of the two degenerate modes, LP11a, LP11b, LP21a and LP21b.\nFig. R5. Measured BER scatter plots of (a) five-mode and (b) four-mode transmission systems in the C+L bands.\nSubsequently, coherent transmission experiments based on these two designs were conducted in the C band (1530.33−1561.43 nm) and L band (1570−1608 nm) respectively. These experiments adhered to the same system configuration as that described in the original manuscript. As depicted in Fig. R5, the measured bit error rate (BER) values for all channels were found to remain below the forward error correction (FEC) threshold of 2.4 *10^-2. This indicates that a mode-division multiplexing transmission system that incorporates both LP21a and LP21b degenerate modes is still capable of achieving coherent detection at the receiver.\nb. How to deal with the rotation, coupling, differential mode gain/loss in the degenerated LP21 mode in the DSP algorithm?\nWe thank the reviewer for the professional comment. Regarding mode rotation and coupling effects, we address them by building a multiple-input multiple-output (MIMO) model. The signal of each mode interferes with others through a coupling matrix. The MIMO algorithm, based on an LMS adaptive filter, adjusts the received signals of each mode to progressively eliminate interference between modes. By continuously updating the filter’s weight coefficients, the MIMO algorithm restores the original signals, effectively decoupling the mode coupling.\nRegarding differential mode gain/loss issues, since the length of the few-mode fiber in the four-mode multiplexing system based on coherent detection in this paper is only 20 km, the mode power differences induced during transmission are not significant. Furthermore, before entering the 20 km FMF, we split the signal into four paths using a 1*4 optical coupler and placed a variable optical attenuator (VOA) in each mode channel as shown in Fig. R6, which allows for optimization of the power across all four modes, ensuring consistent power levels. For long-distance few-mode multiplexing optical communication systems, the differential mode gain issue raised by the reviewer is indeed critical. We can further compensate for this by using space-time coding (STC) and interference cancellation (IC) algorithms in the DSP process.\nFig. R6. Schematic of the experimental setup for the communication transmission system in C/L band.\nWe have replaced Fig. 5 with Fig. R6 for a more comprehensive description of the experimental setup and highlighted in our revised manuscript as follows:\nFig. 5 Schematic of the experimental setup for the WDM-MDM system in C/L band.\nAdditionally, we have added the related description and highlighted in section [Methods] (Paragraph 3) in our revised manuscript as follows:\n“Meanwhile, a variable optical attenuator (VOA) was placed after each branch of the 1*4 optical coupler to equalize the optical power of the four modes.”\nc. Is 4X4 or 8X8 MIMO-DSP applied at the receiver?\nWe thank the reviewer for raising this question. In the O+E bands, we selected only one mode from LP11a and LP11b for IMDD-based mode multiplexing communication testing. The isolation between the LP01, LP11, and LP21 modes is sufficiently small to support weakly coupled mode multiplexing transmission, so MIMO-DSP is not necessary in this scenario.\nIn the C+L bands, and as required in reviewer’s question 5d regarding the S band, we conducted coherent detection-based mode multiplexing communication testing with four modes: LP01, LP11a, LP11b, and LP21. In this case, four modes and two polarization states were multiplexed. Since each polarization channel’s I&Q signals at the receiver are combined into complex values for MIMO equalization in the DSP, an 8*8 MIMO-DSP was used.\nIn response to reviewer’s question 5d regarding the U-band, we selected the LP01, LP11, and LP21 modes for MDM transmission testing. Due to the lack of compatible optical amplifiers in the U-band, the received optical power was below -22 dBm. At the receiver, a 3*3 MIMO-DSP was implemented.\nFurthermore, as requested in reviewer’s question 5a, we conducted communication testing with five-mode multiplexing (LP01, LP11a, LP11b, LP21a, and LP21b) in the C+L band. In this scenario, a 10*10 MIMO-DSP was used.\nd. Are the transmission data absent for the S and U bands? If they are, the claim of “first OESCLU MDM transmission” will be inappropriate.\nWe sincerely thank the reviewer for the constructive remarks. We have added transmission experiments in the S and U bands to the revised manuscript.\nFor the broadband MDM transmission experiment in the S-band, we adopted an experimental setup based on coherent detection, similar to that used in the L-band, as shown in Fig. R7(a). At the transmitter, an external cavity laser (ECL) operating in the S-band was used to generate 26 optical carriers at different wavelengths, which were sequentially fed into an IQ modulator to test the transmission performance of the ultra-wideband mode multiplexer in the S-band. The wavelength range of the optical carriers was 1460−1510 nm, with a wavelength spacing of 2 nm. An AWG with a sampling rate of 64 GSa/s was used to drive the IQ modulator for electro-optic modulation. Considering the wavelength response characteristics of the modulator, the signal modulation format was set to DP-QPSK with a symbol rate of 32 GBaud. Subsequently, the modulated signals traversed an analog polarization multiplexer (PM) containing a 1 m length of polarization-maintaining (PM) fiber serving as a delay line for decorrelation between the two polarization signals. They were then amplified in a thulium-doped fiber amplifier (TDFA) and divided into four channels by a 1*4 optical coupler to be incident on the input ports of a mode multiplexer. The four channels of fundamental mode signals were further modulated by the wideband mode multiplexer to generate LP01, LP11a, LP11b, and LP21 four spatial modes of broadband signals, which were transmitted over a 20 km four-mode fiber. Single-mode fiber delay lines, with lengths of 10 m, 6 m, and 3 m, were introduced to three of the input optical signal channels of the mode multiplexer to facilitate decorrelation between signals of different modes. Meanwhile, a variable optical attenuator (VOA) was placed after each branch of the 1*4 optical coupler to equalize the optical power of the four modes. At the receiver, the LP modes signals were split into four channels by the wideband mode demultiplexer. A time-division multiplexing (TDM) receiver was employed, receiving a single channel of four-mode broadband signals output from the mode demultiplexer for each time slot. Finally, the signals were received and demodulated by an integrated coherent receiver (ICR), and the waveform was displayed on an oscilloscope (OSC).\nFig. R7. (a) Schematic of the experimental setup for the MDM system in S band. (b) Measured BER scatters in S band.\nThe experimental results are shown in the Fig. R7(b). It can be observed that within the wavelength range of 1460−1510 nm, the BER values for all measured channels remain below the FEC threshold of 2.4E-2, confirming that the proposed ultra-wideband mode multiplexer effectively supports mode-multiplexed transmission in the S-band. Furthermore, in the communication test conducted in the S-band, the data rate for a single wavelength channel reached 128 Gbps.\nIn the broadband MDM transmission experiment in the U-band, an IMDD-based experimental setup was used, as shown in Fig. R8(a). At the transmitter, the external cavity laser (ECL) was utilized to generate 5 optical carriers in the range of 1630nm−1670nm with a 10 nm wavelength spacing. The OOK signals modulated by the DSP were loaded onto the arbitrary waveform generator (AWG) to generate the electrical waveform, which was amplified by the electrical amplifier (EA) to drive a Mach-Zehnder modulator (MZM) biased at the quadrature point to implement intensity modulation. Then, the signals were divided into three channels by a 1*4 optical coupler to be incident on the input ports of a mode multiplexer. The three channels of fundamental mode signals were further modulated by the wideband mode multiplexer to generate LP01, LP11, and LP21 spatial modes of broadband signals, which were transmitted over a 5 km four-mode fiber. At the receiver, followed by the fiber was a polarization controller (PC) to adjust the coupling of modes in fiber transmission. Then, the LP modes signals were split into three channels by the wideband mode demultiplexer. Photoelectric signal conversion detection was performed by a photodiode (PD) and the waveform was displayed on the mixed signal oscilloscope (MSO).\nFig. R8. (a) Schematic of the experimental setup for the MDM system in U band. (b) Measured BER curves in U band.\nThe data rate of a single-mode signal was 10 Gbps, while the three-mode multiplexed signal achieved a total rate of 30 Gbps. The measured bit error rate (BER) values of all channels in the MDM transmission scenario were depicted in Fig. R8(b). All BER values are below the forward error correction (FEC) threshold of 2.4E-2, indicating successful data signal transmission in the U band.\nFigures R7(a) and R8(a) are added and highlighted in section [Supplementary Figures] (Fig. S9 and Fig. S8) of the revised Supporting Information. We have added and highlighted the related description of transmission experiments for the U and S bands in section [Methods] in our revised manuscript as follows:\nParagraph 2: “In the broadband MDM transmission experiment in the U-band, an IMDD-based experimental setup was used. At the transmitter, the external cavity laser (ECL) was utilized to generate 5 optical carriers in the range of 1630nm−1670nm with a 10 nm wavelength spacing. The OOK signals modulated by the DSP were loaded onto the arbitrary waveform generator (AWG) to generate the electrical waveform, which was amplified by the electrical amplifier (EA) to drive a Mach-Zehnder modulator (MZM) biased at the quadrature point to implement intensity modulation. Then, the signals were divided into three channels by a 1*4 optical coupler to be incident on the input ports of a mode multiplexer. The three channels of fundamental mode signals were further modulated by the wideband mode multiplexer to generate LP01, LP11, and LP21 spatial modes of broadband signals, which were transmitted over a 5 km four-mode fiber. At the receiver, followed by the fiber was a PC to adjust the coupling of modes in fiber transmission. Then, the LP modes signals were split into three channels by the wideband mode demultiplexer. Photoelectric signal conversion detection was performed by a photodiode (PD) and the waveform was displayed on the mixed signal oscilloscope (MSO). (see Supplementary Fig. S8)”\nParagraph 4: “In the implementation of broadband MDM transmission in the S band, ECL generated 26 optical carriers with varying wavelengths intended for transmission into the IQ modulator. These carriers spanned the wavelength range of 1460−1510 nm with a wavelength spacing of 2 nm. Unlike the facilities used in C band transmission, a thulium-doped fiber amplifier (TDFA) is adopted to amplify the signals, which can be better adapted to the S band transmission. The sample rate of arbitrary waveform generator (AWG) is set at 64 GSa/s. The modulation format of the generated optical signals was DP-QPSK with a symbol rate of 32 Gbaud. (see Supplementary Fig. S9).”\nFigures R7(b) and R8(b) are merged and added to Fig. 4d and 4c in our revised manuscript, respectively. We have added and highlighted the main conclusion in section [Results] (Ultra-wideband mode multiplexing communication transmission experiments) as follows:\nParagraph 1: “Experimental validation of mode multiplexing and wavelength-mode hybrid multiplexing communication transmission in the O, E, S, C, L, and U bands was conducted under two typical optical transceiver mechanisms.”\nParagraph 3: “Based on an intensity modulation direct detection (IMDD) system, a three-mode multiplexed transmission of OOK signals with a channel rate of 10 Gbps over a 5 km four-mode fiber was conducted in the U band. As shown in Fig. 4c, the measured BER values in the wavelength range of 1630−1670 nm are below the forward error correction (FEC) threshold of 2.4E-2, indicating successful data signal transmission in the U band.”\nParagraph 4: “...the C band dense WDM (DWDM)-MDM system, S band and L band broadband MDM transmission systems were constructed...” “... 4-mode multiplexing transmission in the S band (1460−1510 nm) and L band (1570−1608 nm).” “...ultra-wideband mode multiplexer effectively supports mode multiplexing transmission in the S+C+L bands.”\nThe changes in section [Conclusion and Discussion] (Paragraph 2) as follows:\n“In our experiments, communication transmissions were carried out in four distinct trials, encompassing the O+E+U and S+C+L bands.”\n“... in the S+C+L bands, leveraging coherent detection. For the U band, due to limitations of the associated components, it is primarily used in direct detection scenarios.”\nThe changes in section [Abstract] as follows:\n“Validations across O, E, S, C, L, and U bands demonstrate...”\nThe Fig. 4d of BER-OSNR curves in the original manuscript is removed, added and highlighted in section [Supplementary Figures] (Fig. S6) of the revised Supporting Information and noted and highlighted in section [Results] (Ultra-wideband mode multiplexing communication transmission experiments) of the revised manuscript as follows:\nParagraph 4: “At the same optical signal-to-noise ratio (OSNR), measured BER values for all mode channels are similar with low wavelength dependence, demonstrating the system's ability to transmit high-quality broadband signals within a given OSNR range (see Supplementary Fig. S6).”\nWe have highlighted Fig. 4 in our revised manuscript as follows:\nFig. 4 The characterization of WDM-MDM and broadband MDM communication transmission. a Schematic of ultra-wideband mode multiplexing communication system. Measured BER curves b in O+E bands and c in U band. d Measured BER scatter in S+C+L bands.\ne. I suggest the authors provide the typical back-to-back mode crosstalk at C-band, O-band, U-band.\nWe sincerely thank the reviewer for the constructive feedback and valuable comments. The mode crosstalk is characterized through the measurement of the output power matrix of a back-to-back system. This system consists of a mode multiplexer (MUX), a 3-meter six-mode fiber, a polarization controller (PC) which is used for adjusting the coupling of modes within the fiber, and a mode demultiplexer (DEMUX). As illustrated in Fig. R9(a), the laser beam is directly incident on the input single-mode fiber of the fiber-microlens array and then passes through the mode multiplexer. After propagating through the 3-meter six-mode fiber, with the coupling of modes adjusted by the polarization controller, the beam passes through the mode demultiplexer and is output from the single-mode fiber array.\nFig. R9. (a) Experimental setup diagram of the back-to-back multiplexing/demultiplexing system. (b) Maximum mode crosstalk for the back-to-back system with three-mode across O+E+S+C+L+U bands.\nFor each input channel of the multiplexer, the power of all output channels of the demultiplexer are detected using an optical power meter (PM). The crosstalk is defined as the power in the desired output channel related to the total power among other output channels. The crosstalk matrix at different wavelengths can be obtained by varying the wavelength of the incident laser beam. Due to the degeneration of the LP11a and LP11b modes, the three-mode (LP01, LP11a, and LP21) are measured across the O+E+S+C+L+U bands. As shown in Fig. R9(b), the maximum mode crosstalk is below -8.87 dB.\nThe Fig. R9 are added and highlighted in section [Supplementary Figures] (Fig. S5) of the revised Supporting Information. We have added and highlighted the main conclusion in section [Results] (Ultra-wideband mode multiplexing communication transmission experiments) in our revised manuscript as follows:\nParagraph 1: “The Measured maximum mode crosstalk of a typical back-to-back system with three modes (LP01, LP11a, and LP21) across the O+E+S+C+L+U bands is below -8.87 dB (see Supplementary Fig. S5).”\n[1]. Y. Zhang et al., \"An Ultra-Broadband Polarization-Insensitive Optical Hybrid Using Multiplane Light Conversion,\" in Journal of Lightwave Technology, vol. 38, no. 22, pp. 6286-6291, 15 Nov.15, 2020.\n[2]. B. J. Puttnam et al., \"402 Tb/s GMI data-rate OESCLU-band Transmission,\" 2024 Optical Fiber Communications Conference and Exhibition (OFC), San Diego, CA, USA, 2024, pp. 1-3.\n[3]. N. K. Fontaine, R. Ryf, H. Chen, D. Neilson and J. Carpenter, \"Design of High Order Mode-Multiplexers using Multiplane Light Conversion,\" 2017 European Conference on Optical Communication (ECOC), Gothenburg, Sweden, 2017, pp. 1-3.\nResponse to reviewer 2:\nReviewer #2: This paper described a design methodology, a prototyped device and its characterizations, and proof-of-concept system demonstration of a ultra-wideband mode multiplexer across 420 nm spectrum range, covering almost all the telecom wavelengths of interest. The motivation of this work is valid and considered very timely, and the characteristics and performance of the device are quite good. The authors also performed careful review on the state-of-the-art technologies that are relevant to this study. The transmission experiments are carefully carried out with stringent configurations to de-correlate all the adjacent channels, which is highly appreciated. The design methodology is also considered valuable takeaway information from this paper. This work is indeed, to my knowledge, the first report on a close-to-feasible miniaturized mode mux for such a wide range of spectrum coverage, which can directly support multi-band transmission system upgrade. Therefore, I consider this work highly novel and quite significant in driving forward mode multiplexing on top of ultra-wide multiband optical communications beyond the C-band.\nBefore recommending accepting the manuscript for publication, I have a list of comments for the authors to address, detailed as follow.\n1. Figure 1a shows the end-to-end inverse design framework. The way showing such a design is intuitive and clear. However, considering each layer of the later fabricated device in practice is reflective rather than transmissive, it's suggested to clarify this early in the text when describing the design concept in order to avoid confusion.\nWe thank the reviewer for the insightful and positive comments on our manuscript. We agree that distinguishing between transmissive and reflective layers is an important detail that should be emphasized early in the paper to prevent any confusion. In response, we have revised the manuscript to explicitly clarify this point. Specifically, in Section [Result] (Ultra-wideband optical diffraction network and chromatic dispersion compensation architecture), we have added the following statement:\nParagraph 2: “It is worth noting that all layers in the described framework are transmissive, although reflective layers can also be employed in practical experiments. The phase profiles of all the layers can be printed on a single reflective phase plate. In this work, the subsequently fabricated device is experimentally realized by utilizing a multi-pass reflective cavity formed by the phase plate and a mirror.”\nWe hope that this additional clarification addresses the concern and provides a clear understanding of the design concept for the readers. Thank you again for pointing this out.\n2. In the Subsection \"Experimental characterization for ultra-wideband mode multiplexing\", the authors described briefly the characterization setup. For a better understanding, please elaborate on the fiber-microlens array configuration. Does each input fiber carry the same ultra-wideband fundamental mode Gaussian light? How was this input light generated? Was it also a WDM muxed signal similar to the transmission experiment, or was it a single ultra-wideband light source? What's the distance between the array and the reflective cavity? Please clarify.\nWe thank the reviewer for their thoughtful questions regarding the experimental setup and fiber-microlens array configuration. We understand that providing additional details on these aspects will contribute to a clearer illustration of the experimental methodology.\nThe fiber-microlens array configuration employed in our experiments is structured as a 2-dimensional array of single-mode fibers that are coupled to a 2-dimensional array of microlenses, with a standard pitch of 250 um. The fiber array is a mechanical transfer (MT) array composed of Corning SMF28E fibers, while the microlens array is fabricated from fused silica circular lenses.\nIn the experimental characterization section, given the absence of a single laser light source possessing a bandwidth wide enough to span the entire spectral range, we use four tunable laser light sources. These lasers cover distinct spectral bands, and measurements are conducted separately within each band to encompass the full range. Each input fiber carries fundamental mode Gaussian light within the identical wavelength band.\nThe distance between the fiber-microlens array and the reflective cavity is approximately 15 mm.\nWe have added and highlighted the main configuration in section [Results] (Experimental characterization for ultra-wideband mode multiplexing) in our revised manuscript as follows:\nParagraph 1: “The experimental setup comprises a 2*2 fiber-microlens array with a standard 250 um pitch.” “The distance between the fiber-microlens array and the reflective cavity was approximately 15 mm.”\nParagraph 2: “To cover the entire bandwidth range, measurements were conducted separately using four tunable laser light sources, each operating in a different spectral band.”\n3. In the same subsection, the authors tried to explain the performance variations in the short wavelength range, which is still not very clear to me. What does it exactly mean by enhanced short-wavelength attenuation due to free-space propagation? Could you please quantify or provide references?\nWe thank the reviewer for their valuable feedback and for highlighting the need for further clarification regarding the performance variations in the short wavelength range. We understand that the term “enhanced short-wavelength attenuation due to free-space propagation” may not have been sufficiently clear, and we appreciate the opportunity to elaborate on this point.\nFrom the perspective of angular spectral propagation theory, the fundamental reason for the short-wavelength attenuation enhanced by free-space propagation is closely associated with the propagation and diffraction characteristics of high-frequency light waves. The angular spectrum theory accomplishes a mathematical description of the propagation of light or electromagnetic waves in free space by depicting the wave field as a superposition of plane waves at diverse angles. The formulation of the angular spectrum is achieved through the use of the Fourier transform, which creates the connection between the spatial domain and the wave number domain.\nAs depicted in Fig. R10, in accordance with the angular spectrum theory, the propagation of the electromagnetic field can be achieved through a two-dimensional Fourier transform. Given the wavefront of the initial optical field u (x, y, 0) at z=0, its propagating wavefield u (x, y, z) at distance z can be expressed by the following Fourier integral:\n+inf +inf\n(u x,y,z)=integral integral U(f , f ,0)*exp[j2*pi*(f *x+f *y)]*exp(j2*pi*f *z)*df *df , (R16)\nx y x y z x y\n-inf -inf\nwhere U (f , f , 0) represents the two-dimensional Fourier transform of u (x, y, 0). f and f are the spatial frequency components. f is the wave number component in the z-propagation direction and satisfies the dispersion relation for wave number plane waves:\nf = k^2 - f^2 - f^2. (R17)\nz x y\nWhere k=2*pi/lambda is the wave number and lambda is wavelength.\nFig. R10 Diagram of angular spectrum propagation.\nThe high-frequency components correspond to short wavelengths, and such components are prone to greater diffraction and diffusion effects during the process of propagation. Specifically, based on the angular spectrum theory, we can summarize the root cause of short-wavelength attenuation as follows:\nPropagation losses of high frequency components: The larger wave numbers of high-frequency components (i.e., shorter wavelengths) imply that their propagation paths are more curved and diffused. During propagation, these high-frequency components undergo stronger diffraction, which causes their energy to disperse across other modes in space, thereby resulting in attenuation.\nThe effect of propagation distance: In the case of free-space propagation, diffraction effects are more conspicuous for components with shorter wavelengths, particularly over longer propagation distances. This leads to the dispersion and diffusion of energy in space, which manifests as intensity decay.\nIn conclusion, the enhanced short-wavelength attenuation is mainly attributable to the stronger diffraction effects that high-frequency components undergo during free-space propagation. As the propagation distance increases, these high-frequency components decay rapidly, whereas low-frequency components decay at a slower pace. Consequently, short-wavelength beams experience more significant attenuation in free space compared to long-wavelength beams, demonstrating stronger attenuation losses.\n4. A follow up question on the previous comment, is it possible to include such performance variation aspect in the analytical expressions Eq.4？This can provide more precise guidelines on selecting the reference wavelength. Beside, could you please explain how exactly did you choose the reference wavelength of 1380 nm? Is it manually optimized?\nThank you for your insightful follow-up questions. Below are our responses to the points raised:\nRegarding the selection of the reference wavelength lambda =1380 nm, this choice is not simply manually optimized. In fact, the reference wavelength was determined through a comprehensive analysis that took into account the system’s design and performance requirements. Initially, the reference wavelength is derived from the reference wavelength equation when free-space propagation is not considered. Subsequently, it is obtained through a balanced manual optimization process based on simulation results. As previously stated, shorter wavelengths are subject to stronger diffraction effects during free-space propagation. Hence, a reference wavelength that can strike a balance among these effects is more desirable. In our specific situation, lambda =1380 nm was chosen as it provides a satisfactory compromise across the working wavelength range.\nRegarding the possibility of incorporating the performance variation in the short wavelength range aspect into the analytical expression in Eq. 4, we indeed endeavored to take into account the influence of propagation effects on the reference wavelength selection. However, given that the propagation distance, diffraction layer, and specific wavelength range are all subject to uncertainties within the training models for diverse scenarios, it proves to be rather arduous to comprehensively incorporate these variations into a single analytical expression. Consequently, it becomes challenging to present an accurate and quantifiable formula for the selection of a reference wavelength when free-space propagation is taken into consideration.\n5. Figure 3c shows the crosstalk results through measurements using a digital holography setup, which is not described in the manuscript. Can the authors provide the descriptions of this experimental configuration or provide a reference?\nWe thank the reviewer for highlighting the absence of a description regarding the digital holography setup utilized for the crosstalk measurements in Figure 3c. We apologize for the oversight and are pleased to supply the details about the experimental configuration as requested.\nThe crosstalk measurements were carried out by employing the digital holography system illustrated in Fig. R11(a). In this setup, the laser beam is divided into two separate beams by a fiber-optic splitter. One of the beams is guided through a mode multiplexer to generate the corresponding mode light. Subsequently, this mode light is projected vertically onto the holographic recording plane as the object light (S) with the help of a free-space beam-splitting prism (BS). The second beam is collimated by a collimator and then reflected by the beam splitter onto the holographic recording plane, serving as the reference light (R). The object and reference light fields interfere on the holographic recording plane, and a CCD camera is used to capture the intensity of the resulting interfering light.\nThe light intensity of the recorded hologram is given by |S+R|^2. As shown in Fig. R11(b), the mode intensity reconstruction process begins with the Fourier transform of the hologram to obtain its spatial spectrum. This spectrum contains a zero-frequency component at the center, and the +1 and -1 order spectra on either side, which correspond to the real and conjugate images of the recorded light field, respectively. A suitable frequency-domain filter is applied to retain either the +1 or -1 order spectra while filtering out the others. The complex conjugate distribution of the object light field in the holographic recording plane is then obtained by performing an inverse Fourier transform of the filtered spectrum. The reconstructed image can be obtained via diffraction calculation.\nFig. R11 (a) Interferogram characterization apparatus of digital holography. (b) Mode field reconstruction process based on the Fourier transform.\nOnce the reconstructed numerical mode field is obtained, the mode crosstalk is calculated by reintegrating the different modes, as described in Eq. R18.\nrho = ... (R18)\nThe digital holography technique utilized here is analogous to the ones expounded upon in [Reference 38] (Fontaine, N. K. et al. Laguerre-Gaussian mode sorter. Nature Communications 10, 1865, 2019), within our revised manuscript.\nWe have cited the relevant reference in section [Results] (Experimental characterization for ultra-wideband mode multiplexing) in our revised manuscript as follows:\nParagraph 2: “…captured by a digital holography experiment38.”\n6. In the same section, the authors stated that \"This characteristic significantly reduces power consumption in the transmission system.\" Please explain with which reference schemes is it compared to draw such a conclusion?\nThank you for your insightful question. We appreciate the opportunity to clarify the comparison regarding power consumption in the transmission system.\nThe statement “This characteristic significantly reduces power consumption in the transmission system” in our manuscript is based on a comparison between our work and researches on other broadband mode multiplexers utilized in broadband transmission systems. To offer greater clarity, we’ve incorporated a comparison with several pivotal studies in Table R1. Researches on MPLC-based multiplexers featuring 6-mode [reference 46], 10-mode [reference 47], and 15-mode [reference 26] configurations operate within bandwidths that do not exceed the C+L bands. Their maximum fiber-to-fiber losses are approximately 5 dB, 5.5 dB, and 12 dB, respectively. Moreover, a 4-mode multiplexer based on a three-dimensional waveguide structure operates within the 1540–1620 nm wavelength range, yet it experiences a notable loss of 9.9 dB at just 1550 nm [reference 48].\nIn contrast, our ultra-wideband mode multiplexer achieves fiber-to-fiber losses of less than 4.9 dB for four-mode fiber) and 4.4 dB for six-mode fiber across the 1260–1675 nm wavelength range. These results demonstrate that our approach outperforms the referenced studies in both bandwidth and loss performance. These low loss values over a broad wavelength range are crucial for efficient signal transmission in various optical communication applications.\nTable R1. Comparison of fiber-to-fiber loss in our study and other researches\nThese relevant references have been added to our revised manuscript as references [26] and [46-48] and highlighted as follows:\n26. Rademacher, G. et al. Peta-bit-per-second optical communications system using a standard cladding diameter 15-mode fiber. Nature Communications 12, 4238, (2021).\n46. Genevaux, P. et al. 6-mode Spatial Multiplexer with Low Loss and High Selectivity for Transmission over Few Mode Fiber. In Optical Fiber Communication Conference. W1A.5. (2015).\n47. Labroille, G. et al. Mode Selective 10-Mode Multiplexer based on Multi-Plane Light Conversion. In Optical Fiber Communication Conference, Th3E.5. (2016).\n48. Wu, Y. F. et al. Ultra-broadband mode multiplexers based on three-dimensional asymmetric waveguide branches. Optics Letters 42, 407-410, (2017).\nWe have cited these references at this statement in our revised manuscript as follows:\n“This characteristic significantly reduces power consumption in the transmission system26,46-48.”\n7. Also the authors stated that the mode dispersion becomes more severe in the short wavelength region. Please show the characterized result to support this statement, or explain qualitatively how you made such an observation.\nWe appreciate the reviewer’s insightful comment regarding the mode dispersion in the short wavelength region. After a thorough review of the relevant literature, we aim to clarify the exact definition of mode dispersion. Mode dispersion is the phenomenon in which, when light waves propagate through an optical fiber, different modes encounter different transmission times because of the fiber’s properties. This disparity in transmission times causes delays in the arrival of light waves, thereby resulting in dispersion during the transmission process.\nWe recognize that the statement “…the mode dispersion of the four-mode fiber becomes more severe in the short wavelength range…” is not entirely precise, and we apologize for any confusion this may have caused. What we actually intended to convey is that, as the transmission distance increases, modes degradation in the four-mode fiber becomes more pronounced in the short wavelength range. This conclusion is based on qualitative observations made during our experiments. Specifically, after the light has been transmitted through 5 km of four-mode fiber, the intensity distributions of the output modes deteriorate significantly at shorter wavelengths.\nWe have corrected “mode dispersion” to “mode degradation” and highlighted in our revised manuscript in section [Results] (Experimental characterization for ultra-wideband mode multiplexing) in our revised manuscript as follows:\nParagraph 2: “Furthermore, it was observed in the experiments that the mode degradation of the four-mode fiber becomes more severe in the short wavelength range as the transmission distance increases.”\n8. The caption of Figure 4d states that BER curves are shown. These are not BER curves. Please rephrase.\nWe are grateful to the reviewer for the meticulous examination of our manuscript. We acknowledge the confusion caused and concur that the term “BER curves” was unsuitable for the data depicted in that particular figure. To rectify this, we have revised the caption to precisely describe the presented data. Given the specific characteristics of the data, we have replaced the term “BER curves” with “BER scatter” in the description. This adjustment ensures that the caption accurately represents the data, enhancing the clarity and accuracy of our manuscript.\n9. In the Subsection \"Ultra-wideband mode multiplexing communication transmission experiments\", I would suggest to include the data rates and modulation formats for both the O-band CWDM and C+L band DWDM transmissions already here in the text, besides the descriptions later in the Methods section. It helps to better understand the transmission results.\nWe thank the reviewer for this helpful suggestion. We agree that providing details about the data rates and modulation formats in the main text, in addition to the descriptions in the Methods section, would improve the clarity and context of the transmission results.\nWe have added and highlighted specific details about the data rates and modulation formats in section [Results] (Ultra-wideband mode multiplexing communication transmission experiments) in our revised manuscript as follows:\nParagraph 2: “Signals modulated with on-off keying (OOK) were transmitted at a data rate of 10 Gbps per channel.”\nParagraph 4: “The S band employed dual-polarization quadrature phase shift keying (DP-QPSK) modulation, whereas DP 16-quadrature amplitude modulation (DP-16QAM) was utilized in the C and L bands, both operating at a symbol rate of 32 Gbaud.”\nResponse to reviewer 3:\nReviewer #3: Dear Authors,\nThank you submitting this very interesting paper. It presents very important research results on enhancing the capacity of fiber-optic telecom networks by means of mode multiplexing and addressing several optical transmission bands. The results are very readable and convincing, which pave the road for further development in the SDM subject area.\nI have nothing to suggest regarding methodology, analysis of the obtained results.\nI recommend this paper to be accepted provided:\n1. It is thoroughly checked in terms of English language use. There are some minor mistakes regarding punctuation or grammar e.g. 5-6 line of the abstract \"... system to enhancing...\";\nWe thank the reviewer for a careful survey of our manuscript. We greatly appreciate the suggestion to improve the manuscript’s language clarity. We have thoroughly revised the manuscript to address the minor punctuation and grammar issues, including the specific example mentioned in the abstract (“... system to enhancing...”). We have corrected this phrase to “... system to enhance...”.\nAdditionally, we have conducted a comprehensive review of the entire manuscript to identify and correct any other minor language issues, such as punctuation, grammar, and sentence structure, ensuring that the manuscript reads smoothly and professionally.\n2. Some graphics need improvement to be better readable. Perhaps, it is enough to take advantage of the complete column width e.g. Fig.1, 2b (indexes are hardly visible).\nWe greatly appreciate the reviewer’s suggestion to improve the readability of the graphics. We agree that some figures, particularly Fig. 1 and Fig. 2b, could benefit from adjustments to enhance their clarity.\nWe have increased the size of Fig. 1 and Fig. 2b to take advantage of the full column width and have also improved the font size for the axis labels, indices, and other text in the figures to ensure better readability, especially for smaller text elements.\nAdditional revisions\nIn addition to addressing the specific comments and suggestions put forward by the reviewers, we have made several other updates to the manuscript. These updates aim to emphasize the additional contributions that emerged from the extra simulations and experiments carried out during the revision process. A summary of these changes is presented below:\n1. Addition of affiliations\nDue to career relocations, Aru Kong and Bo Liu completed the revised manuscript at the Great Bay University & Great Bay Institute for Advanced Study and Tsinghua University respectively. Incorporating these new affiliations in the revised manuscript accurately reflects the current working environments where the revised work was accomplished.\n2. Addition of authors and their affiliations\nWe have added the following co-authors to the manuscript: Rongyi Lin, Chen Wang, Jianyu Long, Fufei Pang, Xueli Chen, Xiaoyu Yan, and Zhifeng Wang. Their contributions are as follows:\nRongyi Lin (Shenzhen University): In the supplementary experiments, Rongyi Lin made substantial contributions to device fabrication. This work was of utmost importance as it was fundamental for the subsequent performance tests. The high-quality devices fabricated under Lin's efforts ensured the reliability and accuracy of the tests that followed.\nChen Wang and Jianyu Long (Fudan University): They provided assistance in partial communication measurements. The contribution of a significant amount of communication data was crucial for the refinement of the study and the revision of the manuscript.\nFufei Pang, Xueli Chen, Xiaoyu Yan, and Zhifeng Wang (Shanghai University): Shanghai University provided the necessary facilities for device processing. These authors actively participated in exploring processing parameters and fabricating devices during the additional experiments. Their collective efforts in optimizing the device-processing procedures led to the production of well-performing experimental devices, thereby strengthening the experimental foundation of our study.\n3. Removing some sources of funding\nWe have removed two funding sources in section [Acknowledgments]: Stable Support Project of Shenzhen (No. 20220810152651001) and Shenzhen University (No. GFPY-YB-2024-01). The reason for this removal is that these funding sources were not directly involved in supporting this research, and their inclusion was an oversight.\nWe have added and highlighted the addition of authors and affiliations in first page in our revised manuscript as follows:\nAru Kong1,2,#, Jianxin Ren3,#, Kaihui Wang4, Yongyi Yu3, Rongyi Lin1, Chen Wang4, Jianyu Long4, Fufei Pang5, Xueli Chen5, Xiaoyu Yan5, Zhifeng Wang5, Bo Liu3,6*, Ting Lei1,*, Xiaocong Yuan1,*.\n1Nanophotonics Research Centre, Institute of Microscale Optoelectronics & State Key Laboratory of Radio Frequency Heterogeneous Integration, Shenzhen University, Shenzhen 518060, China\n2School of Physical Sciences, Great Bay University & Great Bay Institute for Advanced Study, Dongguan, 523000, China\n3Institute of Optoelectronics, Nanjing University of Information Science and Technology, Nanjing 210044, China\n4Key Laboratory for Information Science of Electromagnetic Waves (MoE), School of Information Science and Technology, Fudan University, Shanghai 200433, China\n5Key Laboratory of Specialty Fiber Optics and Optical Access Networks, Joint International Research Laboratory of Specialty Fiber Optics and Advanced Communication, Shanghai University, Shanghai 200444, China\n6Department of Electronic Engineering, Tsinghua University, Beijing 100084, China\nTo ensure a more accurate representation of the corresponding author’s contributions, we have updated and highlighted a clearer and more precise description of their role in section [Author contributions] in our revised manuscript as follows:\n“X.Y. led this work. X.Y., T.L. and B.L. conceived the original idea, initiated the research and coordinated instruments.”\nWe have added and highlighted additional author contributions in section [Author contributions] in our revised manuscript as follows:\n“K.W., Y.Y., C.W. and Y.L. assisted in partial measurements. Y.L., F.P., L.C., Y.Y. and F.W. assisted in device fabrication.”\nResponse to the reviewers’ comments (Nature Communications, Research Article, No. NCOMMS-24-56122A)\nDear Reviewers,\nWe sincerely appreciate the opportunity to revise again our manuscript entitled \"Ultra-wideband optical diffractive network for mode multiplexing across the entire telecommunication range\" by Aru Kong et al. We would like to express our deepest gratitude for your invaluable insights and time invested in evaluating our work. We have carefully incorporated all suggestions and made supplementary revisions to address the reviewers’ concerns. Below, we present a detailed, point-by-point response to each reviewer’s comments, accompanied by specific changes made to the manuscript.\nBlue Italic: Original comments from reviewers.\nBlack: Our response including the action taken. The position of the changes made in the manuscript has been highlighted in this response.\nResponse to reviewer 1:\nReviewer #1: I believe that the revised manuscript has been greatly improved. However, I still have some question as below:\n1. Although the authors have replied most of the comments, I notice that not all the responses have been updated to the manuscript. I suggest that more revision should be made to improve the quality of the manuscript.\nThank you for your constructive feedback. We concur that ensuring more revisions are incorporated into the manuscript is critical. To address this concern, we have thoroughly re-examined all revisions against the manuscript. More revisions from the first round of review have been incorporated, including the following updates:\nReferences [3] and [1] of Comment 1 are added and highlighted in section [References] of our revised manuscript as references [41] and [48] as follows:\n“41. Fontaine, N. K. et al. Design of High Order Mode-Multiplexers using Multiplane Light Conversion. In European Conference on Optical Communication. 1-3 (2017).\n48. Zhang. Y. et al. An Ultra-Broadband Polarization-Insensitive Optical Hybrid Using Multiplane Light Conversion. Journal of Lightwave Technology, 38, 6286-6291, (2020).”\nWe have cited and highlighted the above references in paragraph 4 of the main text of the revised manuscript as follows:\n“... and multi-layer optical networks38-42.”\n“... and ultra-wideband 90-degree optical hybrid48.”\nThe algorithm-related details of Comment 3(d) are added and highlighted in section [Methods] in our revised manuscript as follows:\n“Optical propagation model of the ultra-wideband diffractive network. To model the diffractive network, its diffractive layers are treated as thin planar elements. For the i-th diffractive feature of the k-th layer at the spatial coordinates (x, y, z ), the wavelength-dependent complex transmission coefficient tk (x, y, z ) at the c-th mode channel can be represented as a function of the wide-spectrum modulation phase function Psi(lambda) at the spatial coordinates (x, y, z ). The relationship can be mathematically expressed as:\ntk (x,y,z )=ak(x,y,z )*exp(j*psi_k (x,y,z )), (8)\nlambda_c i i k i i k lambda_c i i k\nhere, a and Psi represent the amplitude and phase coefficients, respectively. The diffractive layers are connected by free-space propagation, which is modeled through the band-limited angular approach. Consequently, the transfer function for free-space propagation over a distance d between two successive layers is given by:\nHk(f , f ,d,lambda)=... (9)\nwhere f and f represent the spatial frequencies along the x and y directions, respectively.\nAssuming uk (x, y, z ) represents the input complex light field of wavelength lambda at the c-th mode channel on the k-th optical layer. The input complex light field of the k-th optical layer is obtained through the convolution operation between the scattered field generated by the (k-1)-th optical layer and the diffraction kernel. For the k-th layer, the forward optical field uk (x, y, z ) can be expressed as:\nlambda_c i i k\nU_lambda_c^k-1(f_x, f_y,d)=... (10)\nuk (x,y,z )=... (11)\nwhere F{·} and F^-1{·} denote the Fourier transform and inverse Fourier transform operations, respectively. lambda_r represents the reference wavelength.\nIt is assumed that gk (x, y, z ) represents the ouput complex light field of a wavelength of lambda at the c-th mode channel on the k-th optical layer. The ouput complex light field of the k-th optical layer is obtained through the convolution between the scattered field generated by the (k+1)-th optical layer and the diffraction kernel. The backward optical field gk (x, y, z ) of the k-th layer can be written as follows:\nG_lambda_c^k+1(f_x, f_y,d)=... (12)\ngk (x,y,z )=... (13)\nThe correlation function is described as follows:\nrho = ... (14)\nThe function is defined as mode purity, which is used to evaluate modulation efficiency. Here, O and T denote the generated and target modes light fields of wavelength lambda at the c-th mode channel on the output plane, respectively.”\nFigure R5 of Comment 5(a) is added and highlighted in section [Supplementary Figures] (Fig. S10) of the revised Supporting Information.\nThe MIMO-DSP dimensionality-related descriptions addressed in Comment 5(c) have been incorporated into section [Results] (Ultra-wideband mode multiplexing communication transmission experiments) of the revised manuscript as follows:\nParagraph 3: “In the absence of optical amplifiers, a 3*3 multiple-input multiple-output (MIMO) architecture with digital signal processing (DSP) was implemented at the receiver. The measured BER values remained below the FEC threshold of 2.4E-2 across the 1630−1670 nm wavelength range, as demonstrated in Fig. 4c.”\nParagraph 4: “All three systems employed 8*8 MIMO-DSP configuration at the receiver.”\n2. I think the authors has not clearly explained the novelty for the design of MPLC device. The author has compared the method in the manuscript to that in the reference, and said “Our research utilizes multi-layer diffraction network technology to advance combined WDM+MDM multiplexing technology, enabling the parallel transmission of multiple wavelengths and spatial modes within a single optical fiber.” However, supporting both WDM and MDM may be a well-known design requirement instead of new concept for MPLC. For example, the function of multi-wavelength multiplexer has been proposed in Ref. A. If it’s a whole new design for WDM+MDM, please state the novelty clearly with detailed processing methods, simulation algorithms, etc.\nRef. A. G. Labroille, N. Barre, P. Jian, and J. Morizur, \"Multi-Wavelength Multiplexer with Independent Mode Control Based on Multi-Plane Light Conversion,\" in ECOC 2016.\nWe sincerely appreciate reviewer’s insightful feedback on the novelty of our WDM+MDM design. To better highlight the innovative aspects of our work, we present a systematic analysis of these issues alongside comprehensive descriptions of our detailed processing methodologies and computational algorithms.\nIt is true that supporting both WDM and MDM has been a well-known design with MPLC since 2016. The key issue inspired us in this scenario is to find out a new design with a capability of covering the entire working wavelength for optical communication.\nTo overcome the chromatic dispersion bottleneck in ultra-wideband MDM, we propose a wide-spectrum dispersion compensation mechanism by introducing a spectral modulation function and constructing the multi-physics-coupled, multi-objective cooperative optimization model. By developing a wavelength-mode joint end-to-end phase-matching design framework, we achieve synchronous extraction of global multi-parameter errors and coordinated correction of local features, thereby enabling low insertion loss, high mode purity, and low crosstalk performance across the entire working bandwidth.\nWide-spectrum dispersion compensation mechanism\nIn the proposed ultra-wideband multi-layer diffractive network, we establish a multi-parameter space incorporating wavelength, mode, spatial coordinates, free-space transfer functions, and the number of diffractive layers. For a given set of working wavelengths lambda in {lambda_1, lambda_2, ..., lambda_N}, the modulation efficiency function is designed as follows:\neta{purity(lambda),crosstalk(lambda),loss(lambda)}, (r1)\nwhere eta integrates key performance metrics, including wavelength-dependent mode purity, mode crosstalk and insertion loss.\nTo achieve high-efficiency chromatic dispersion compensation, we introduce a wide-spectrum modulation phase function Psi(lambda), establishing a nonlinear mapping between the multi-parameter optimization space and the modulation efficiency function:\nPsi(lambda) = ... (r2)\nHere, normalization factor (lambda/lambda_r) maps phase compensation across wavelengths to a reference wavelength (lambda_r), eliminating absolute phase discrepancies. Training bandwidth (W) and reference wavelength (lambda_r) are optimized to ensure balanced dispersion compensation over the target working bandwidth (B). Coefficient K related to the number of diffractive layers, governs overall efficiency.\nWavelength-mode joint end-to-end phase matching\nTo model the diffractive network, its diffractive layers are treated as thin planar elements. For the i-th diffractive feature of the k-th layer at the spatial coordinates (x, y, z ), the wavelength-dependent complex transmission coefficient tk (x, y, z ) at the c-th mode channel can be represented as a function of the wide-spectrum modulation phase function Psi_k (lambda) at the spatial coordinates (x, y, z ). The relationship can be mathematically expressed as:\ntk (x,y,z )=ak(x,y,z )*exp(j*psi_k (x,y,z )), (r3)\nlambda_c i i k i i k lambda_c i i k\nhere, a and Psi represent the amplitude and phase coefficients, respectively. The diffractive layers are connected by free-space propagation, which is modeled through the band-limited angular approach. Consequently, the transfer function for free-space propagation over a distance d between two successive layers is given by:\nHk (f , f ,d,lambda)=... (r4)\nlambda_c x y\nwhere f and f represent the spatial frequencies along the x and y directions, respectively.\nAssuming uk (x, y, z ) represents the input complex light field of wavelength lambda at the c-th mode channel on the k-th optical layer. The input complex light field of the k-th optical layer is obtained through the convolution operation between the scattered field generated by the (k-1)-th optical layer and the diffraction kernel. For the k-th layer, the forward optical field uk (x, y, z ) can be expressed as:\nlambda_c i i k\nU_lambda_c^k-1(f_x, f_y,d)=... (r5)\nuk (x,y,z )=... (r6)\nwhere F{·} and F^-1{·} denote the Fourier transform and inverse Fourier transform operations, respectively. lambda_r represents the reference wavelength.\nIt is assumed that gk (x, y, z ) represents the ouput complex light field of a wavelength of lambda at the c-th mode channel on the k-th optical layer. The ouput complex light field of the k-th optical layer is obtained through the convolution between the scattered field generated by the (k+1)-th optical layer and the diffraction kernel. The backward optical field gk (x, y, z ) of the k-th layer can be written as follows:\nG_lambda_c^k+1(f_x, f_y,d)=... (r7)\ngk (x,y,z )=... (r8)\nBy comparing the forward-propagated optical field uk (x, y) and the backward-propagated optical field gk (x, y), the local error field at each spatial coordinate (x, y) is computed as:\nError_k (x,y) = ... (r9)\nthe wavelength-mode-dependent global phase error at each diffractive layer is quantified by the sum of local phase errors across all spatial coordinates as follows:\ndelta_k(lambda,c) = ... (r10)\nlocal phase correction and update of each diffractive layer is iteratively refined through the following global phase errors gradient-based optimization:\nL_k(x,y,z) = ... (r11)\nBuilding upon these theoretical foundations, simulation validation through parametric optimization in Figure R1 further elucidates the interplay between design parameters and ultra-wideband performance.\nAs shown in Figure R1(a), within the fixed bandwidth of 1260–1680 nm, training bandwidth W changes from 100 nm to 1000 nm. Simulation results reveal that while the peak mode purity decreases with increasing W, high mode purity is maintained across the ultra-wideband range, and wavelength dependence is reduced. Specifically, when W is set within the range of 600 nm to 800 nm, the mode purity remains consistently above 90%, with a standard deviation of approximately 0.01.\nAs demonstrated in Figure R1(b), under a 700 nm training bandwidth configuration, adjusting the reference wavelength lambda_r from the 1440 nm to shorter wavelengths significantly improves the performance balance across the entire working bandwidth. Specifically, when the reference wavelength is set to 1380 nm, the device exhibits enhanced tolerance to short-wavelength attenuation.\nFigure R1(c) indicates that increasing the number of diffraction layers substantially enhances the device’s overall performance across the entire working bandwidth.\nThe proposed ultra-wideband mode (de)multiplexing scheme covering the full O+E+C+L+U (1260–1675 nm) communication bands fundamentally in design objectives differs from the existing MPLC-based functional devices mentioned by the reviewers (e.g., The ultra-broadband optical hybrid in reference [1] of Comment 1 from the first round of review or the dual-wavelength design at 1550 nm + 980 nm in Ref. A). The study in reference [1] of Comment 1 employed MPLC primarily for phase-error compensation in ultra-broadband optical hybrids, without addressing mode-dimensional control. The work in Ref.A focused on specific wavelength combinations (e.g., 1550 nm for communication + 980 nm for pumping), concentrating on single-wavelength multi-mode multiplexing. In this approach, a single wavelength was fixedly input into the same incident channel, without exploring multi-wavelength transmission within the same mode channel.\nFigure R1. The curves of mode purity within the working bandwidth ranges of 1260 nm—1680 nm with different (a) training spectral width W, (b) reference wavelengths lambda_r and (c) number of diffractive layers.\nBroadband MDM requires simultaneous mode-dimensional control and multi-wavelength coordination within a single mode channel. Prior MPLC-based mode multiplexing research has been confined to the C/L bands, limited by wavelength-dependent response models or chromatic dispersion, with reported bandwidths below 100 nm [references 26, 27, 38, 40]. Due to structural dispersion in diffractive elements, phase errors accumulate when the operating wavelength deviates from the design value. These results in significant degradation of mode purity, crosstalk and insertion loss as wavelength shifts, making bandwidth expansion challenging. In our work, with the core objective of achieving seamless full-band coverage from the O to U bands (1260–1675 nm), active compensation for chromatic dispersion is implemented to reduce the dependency of mode channels on specific wavelengths. Mode-manipulation capability is extended to the entire communication spectrum, breaking the traditional C/L-band constraints. The proposed device synergizes with WDM technology to support hybrid WDM+MDM communication systems, facilitating coordinated multi-band (O/E/C/L/U) transmission.\nFigure R2. (a) The curves and (b) average values and standard deviation of within the working bandwidth ranges of 1260 nm—1680 nm with different training wavelengths. Case1:1260 nm, Case 2:1675 nm, Case 3:1260 nm and 1675 nm with reference wavelength 1260 nm, Case 4: 1260 nm and 1675 nm with reference wavelength 1380 nm, Case 5: 1260 nm and 1675 nm with reference wavelength 1675 nm, Case 6: this work.\nAs shown in Figure R2(a), the fixed single-wavelength designs (Case 1 and Case 2) exhibit approximately linear degradation of mode purity characteristics across the entire wavelength range. Comparative analysis of Case 3, Case 4 and Case 5 demonstrates that the ultra-wideband dual-wavelength design, when integrated with an appropriate reference wavelength that our strategically emphasized, can achieve enhanced uniformity in performance optimization across the entire wavelength spectrum. The proposed wide-spectrum dispersion compensation scheme (Case 6) maintains mode purity exceeding 90% even under operating wavelength deviations from the reference wavelength to 1260 nm or 1680 nm. Figure R2(b) demonstrates that our design simultaneously achieves performance balancing (standard deviation approx 1.37%) and comprehensive performance enhancement (average mode purity >92%) across the entire telecommunication range.\nThese relevant references [references 26, 27, 38, 40] are from the following original references in our revised manuscript:\n“26. Rademacher, G. et al. Peta-bit-per-second optical communications system using a standard cladding diameter 15-mode fiber. Nature Communications 12, 4238, (2021).\n27. Rademacher, G. et al. 1.53 Peta-bit/s C-Band Transmission in a 55-Mode Fiber. In European Conference on Optical Communication. 1-4 (2022).\n38. Fontaine, N. K. et al. Laguerre-Gaussian mode sorter. Nature Communications 10, 1865, (2019).\n40. Thapa, S. R., Smith-Dryden, S., Zhu, Z., Pang, S. S. & Li, G. Experimental Demonstration and Characterization of a Non-Mode Selective (De)Multiplexer Using Multi-Plane Light Converter (MPLC). IEEE Photonics Journal 16, 1-6, (2024).”\nRef. A is added and highlighted in section [References] of our revised manuscript as reference [42] as follows:\n“42. Labroille. G. et al. Multi-wavelength multiplexer with independent mode control based on multi-plane light conversion. In 42nd European Conference on Optical Communication. 1-3 (2016).”\nWe have cited and highlighted the above reference in paragraph 4 of the main text of the revised manuscript as follows:\n“... and multi-layer optical networks38-42.”\nWe have revised and highlighted the above main discussion in section [Results] (Ultra-wideband optical diffraction network and chromatic dispersion compensation architecture) of the revised manuscript in context as follows:\nParagraph 3: “To achieve high-efficiency chromatic dispersion compensation, a mapping between the multi-parameter space and modulation efficiency function eta is established based on the modulation phase function Psi(lambda). Here, normalization factor (lambda/lambda_r) maps phase compensation across wavelengths to a reference wavelength (lambda_r), eliminating absolute phase discrepancies. Training bandwidth (W) and reference wavelength (lambda_r) are optimized to ensure balanced dispersion compensation over the target working bandwidth (B). Coefficient K related to the number of diffractive layers, governs overall efficiency.”\nThe algorithm-related details and above main conclusions are added and highlighted in section [Results] (Free-space architecture design of cascaded optical layers constructed by the multi-focal optical system) of the revised manuscript in context as follows:\nParagraph 2: “These diffraction units in each layer exhibit varying phase responses Psi_k (x, y, z) for different wavelength channels lambda and mode channels c of light. To execute both forward and backward design simultaneously, the outputs of each layer are remapped as inputs to the subsequent layer, except for the final one. Each layer is constructed through wavelength-mode joint end-to-end phase matching to achieve synchronous extraction of global multi-parameter errors and coordinated correction of local features, thereby enabling high mode purity, low mode crosstalk and low insertion loss performance across the entire working bandwidth. The local error field at each spatial coordinate (x, y) is computed by comparing the forward optical field uk (x, y, z ) and the backward optical field gk (x, y, z ):\nError_k (x,y) = ... (5)\nthe wavelength-mode-dependent global phase error at each layer is then quantified by summing these local phase errors across all spatial coordinates as follows:\ndelta_k(lambda,c) = ... (6)\nLocal correction phase of each layer is iteratively refined through the following gradient-based optimization of global phase errors across all wavelength channels and mode channels:\nL_k(x,y,z) = ... (7)\n”\nParagraph 3: “This design simultaneously achieves performance balancing and comprehensive performance enhancement across the entire telecommunication range, advancing MDM technology from fixed-wavelength/band or narrowband dedicated devices toward an ultra-wideband universal platform (see Supplementary Fig. S5).”\nFigures R1(b) and R2 are added and highlighted in section [Supplementary Figures] (Fig. S4 and Fig. S5) of the revised Supporting Information.\n3. The authors have not fully replied my previous question about degenerate mode reception in Comment 5(b). For example, the authors may have excited the LP21a mode, and only received LP21a or LP21b mode at the receiver for coherent detection. However, the rotation and coupling between degenerate modes is inevitable during FMF transmission, and joint MIMO-DSP for both should be carried out to combat the frequency-selective fading, otherwise the performance will be greatly degraded. If the joint reception is not performed limited by available experimental equipment, the authors should describe it as it is.\nWe sincerely thank the reviewer for raising this important point again and for your patience regarding our previous response, and we provide further clarification here. In the specific experiment described in the main text, we only excited the LP21a mode (together with LP01, LP11a, and LP11b) and performed coherent detection solely on these four modes. As such, the DSP algorithm did not include any compensation for degenerate mode rotation, coupling, or differential modal gain/loss between LP21a and LP21b. This is an important point, and we thank the reviewer for highlighting it.\nWe fully agree with the reviewer that joint detection of degenerate modes would provide better system robustness and improve performance, especially in the presence of frequency-selective fading caused by mode coupling. However, in this particular setup, the same device used for multiplexing was applied in reverse as the demultiplexer. Since the device was designed to excite only the LP21a mode, it inherently implemented selective demultiplexing of LP21a at the receiver, making joint reception of LP21a and LP21b infeasible in this configuration.\nDuring transmission, unintentional coupling between LP21a and LP21b may occur due to fiber imperfections and perturbations, potentially causing partial energy leakage. To mitigate this, we inserted a polarization controller before the demultiplexer to suppress coupling and retain more energy in the LP21a component.\nIn contrast, when the multiplexer is specifically designed to excite both LP21a and LP21b modes simultaneously, the system enables full joint detection at the receiver. As shown in the supplementary experiment added in response to Comment 5(a), we constructed 4-mode and 5-mode devices that excited both degenerate LP21 modes, allowing coherent joint reception without additional polarization controller. These configurations more fully align with the reviewer’s recommended approach and demonstrate its performance benefits.\nWe have added and highlighted the related description in section [Results] (Ultra-wideband mode multiplexing communication transmission experiments) in our revised manuscript as follows:\nParagraph 4: “Since the device was designed to excite only the LP21a mode, it inherently implemented selective demultiplexing of LP21a at the receiver for coherent detection. When the multiplexer is specifically designed to excite both LP21a and LP21b modes simultaneously, the system enables joint detection of degenerate LP21 modes at the receiver without additional polarization controller for suppressing coupling (see Supplementary Fig. S10).”\n4. “The isolation between the LP01, LP11, and LP21 modes is sufficiently small” should be “sufficiently large”.\nWe sincerely appreciate your meticulous scrutiny for our response. It was indeed an error on our part and we deeply apologize for any confusion caused.\n5. The authors have provided more information for MIMO-DSP. Please explain your considerations for the different dimensions of MIMO-DSP such as 8*8, 10*10, 3*3.\nWe sincerely thank the reviewer for your insightful question regarding the considerations for the different dimensions of MIMO-DSP in our study. Below, we provide a detailed explanation in our considerations:\nIn our experiments, different MIMO-DSP dimensions were adopted according to the specific number of spatial channels involved in each transmission scenario, which include spatial modes and polarizations. Specifically, 8*8 MIMO-DSP was applied in our main transmission experiments involving four spatial modes: LP01, LP11a, LP11b, and LP21a, each with two orthogonal polarizations (4 modes * 2 polarizations = 8 spatial channels). This configuration reflects the actual multiplexed channels in the system and requires full 8*8 equalization for proper mode demultiplexing and polarization tracking.\nThe 10*10 MIMO-DSP was used in extended experiments where we included both degenerate LP21 modes (LP21a, LP21b), resulting in 5 modes * 2 polarizations = 10 spatial channels. This setup allowed us to evaluate the impact of including LP21b in the joint processing and confirmed its benefit in mitigating inter-modal interference.\nThe 3*3 MIMO-DSP was used in the U-band few-mode multiplexed transmission experiment based on IMDD. In this setup, three spatial modes (LP01, LP11a, and LP21a) were multiplexed and transmitted. Since only a single polarization state was used for each spatial mode in this IMDD-based experiment, the total number of spatial channels was three, leading to the use of a 3*3 MIMO equalization structure.\nWe have added and highlighted the above main explanation in section [Methods] (Experimental setup for the communication transmission demonstration) in our revised manuscript as follows:\nParagraph 2: “Given the use of a single polarization state per spatial mode and a total of three spatial channels, a 3*3 MIMO equalization structure was implemented.”\nParagraph 3: “Given the total 8 spatial channels of 4 modes and 2 polarization states, an 8*8 MIMO-DSP configuration was employed.”\n6. I notice that only 20-km transmission reach could be achieved (above the FEC limit) in the experiment using 8*8 MIMO-DSP. It’s much less than a typical strongly-coupled MDM transmission reach. please analyze it. Dos the limit come from the MPLC device?\nWe sincerely appreciate the reviewer’s valuable question regarding the transmission distance in our experiment. We acknowledge that the 20-km reach achieved in our 8*8 MIMO-DSP experiment (while remaining above the FEC threshold) is indeed shorter than typical transmission distances reported for strongly-coupled MDM systems.\nThe limitation in transmission distance does not stem from the device itself. Our device exhibits low insertion loss, excellent mode selectivity and stable performance throughout our experiments. The current restriction arises primarily from algorithmic trade-offs in our DSP implementation. Since this work focuses on validating the multiplexer’s feasibility, we prioritized functional demonstration over long-haul optimization. Thus, parameters such as filter tap count and convergence behavior were optimized for practicality rather than maximal reach.\nIn our experimental configuration, the 8*8 MIMO-DSP system with 90 taps was employed for channel equalization. To evaluate the relationship between tap count and bit error rate (BER), we quantified BER performance across all four mode channels (at 1550 nm) under varying tap configurations. As illustrated in Figure R3, a consistent inverse correlation was observed between tap count and BER, wherein increasing the number of taps progressively reduced BER values across all mode channels. Notably, at 240 taps, all four mode channels achieved BER values below the HD-FEC threshold (BER < 3.8*10^-3), with further asymptotic improvements beyond this point.\nFigure R3. The BER values of four mode channels at a wavelength of 1550 nm with different number of taps.\nFor long-haul transmission, signal degradation results from the combined effects of: (1) accumulated inter-modal crosstalk over longer fiber spans, (2) OSNR degradation due to multi-stage optical amplification and (3) nonlinear impairments induced by fiber nonlinearity. With deeper MIMO equalizer optimization (e.g., increased taps and advanced nonlinear compensation), our system could support extended distances while retaining the current multiplexer.\nWe have added and highlighted the above main discussion in section [Results] (Ultra-wideband mode multiplexing communication transmission experiments) in our revised manuscript as follows:\nParagraph 4: “Deeper MIMO equalizer enhancements (e.g., increased taps or advanced nonlinear compensation) demonstrate effective BER reduction (see Supplementary Fig. S9), enabling our system to achieve longer-distance transmission potential while using the current multiplexer.”\nFigures R3 is added and highlighted in section [Supplementary Figures] (Fig. S9) of the revised Supporting Information.\n7. In the response to Comment 5(e), the authors claim that the b2b modal crosstalk for a pair of mode MUX/DEMUX is -8.87 dB. However, I think there is contradiction between it and the claim “achieves over 90% mode purity and mode crosstalk less than -17 dB” in the Abstract. Please clarify it.\n8. and mode crosstalk less than -17 dB” parameter may be\nWe sincerely thank the reviewer for your meticulous attention to detail. We appreciate the opportunity to clarify the distinction between the values mentioned in the response to Comment 5(e) and the Abstract. The modal crosstalk of less than -8.87 dB mentioned in the response to Comment 5(e) refers to the back-to-back (b2b) testing results of the mode multiplexer-demultiplexer system. This value represents the combined performance of the MUX and DEMUX under specific few-mode fiber connection experimental conditions. In contrast, the claim of “mode crosstalk less than -17 dB” in the Abstract refers to the performance of a single mode multiplexer (MUX) in experimental characterization. This value highlights the performance achieved by the individual component. Furthermore, we notice that Question 8 is incomplete but appears to be related to Question 7. We hope that above clarification addresses both concerns.\nTo prevent confusion, we have revised and highlighted the description in section [Results] (Ultra-wideband mode multiplexing communication transmission experiments) in our revised manuscript as follows:\nParagraph 1: “…a typical back-to-back mode multiplexer-demultiplexer system…”\nResponse to reviewer 2:\nReviewer #2: The authors have properly addressed my previous comments with detailed responses and revisions. It is my pleasure to recommend accepting the paper in its current form.\nWe are delighted to hear that the revisions and detailed responses meet your expectations, and we appreciate your recommendation to accept the paper in its current form.\nResponse to reviewer 3:\nReviewer #3: The paper has been substantially improved. I suggest acceptance of the paper.\nWe are grateful for your recommendation to accept the paper and appreciate your valuable contributions to enhancing the quality of our work.\nResponse to the reviewers’ comments (Nature Communications, Research Article, No. NCOMMS-24-56122C)\nDear Reviewers,\nWe would like to express our sincere gratitude for your valuable time and consideration of our manuscript. Your efforts are essential in upholding the journal's high standards.\nThe author proposes a wide-spectrum, dispersioncompensated, multi-layer diffractive network design method to create ultra-wideband optical components, based on which an ultra-wideband, 4-LP-mode MPLC mode multiplexer is designed and fabricated for proof-of-concept demonstration. The MPLC component can achieve high efficiency over a very broad bandwidth ranging from 1260 nm to 1680 nm. However, I do not recommend this paper to be published by Nature Communications for the following reasons:\nultra-wideband MPLC-based devices and transmission experiments have been experimentally demonstrated in previous works. Its novelty, progress, and practical significance must be clarified and highlighted before it be considered for publication. Here are my comments:\n1. The authors claim a wide-spectrum device design method for multi-layer diffraction network by compensating the spatial dispersion. Please clarify the innovations of your approach by comparing it with previous work in reference [1], in which an ultra-wideband MPLC with a bandwidth over 900 nm is achieved by optimizing phase planes at various wavelengths. Mathematical and simulation descriptions must be added to highlight the practical significance of this work.\n2. The ultra-wideband transmission covering OESCLU bands has been demonstrated in reference [2]. Please clarify the novelty of your experimental transmission work.\n3. The descriptions of this algorithm should be clarified for it’s a main innovation of this work, such as:\na. Which phase plane is the spatial dispersion compensation algorithm applied to?\nb. How does the algorithm be applied? Is it just in the judgment step after calculating the correlation?\nc. The significance of this algorithm in reducing the number of phase planes and decreasing computational complexity?\nd. I suggest that the detailed mathematical algorithm should be described like that in the reference [2].\n4. With the same number of phase planes, the optimization of bandwidth will cause a deterioration in assembly tolerance such as the error planes or incident angle. Please add mathematical and simulation descriptions for them.\n5. Please provide some details for the transmission experiment:\na. Are both LP21a and LP21b modes demultiplexed at the receiver?\nb. How to deal with the rotation, coupling, differential mode gain/loss in the degenerated LP21 mode in the DSP algorithm?\nc. Is 4X4 or 8X8 MIMO-DSP applied at the receiver?\nd. Are the transmission data absent for the S and U bands? If they are, the claim of “first OESCLU MDM transmission” will be inappropriate.\ne. I suggest the authors provide the typical back-to-back mode crosstalk at C-band, O-band, U-band.\n[1]. Y. Zhang et al., \"An Ultra-Broadband Polarization-Insensitive Optical Hybrid Using Multiplane Light Conversion,\" in Journal of Lightwave Technology, vol. 38, no. 22, pp. 6286-6291, 15 Nov.15, 2020.\n[2]. B. J. Puttnam et al., \"402 Tb/s GMI data-rate OESCLU-band Transmission,\" 2024 Optical Fiber Communications Conference and Exhibition (OFC), San Diego, CA, USA, 2024, pp. 1-3.\n[3]. N. K. Fontaine, R. Ryf, H. Chen, D. Neilson and J. Carpenter, \"Design of High Order Mode-Multiplexers using Multiplane Light Conversion,\" 2017 European Conference on Optical Communication (ECOC), Gothenburg, Sweden, 2017, pp. 1-3.\nI believe that the revised manuscript has been greatly improved. However, I still have some question as below:\n1. Although the authors have replied most of the comments, I notice that not all the responses have been updated to the manuscript. I suggest that more revision should be made to improve the quality of the manuscript.\n2. I think the authors has not clearly explained the novelty for the design of MPLC device. The author has compared the method in the manuscript to that in the reference, and said “Our research utilizes multi-layer diffraction network technology to advance combined WDM+MDM multiplexing technology, enabling the parallel transmission of multiple wavelengths and spatial modes within a single optical fiber.” However, supporting both WDM and MDM may be a well-known design requirement instead of new concept for MPLC. For example, the function of multi-wavelength multiplexer has been proposed in Ref. A. If it’s a whole new design for WDM+MDM, please state the novelty clearly with detailed processing methods, simulation algorithms, etc.\nRef. A. G. Labroille, N. Barre, P. Jian, and J. Morizur, \"Multi-Wavelength Multiplexer with Independent Mode Control Based on Multi-Plane Light Conversion,\" in ECOC 2016.\n3. The authors have not fully replied my previous question about degenerate mode reception in Comment 5(b). For example, the authors may have excited the LP21a mode, and only received LP21a or LP21b mode at the receiver for coherent detection. However, the rotation and coupling between degenerate modes is inevitable during FMF transmission, and joint MIMO-DSP for both should be carried out to combat the frequency-selective fading, otherwise the performance will be greatly degraded. If the joint reception is not performed limited by available experimental equipment, the authors should describe it as it is.\n4. “The isolation between the LP01, LP11, and LP21 modes is sufficiently small” should be “sufficiently large”.\n5. The authors have provided more information for MIMO-DSP. Please explain your considerations for the different dimensions of MIMO-DSP such as 8x8, 10x10, 3x3.\n6. I notice that only 20-km transmission reach could be achieved (above the FEC limit) in the experiment using 8x8 MIMO-DSP. It’s much less than a typical strongly-coupled MDM transmission reach. please analyze it. Dos the limit come from the MPLC device?\n7. In the response to Comment 5(e), the authors claim that the b2b modal crosstalk for a pair of mode MUX/DEMUX is -8.87 dB. However, I think there is contradiction between it and the claim “achieves over 90% mode purity and mode crosstalk less than -17 dB” in the Abstract. Please clarify it.\n8. and mode crosstalk less than -17 dB” parameter may be\nThe authors have replied all the comments. But I do not think that all the modifications are satisfying.\n1. The authors have explained some details of the “wide-spectrum dispersion compensation mechanism”, in which the key technique point is to “establish a multi-parameter space incorporating wavelength, mode, spatial coordinates, free-space transfer functions, and the number of diffractive layers”. However, all these factors are widely considered in previous MPLC designs. So, it confirms my concerns for the innovations of the MPLC design.\n2. The authors state that the main difference between the manuscript and the Ref. A (another MPLC design covering beyond-400 nm spectrum) is that they have different design objectives, specifically, two fixed wavelengths V.S. multiple wavelengths. But I do not think that this kind of optimization could be a significant innovation.\n3. The authors have explained the reception method of degenerate modes. It’s said that “During transmission, unintentional coupling between LP21a and LP21b may occur due to fiber imperfections and perturbations, potentially causing partial energy leakage. To mitigate this, we inserted a polarization controller before the demultiplexer to suppress coupling and retain more energy in the LP21a component.”\nHowever, I disagree this experimental method of applying a PC before the receiver, only detecting on of a pair of degenerate modes, and choosing the favorable results.\n4. The authors have explained my questions about both the different dimensions of MIMO-DSP and the b2b modal crosstalk level. Based on the responses, I believe that the statement in the Abstract of “A 4-linearly polarized mode multiplexer ,……, achieves over 90% mode purity and mode crosstalk less than -17 dB” is misleading. There are both strongly-coupled and weakly-coupled mode-division-multiplexing schemes for FMF transmission systems. The authors’ claim of high-purity MPLC device (only the multiplexer instead of both multiplexer/demultiplexer) may mislead the readers that it achieves so excellent performance that weakly-coupled systems could be supported by the MPLC devices, which is much more difficult than those for typical strongly-coupled systems. However, the b2b modal crosstalk is high and inter-mode MIMO-DSP for strongly-coupled transmission has to be adopted. So, I suggest modifications to this statement.\nBased on the above concerns, I’m sorry to say that I still think the two innovations of the manuscript including the MPLC design and the transmission experiment are not significant enough for the publication of NC."}, {"round": 2, "reviewer_comments": {"reviewer_2": "The authors have properly addressed my previous comments with detailed responses and revisions. It is my pleasure to recommend accepting the paper in its current form.", "reviewer_3": "The paper has been substantially improved. I suggest acceptance of the paper."}, "author_response": "2 June, 2025\nResponse to the reviewers’ comments (Nature Communications, Research Article, No. NCOMMS-24-56122A)\nDear Reviewers,\nWe sincerely appreciate the opportunity to revise again our manuscript entitled \"Ultra-wideband optical diffractive network for mode multiplexing across the entire telecommunication range\" by Aru Kong et al. We would like to express our deepest gratitude for your invaluable insights and time invested in evaluating our work. We have carefully incorporated all suggestions and made supplementary revisions to address the reviewers’ concerns. Below, we present a detailed, point-by-point response to each reviewer’s comments, accompanied by specific changes made to the manuscript.\nBlue Italic: Original comments from reviewers.\nBlack: Our response including the action taken. The position of the changes made in the manuscript has been highlighted in this response.\nResponse to reviewer 1:\nReviewer #1: I believe that the revised manuscript has been greatly improved. However, I still have some question as below:\n1. Although the authors have replied most of the comments, I notice that not all the responses have been updated to the manuscript. I suggest that more revision should be made to improve the quality of the manuscript.\nThank you for your constructive feedback. We concur that ensuring more revisions are incorporated into the manuscript is critical. To address this concern, we have thoroughly re-examined all revisions against the manuscript. More revisions from the first round of review have been incorporated, including the following updates:\nReferences [3] and [1] of Comment 1 are added and highlighted in section [References] of our revised manuscript as references [41] and [48] as follows:\n“41. Fontaine, N. K. et al. Design of High Order Mode-Multiplexers using Multiplane Light Conversion. In European Conference on Optical Communication. 1-3 (2017).\n48. Zhang. Y. et al. An Ultra-Broadband Polarization-Insensitive Optical Hybrid Using Multiplane Light Conversion. Journal of Lightwave Technology, 38, 6286-6291, (2020).”\nWe have cited and highlighted the above references in paragraph 4 of the main text of the revised manuscript as follows:\n“... and multi-layer optical networks38-42.”\n“... and ultra-wideband 90-degree optical hybrid48.”\nThe algorithm-related details of Comment 3(d) are added and highlighted in section [Methods] in our revised manuscript as follows:\n“Optical propagation model of the ultra-wideband diffractive network. To model the diffractive network, its diffractive layers are treated as thin planar elements. For the i-th diffractive feature of the k-th layer at the spatial coordinates (x, y, z ), the wavelength-dependent complex transmission coefficient tk (x, y, z ) at the c-th mode channel can be represented as a function of the wide-spectrum modulation phase function Psi(lambda) at the spatial coordinates (x, y, z ). The relationship can be mathematically expressed as:\ntk (x,y,z )=ak(x,y,z )*exp(j*psi_k (x,y,z )), (8)\nlambda_c i i k i i k lambda_c i i k\nhere, a and Psi represent the amplitude and phase coefficients, respectively. The diffractive layers are connected by free-space propagation, which is modeled through the band-limited angular approach. Consequently, the transfer function for free-space propagation over a distance d between two successive layers is given by:\nHk(f , f ,d,lambda)=... (9)\nwhere f and f represent the spatial frequencies along the x and y directions, respectively.\nAssuming uk (x, y, z ) represents the input complex light field of wavelength lambda at the c-th mode channel on the k-th optical layer. The input complex light field of the k-th optical layer is obtained through the convolution operation between the scattered field generated by the (k-1)-th optical layer and the diffraction kernel. For the k-th layer, the forward optical field uk (x, y, z ) can be expressed as:\nlambda_c i i k\nU_lambda_c^k-1(f_x, f_y,d)=... (10)\nuk (x,y,z )=... (11)\nwhere F{·} and F^-1{·} denote the Fourier transform and inverse Fourier transform operations, respectively. lambda_r represents the reference wavelength.\nIt is assumed that gk (x, y, z ) represents the ouput complex light field of a wavelength of lambda at the c-th mode channel on the k-th optical layer. The ouput complex light field of the k-th optical layer is obtained through the convolution between the scattered field generated by the (k+1)-th optical layer and the diffraction kernel. The backward optical field gk (x, y, z ) of the k-th layer can be written as follows:\nG_lambda_c^k+1(f_x, f_y,d)=... (12)\ngk (x,y,z )=... (13)\nThe correlation function is described as follows:\nrho = ... (14)\nThe function is defined as mode purity, which is used to evaluate modulation efficiency. Here, O and T denote the generated and target modes light fields of wavelength lambda at the c-th mode channel on the output plane, respectively.”\nFigure R5 of Comment 5(a) is added and highlighted in section [Supplementary Figures] (Fig. S10) of the revised Supporting Information.\nThe MIMO-DSP dimensionality-related descriptions addressed in Comment 5(c) have been incorporated into section [Results] (Ultra-wideband mode multiplexing communication transmission experiments) of the revised manuscript as follows:\nParagraph 3: “In the absence of optical amplifiers, a 3*3 multiple-input multiple-output (MIMO) architecture with digital signal processing (DSP) was implemented at the receiver. The measured BER values remained below the FEC threshold of 2.4E-2 across the 1630−1670 nm wavelength range, as demonstrated in Fig. 4c.”\nParagraph 4: “All three systems employed 8*8 MIMO-DSP configuration at the receiver.”\n2. I think the authors has not clearly explained the novelty for the design of MPLC device. The author has compared the method in the manuscript to that in the reference, and said “Our research utilizes multi-layer diffraction network technology to advance combined WDM+MDM multiplexing technology, enabling the parallel transmission of multiple wavelengths and spatial modes within a single optical fiber.” However, supporting both WDM and MDM may be a well-known design requirement instead of new concept for MPLC. For example, the function of multi-wavelength multiplexer has been proposed in Ref. A. If it’s a whole new design for WDM+MDM, please state the novelty clearly with detailed processing methods, simulation algorithms, etc.\nRef. A. G. Labroille, N. Barre, P. Jian, and J. Morizur, \"Multi-Wavelength Multiplexer with Independent Mode Control Based on Multi-Plane Light Conversion,\" in ECOC 2016.\nWe sincerely appreciate reviewer’s insightful feedback on the novelty of our WDM+MDM design. To better highlight the innovative aspects of our work, we present a systematic analysis of these issues alongside comprehensive descriptions of our detailed processing methodologies and computational algorithms.\nIt is true that supporting both WDM and MDM has been a well-known design with MPLC since 2016. The key issue inspired us in this scenario is to find out a new design with a capability of covering the entire working wavelength for optical communication.\nTo overcome the chromatic dispersion bottleneck in ultra-wideband MDM, we propose a wide-spectrum dispersion compensation mechanism by introducing a spectral modulation function and constructing the multi-physics-coupled, multi-objective cooperative optimization model. By developing a wavelength-mode joint end-to-end phase-matching design framework, we achieve synchronous extraction of global multi-parameter errors and coordinated correction of local features, thereby enabling low insertion loss, high mode purity, and low crosstalk performance across the entire working bandwidth.\nWide-spectrum dispersion compensation mechanism\nIn the proposed ultra-wideband multi-layer diffractive network, we establish a multi-parameter space incorporating wavelength, mode, spatial coordinates, free-space transfer functions, and the number of diffractive layers. For a given set of working wavelengths lambda in {lambda_1, lambda_2, ..., lambda_N}, the modulation efficiency function is designed as follows:\neta{purity(lambda),crosstalk(lambda),loss(lambda)}, (r1)\nwhere eta integrates key performance metrics, including wavelength-dependent mode purity, mode crosstalk and insertion loss.\nTo achieve high-efficiency chromatic dispersion compensation, we introduce a wide-spectrum modulation phase function Psi(lambda), establishing a nonlinear mapping between the multi-parameter optimization space and the modulation efficiency function:\nPsi(lambda) = ... (r2)\nHere, normalization factor (lambda/lambda_r) maps phase compensation across wavelengths to a reference wavelength (lambda_r), eliminating absolute phase discrepancies. Training bandwidth (W) and reference wavelength (lambda_r) are optimized to ensure balanced dispersion compensation over the target working bandwidth (B). Coefficient K related to the number of diffractive layers, governs overall efficiency.\nWavelength-mode joint end-to-end phase matching\nTo model the diffractive network, its diffractive layers are treated as thin planar elements. For the i-th diffractive feature of the k-th layer at the spatial coordinates (x, y, z ), the wavelength-dependent complex transmission coefficient tk (x, y, z ) at the c-th mode channel can be represented as a function of the wide-spectrum modulation phase function Psi_k (lambda) at the spatial coordinates (x, y, z ). The relationship can be mathematically expressed as:\ntk (x,y,z )=ak(x,y,z )*exp(j*psi_k (x,y,z )), (r3)\nlambda_c i i k i i k lambda_c i i k\nhere, a and Psi represent the amplitude and phase coefficients, respectively. The diffractive layers are connected by free-space propagation, which is modeled through the band-limited angular approach. Consequently, the transfer function for free-space propagation over a distance d between two successive layers is given by:\nHk (f , f ,d,lambda)=... (r4)\nlambda_c x y\nwhere f and f represent the spatial frequencies along the x and y directions, respectively.\nAssuming uk (x, y, z ) represents the input complex light field of wavelength lambda at the c-th mode channel on the k-th optical layer. The input complex light field of the k-th optical layer is obtained through the convolution operation between the scattered field generated by the (k-1)-th optical layer and the diffraction kernel. For the k-th layer, the forward optical field uk (x, y, z ) can be expressed as:\nlambda_c i i k\nU_lambda_c^k-1(f_x, f_y,d)=... (r5)\nuk (x,y,z )=... (r6)\nwhere F{·} and F^-1{·} denote the Fourier transform and inverse Fourier transform operations, respectively. lambda_r represents the reference wavelength.\nIt is assumed that gk (x, y, z ) represents the ouput complex light field of a wavelength of lambda at the c-th mode channel on the k-th optical layer. The ouput complex light field of the k-th optical layer is obtained through the convolution between the scattered field generated by the (k+1)-th optical layer and the diffraction kernel. The backward optical field gk (x, y, z ) of the k-th layer can be written as follows:\nG_lambda_c^k+1(f_x, f_y,d)=... (r7)\ngk (x,y,z )=... (r8)\nBy comparing the forward-propagated optical field uk (x, y) and the backward-propagated optical field gk (x, y), the local error field at each spatial coordinate (x, y) is computed as:\nError_k (x,y) = ... (r9)\nthe wavelength-mode-dependent global phase error at each diffractive layer is quantified by the sum of local phase errors across all spatial coordinates as follows:\ndelta_k(lambda,c) = ... (r10)\nlocal phase correction and update of each diffractive layer is iteratively refined through the following global phase errors gradient-based optimization:\nL_k(x,y,z) = ... (r11)\nBuilding upon these theoretical foundations, simulation validation through parametric optimization in Figure R1 further elucidates the interplay between design parameters and ultra-wideband performance.\nAs shown in Figure R1(a), within the fixed bandwidth of 1260–1680 nm, training bandwidth W changes from 100 nm to 1000 nm. Simulation results reveal that while the peak mode purity decreases with increasing W, high mode purity is maintained across the ultra-wideband range, and wavelength dependence is reduced. Specifically, when W is set within the range of 600 nm to 800 nm, the mode purity remains consistently above 90%, with a standard deviation of approximately 0.01.\nAs demonstrated in Figure R1(b), under a 700 nm training bandwidth configuration, adjusting the reference wavelength lambda_r from the 1440 nm to shorter wavelengths significantly improves the performance balance across the entire working bandwidth. Specifically, when the reference wavelength is set to 1380 nm, the device exhibits enhanced tolerance to short-wavelength attenuation.\nFigure R1(c) indicates that increasing the number of diffraction layers substantially enhances the device’s overall performance across the entire working bandwidth.\nThe proposed ultra-wideband mode (de)multiplexing scheme covering the full O+E+C+L+U (1260–1675 nm) communication bands fundamentally in design objectives differs from the existing MPLC-based functional devices mentioned by the reviewers (e.g., The ultra-broadband optical hybrid in reference [1] of Comment 1 from the first round of review or the dual-wavelength design at 1550 nm + 980 nm in Ref. A). The study in reference [1] of Comment 1 employed MPLC primarily for phase-error compensation in ultra-broadband optical hybrids, without addressing mode-dimensional control. The work in Ref.A focused on specific wavelength combinations (e.g., 1550 nm for communication + 980 nm for pumping), concentrating on single-wavelength multi-mode multiplexing. In this approach, a single wavelength was fixedly input into the same incident channel, without exploring multi-wavelength transmission within the same mode channel.\nFigure R1. The curves of mode purity within the working bandwidth ranges of 1260 nm—1680 nm with different (a) training spectral width W, (b) reference wavelengths lambda_r and (c) number of diffractive layers.\nBroadband MDM requires simultaneous mode-dimensional control and multi-wavelength coordination within a single mode channel. Prior MPLC-based mode multiplexing research has been confined to the C/L bands, limited by wavelength-dependent response models or chromatic dispersion, with reported bandwidths below 100 nm [references 26, 27, 38, 40]. Due to structural dispersion in diffractive elements, phase errors accumulate when the operating wavelength deviates from the design value. These results in significant degradation of mode purity, crosstalk and insertion loss as wavelength shifts, making bandwidth expansion challenging. In our work, with the core objective of achieving seamless full-band coverage from the O to U bands (1260–1675 nm), active compensation for chromatic dispersion is implemented to reduce the dependency of mode channels on specific wavelengths. Mode-manipulation capability is extended to the entire communication spectrum, breaking the traditional C/L-band constraints. The proposed device synergizes with WDM technology to support hybrid WDM+MDM communication systems, facilitating coordinated multi-band (O/E/C/L/U) transmission.\nFigure R2. (a) The curves and (b) average values and standard deviation of within the working bandwidth ranges of 1260 nm—1680 nm with different training wavelengths. Case1:1260 nm, Case 2:1675 nm, Case 3:1260 nm and 1675 nm with reference wavelength 1260 nm, Case 4: 1260 nm and 1675 nm with reference wavelength 1380 nm, Case 5: 1260 nm and 1675 nm with reference wavelength 1675 nm, Case 6: this work.\nAs shown in Figure R2(a), the fixed single-wavelength designs (Case 1 and Case 2) exhibit approximately linear degradation of mode purity characteristics across the entire wavelength range. Comparative analysis of Case 3, Case 4 and Case 5 demonstrates that the ultra-wideband dual-wavelength design, when integrated with an appropriate reference wavelength that our strategically emphasized, can achieve enhanced uniformity in performance optimization across the entire wavelength spectrum. The proposed wide-spectrum dispersion compensation scheme (Case 6) maintains mode purity exceeding 90% even under operating wavelength deviations from the reference wavelength to 1260 nm or 1680 nm. Figure R2(b) demonstrates that our design simultaneously achieves performance balancing (standard deviation approx 1.37%) and comprehensive performance enhancement (average mode purity >92%) across the entire telecommunication range.\nThese relevant references [references 26, 27, 38, 40] are from the following original references in our revised manuscript:\n“26. Rademacher, G. et al. Peta-bit-per-second optical communications system using a standard cladding diameter 15-mode fiber. Nature Communications 12, 4238, (2021).\n27. Rademacher, G. et al. 1.53 Peta-bit/s C-Band Transmission in a 55-Mode Fiber. In European Conference on Optical Communication. 1-4 (2022).\n38. Fontaine, N. K. et al. Laguerre-Gaussian mode sorter. Nature Communications 10, 1865, (2019).\n40. Thapa, S. R., Smith-Dryden, S., Zhu, Z., Pang, S. S. & Li, G. Experimental Demonstration and Characterization of a Non-Mode Selective (De)Multiplexer Using Multi-Plane Light Converter (MPLC). IEEE Photonics Journal 16, 1-6, (2024).”\nRef. A is added and highlighted in section [References] of our revised manuscript as reference [42] as follows:\n“42. Labroille. G. et al. Multi-wavelength multiplexer with independent mode control based on multi-plane light conversion. In 42nd European Conference on Optical Communication. 1-3 (2016).”\nWe have cited and highlighted the above reference in paragraph 4 of the main text of the revised manuscript as follows:\n“... and multi-layer optical networks38-42.”\nWe have revised and highlighted the above main discussion in section [Results] (Ultra-wideband optical diffraction network and chromatic dispersion compensation architecture) of the revised manuscript in context as follows:\nParagraph 3: “To achieve high-efficiency chromatic dispersion compensation, a mapping between the multi-parameter space and modulation efficiency function eta is established based on the modulation phase function Psi(lambda). Here, normalization factor (lambda/lambda_r) maps phase compensation across wavelengths to a reference wavelength (lambda_r), eliminating absolute phase discrepancies. Training bandwidth (W) and reference wavelength (lambda_r) are optimized to ensure balanced dispersion compensation over the target working bandwidth (B). Coefficient K related to the number of diffractive layers, governs overall efficiency.”\nThe algorithm-related details and above main conclusions are added and highlighted in section [Results] (Free-space architecture design of cascaded optical layers constructed by the multi-focal optical system) of the revised manuscript in context as follows:\nParagraph 2: “These diffraction units in each layer exhibit varying phase responses Psi_k (x, y, z) for different wavelength channels lambda and mode channels c of light. To execute both forward and backward design simultaneously, the outputs of each layer are remapped as inputs to the subsequent layer, except for the final one. Each layer is constructed through wavelength-mode joint end-to-end phase matching to achieve synchronous extraction of global multi-parameter errors and coordinated correction of local features, thereby enabling high mode purity, low mode crosstalk and low insertion loss performance across the entire working bandwidth. The local error field at each spatial coordinate (x, y) is computed by comparing the forward optical field uk (x, y, z ) and the backward optical field gk (x, y, z ):\nError_k (x,y) = ... (5)\nthe wavelength-mode-dependent global phase error at each layer is then quantified by summing these local phase errors across all spatial coordinates as follows:\ndelta_k(lambda,c) = ... (6)\nLocal correction phase of each layer is iteratively refined through the following gradient-based optimization of global phase errors across all wavelength channels and mode channels:\nL_k(x,y,z) = ... (7)\n”\nParagraph 3: “This design simultaneously achieves performance balancing and comprehensive performance enhancement across the entire telecommunication range, advancing MDM technology from fixed-wavelength/band or narrowband dedicated devices toward an ultra-wideband universal platform (see Supplementary Fig. S5).”\nFigures R1(b) and R2 are added and highlighted in section [Supplementary Figures] (Fig. S4 and Fig. S5) of the revised Supporting Information.\n3. The authors have not fully replied my previous question about degenerate mode reception in Comment 5(b). For example, the authors may have excited the LP21a mode, and only received LP21a or LP21b mode at the receiver for coherent detection. However, the rotation and coupling between degenerate modes is inevitable during FMF transmission, and joint MIMO-DSP for both should be carried out to combat the frequency-selective fading, otherwise the performance will be greatly degraded. If the joint reception is not performed limited by available experimental equipment, the authors should describe it as it is.\nWe sincerely thank the reviewer for raising this important point again and for your patience regarding our previous response, and we provide further clarification here. In the specific experiment described in the main text, we only excited the LP21a mode (together with LP01, LP11a, and LP11b) and performed coherent detection solely on these four modes. As such, the DSP algorithm did not include any compensation for degenerate mode rotation, coupling, or differential modal gain/loss between LP21a and LP21b. This is an important point, and we thank the reviewer for highlighting it.\nWe fully agree with the reviewer that joint detection of degenerate modes would provide better system robustness and improve performance, especially in the presence of frequency-selective fading caused by mode coupling. However, in this particular setup, the same device used for multiplexing was applied in reverse as the demultiplexer. Since the device was designed to excite only the LP21a mode, it inherently implemented selective demultiplexing of LP21a at the receiver, making joint reception of LP21a and LP21b infeasible in this configuration.\nDuring transmission, unintentional coupling between LP21a and LP21b may occur due to fiber imperfections and perturbations, potentially causing partial energy leakage. To mitigate this, we inserted a polarization controller before the demultiplexer to suppress coupling and retain more energy in the LP21a component.\nIn contrast, when the multiplexer is specifically designed to excite both LP21a and LP21b modes simultaneously, the system enables full joint detection at the receiver. As shown in the supplementary experiment added in response to Comment 5(a), we constructed 4-mode and 5-mode devices that excited both degenerate LP21 modes, allowing coherent joint reception without additional polarization controller. These configurations more fully align with the reviewer’s recommended approach and demonstrate its performance benefits.\nWe have added and highlighted the related description in section [Results] (Ultra-wideband mode multiplexing communication transmission experiments) in our revised manuscript as follows:\nParagraph 4: “Since the device was designed to excite only the LP21a mode, it inherently implemented selective demultiplexing of LP21a at the receiver for coherent detection. When the multiplexer is specifically designed to excite both LP21a and LP21b modes simultaneously, the system enables joint detection of degenerate LP21 modes at the receiver without additional polarization controller for suppressing coupling (see Supplementary Fig. S10).”\n4. “The isolation between the LP01, LP11, and LP21 modes is sufficiently small” should be “sufficiently large”.\nWe sincerely appreciate your meticulous scrutiny for our response. It was indeed an error on our part and we deeply apologize for any confusion caused.\n5. The authors have provided more information for MIMO-DSP. Please explain your considerations for the different dimensions of MIMO-DSP such as 8*8, 10*10, 3*3.\nWe sincerely thank the reviewer for your insightful question regarding the considerations for the different dimensions of MIMO-DSP in our study. Below, we provide a detailed explanation in our considerations:\nIn our experiments, different MIMO-DSP dimensions were adopted according to the specific number of spatial channels involved in each transmission scenario, which include spatial modes and polarizations. Specifically, 8*8 MIMO-DSP was applied in our main transmission experiments involving four spatial modes: LP01, LP11a, LP11b, and LP21a, each with two orthogonal polarizations (4 modes * 2 polarizations = 8 spatial channels). This configuration reflects the actual multiplexed channels in the system and requires full 8*8 equalization for proper mode demultiplexing and polarization tracking.\nThe 10*10 MIMO-DSP was used in extended experiments where we included both degenerate LP21 modes (LP21a, LP21b), resulting in 5 modes * 2 polarizations = 10 spatial channels. This setup allowed us to evaluate the impact of including LP21b in the joint processing and confirmed its benefit in mitigating inter-modal interference.\nThe 3*3 MIMO-DSP was used in the U-band few-mode multiplexed transmission experiment based on IMDD. In this setup, three spatial modes (LP01, LP11a, and LP21a) were multiplexed and transmitted. Since only a single polarization state was used for each spatial mode in this IMDD-based experiment, the total number of spatial channels was three, leading to the use of a 3*3 MIMO equalization structure.\nWe have added and highlighted the above main explanation in section [Methods] (Experimental setup for the communication transmission demonstration) in our revised manuscript as follows:\nParagraph 2: “Given the use of a single polarization state per spatial mode and a total of three spatial channels, a 3*3 MIMO equalization structure was implemented.”\nParagraph 3: “Given the total 8 spatial channels of 4 modes and 2 polarization states, an 8*8 MIMO-DSP configuration was employed.”\n6. I notice that only 20-km transmission reach could be achieved (above the FEC limit) in the experiment using 8*8 MIMO-DSP. It’s much less than a typical strongly-coupled MDM transmission reach. please analyze it. Dos the limit come from the MPLC device?\nWe sincerely appreciate the reviewer’s valuable question regarding the transmission distance in our experiment. We acknowledge that the 20-km reach achieved in our 8*8 MIMO-DSP experiment (while remaining above the FEC threshold) is indeed shorter than typical transmission distances reported for strongly-coupled MDM systems.\nThe limitation in transmission distance does not stem from the device itself. Our device exhibits low insertion loss, excellent mode selectivity and stable performance throughout our experiments. The current restriction arises primarily from algorithmic trade-offs in our DSP implementation. Since this work focuses on validating the multiplexer’s feasibility, we prioritized functional demonstration over long-haul optimization. Thus, parameters such as filter tap count and convergence behavior were optimized for practicality rather than maximal reach.\nIn our experimental configuration, the 8*8 MIMO-DSP system with 90 taps was employed for channel equalization. To evaluate the relationship between tap count and bit error rate (BER), we quantified BER performance across all four mode channels (at 1550 nm) under varying tap configurations. As illustrated in Figure R3, a consistent inverse correlation was observed between tap count and BER, wherein increasing the number of taps progressively reduced BER values across all mode channels. Notably, at 240 taps, all four mode channels achieved BER values below the HD-FEC threshold (BER < 3.8*10^-3), with further asymptotic improvements beyond this point.\nFigure R3. The BER values of four mode channels at a wavelength of 1550 nm with different number of taps.\nFor long-haul transmission, signal degradation results from the combined effects of: (1) accumulated inter-modal crosstalk over longer fiber spans, (2) OSNR degradation due to multi-stage optical amplification and (3) nonlinear impairments induced by fiber nonlinearity. With deeper MIMO equalizer optimization (e.g., increased taps and advanced nonlinear compensation), our system could support extended distances while retaining the current multiplexer.\nWe have added and highlighted the above main discussion in section [Results] (Ultra-wideband mode multiplexing communication transmission experiments) in our revised manuscript as follows:\nParagraph 4: “Deeper MIMO equalizer enhancements (e.g., increased taps or advanced nonlinear compensation) demonstrate effective BER reduction (see Supplementary Fig. S9), enabling our system to achieve longer-distance transmission potential while using the current multiplexer.”\nFigures R3 is added and highlighted in section [Supplementary Figures] (Fig. S9) of the revised Supporting Information.\n7. In the response to Comment 5(e), the authors claim that the b2b modal crosstalk for a pair of mode MUX/DEMUX is -8.87 dB. However, I think there is contradiction between it and the claim “achieves over 90% mode purity and mode crosstalk less than -17 dB” in the Abstract. Please clarify it.\n8. and mode crosstalk less than -17 dB” parameter may be\nWe sincerely thank the reviewer for your meticulous attention to detail. We appreciate the opportunity to clarify the distinction between the values mentioned in the response to Comment 5(e) and the Abstract. The modal crosstalk of less than -8.87 dB mentioned in the response to Comment 5(e) refers to the back-to-back (b2b) testing results of the mode multiplexer-demultiplexer system. This value represents the combined performance of the MUX and DEMUX under specific few-mode fiber connection experimental conditions. In contrast, the claim of “mode crosstalk less than -17 dB” in the Abstract refers to the performance of a single mode multiplexer (MUX) in experimental characterization. This value highlights the performance achieved by the individual component. Furthermore, we notice that Question 8 is incomplete but appears to be related to Question 7. We hope that above clarification addresses both concerns.\nTo prevent confusion, we have revised and highlighted the description in section [Results] (Ultra-wideband mode multiplexing communication transmission experiments) in our revised manuscript as follows:\nParagraph 1: “…a typical back-to-back mode multiplexer-demultiplexer system…”\nResponse to reviewer 2:\nReviewer #2: The authors have properly addressed my previous comments with detailed responses and revisions. It is my pleasure to recommend accepting the paper in its current form.\nWe are delighted to hear that the revisions and detailed responses meet your expectations, and we appreciate your recommendation to accept the paper in its current form.\nResponse to reviewer 3:\nReviewer #3: The paper has been substantially improved. I suggest acceptance of the paper.\nWe are grateful for your recommendation to accept the paper and appreciate your valuable contributions to enhancing the quality of our work.\nResponse to the reviewers’ comments (Nature Communications, Research Article, No. NCOMMS-24-56122C)\nDear Reviewers,\nWe would like to express our sincere gratitude for your valuable time and consideration of our manuscript. Your efforts are essential in upholding the journal's high standards.\nThe author proposes a wide-spectrum, dispersioncompensated, multi-layer diffractive network design method to create ultra-wideband optical components, based on which an ultra-wideband, 4-LP-mode MPLC mode multiplexer is designed and fabricated for proof-of-concept demonstration. The MPLC component can achieve high efficiency over a very broad bandwidth ranging from 1260 nm to 1680 nm. However, I do not recommend this paper to be published by Nature Communications for the following reasons:\nultra-wideband MPLC-based devices and transmission experiments have been experimentally demonstrated in previous works. Its novelty, progress, and practical significance must be clarified and highlighted before it be considered for publication. Here are my comments:\n1. The authors claim a wide-spectrum device design method for multi-layer diffraction network by compensating the spatial dispersion. Please clarify the innovations of your approach by comparing it with previous work in reference [1], in which an ultra-wideband MPLC with a bandwidth over 900 nm is achieved by optimizing phase planes at various wavelengths. Mathematical and simulation descriptions must be added to highlight the practical significance of this work.\n2. The ultra-wideband transmission covering OESCLU bands has been demonstrated in reference [2]. Please clarify the novelty of your experimental transmission work.\n3. The descriptions of this algorithm should be clarified for it’s a main innovation of this work, such as:\na. Which phase plane is the spatial dispersion compensation algorithm applied to?\nb. How does the algorithm be applied? Is it just in the judgment step after calculating the correlation?\nc. The significance of this algorithm in reducing the number of phase planes and decreasing computational complexity?\nd. I suggest that the detailed mathematical algorithm should be described like that in the reference [2].\n4. With the same number of phase planes, the optimization of bandwidth will cause a deterioration in assembly tolerance such as the error planes or incident angle. Please add mathematical and simulation descriptions for them.\n5. Please provide some details for the transmission experiment:\na. Are both LP21a and LP21b modes demultiplexed at the receiver?\nb. How to deal with the rotation, coupling, differential mode gain/loss in the degenerated LP21 mode in the DSP algorithm?\nc. Is 4X4 or 8X8 MIMO-DSP applied at the receiver?\nd. Are the transmission data absent for the S and U bands? If they are, the claim of “first OESCLU MDM transmission” will be inappropriate.\ne. I suggest the authors provide the typical back-to-back mode crosstalk at C-band, O-band, U-band.\n[1]. Y. Zhang et al., \"An Ultra-Broadband Polarization-Insensitive Optical Hybrid Using Multiplane Light Conversion,\" in Journal of Lightwave Technology, vol. 38, no. 22, pp. 6286-6291, 15 Nov.15, 2020.\n[2]. B. J. Puttnam et al., \"402 Tb/s GMI data-rate OESCLU-band Transmission,\" 2024 Optical Fiber Communications Conference and Exhibition (OFC), San Diego, CA, USA, 2024, pp. 1-3.\n[3]. N. K. Fontaine, R. Ryf, H. Chen, D. Neilson and J. Carpenter, \"Design of High Order Mode-Multiplexers using Multiplane Light Conversion,\" 2017 European Conference on Optical Communication (ECOC), Gothenburg, Sweden, 2017, pp. 1-3.\nI believe that the revised manuscript has been greatly improved. However, I still have some question as below:\n1. Although the authors have replied most of the comments, I notice that not all the responses have been updated to the manuscript. I suggest that more revision should be made to improve the quality of the manuscript.\n2. I think the authors has not clearly explained the novelty for the design of MPLC device. The author has compared the method in the manuscript to that in the reference, and said “Our research utilizes multi-layer diffraction network technology to advance combined WDM+MDM multiplexing technology, enabling the parallel transmission of multiple wavelengths and spatial modes within a single optical fiber.” However, supporting both WDM and MDM may be a well-known design requirement instead of new concept for MPLC. For example, the function of multi-wavelength multiplexer has been proposed in Ref. A. If it’s a whole new design for WDM+MDM, please state the novelty clearly with detailed processing methods, simulation algorithms, etc.\nRef. A. G. Labroille, N. Barre, P. Jian, and J. Morizur, \"Multi-Wavelength Multiplexer with Independent Mode Control Based on Multi-Plane Light Conversion,\" in ECOC 2016.\n3. The authors have not fully replied my previous question about degenerate mode reception in Comment 5(b). For example, the authors may have excited the LP21a mode, and only received LP21a or LP21b mode at the receiver for coherent detection. However, the rotation and coupling between degenerate modes is inevitable during FMF transmission, and joint MIMO-DSP for both should be carried out to combat the frequency-selective fading, otherwise the performance will be greatly degraded. If the joint reception is not performed limited by available experimental equipment, the authors should describe it as it is.\n4. “The isolation between the LP01, LP11, and LP21 modes is sufficiently small” should be “sufficiently large”.\n5. The authors have provided more information for MIMO-DSP. Please explain your considerations for the different dimensions of MIMO-DSP such as 8x8, 10x10, 3x3.\n6. I notice that only 20-km transmission reach could be achieved (above the FEC limit) in the experiment using 8x8 MIMO-DSP. It’s much less than a typical strongly-coupled MDM transmission reach. please analyze it. Dos the limit come from the MPLC device?\n7. In the response to Comment 5(e), the authors claim that the b2b modal crosstalk for a pair of mode MUX/DEMUX is -8.87 dB. However, I think there is contradiction between it and the claim “achieves over 90% mode purity and mode crosstalk less than -17 dB” in the Abstract. Please clarify it.\n8. and mode crosstalk less than -17 dB” parameter may be\nThe authors have replied all the comments. But I do not think that all the modifications are satisfying.\n1. The authors have explained some details of the “wide-spectrum dispersion compensation mechanism”, in which the key technique point is to “establish a multi-parameter space incorporating wavelength, mode, spatial coordinates, free-space transfer functions, and the number of diffractive layers”. However, all these factors are widely considered in previous MPLC designs. So, it confirms my concerns for the innovations of the MPLC design.\n2. The authors state that the main difference between the manuscript and the Ref. A (another MPLC design covering beyond-400 nm spectrum) is that they have different design objectives, specifically, two fixed wavelengths V.S. multiple wavelengths. But I do not think that this kind of optimization could be a significant innovation.\n3. The authors have explained the reception method of degenerate modes. It’s said that “During transmission, unintentional coupling between LP21a and LP21b may occur due to fiber imperfections and perturbations, potentially causing partial energy leakage. To mitigate this, we inserted a polarization controller before the demultiplexer to suppress coupling and retain more energy in the LP21a component.”\nHowever, I disagree this experimental method of applying a PC before the receiver, only detecting on of a pair of degenerate modes, and choosing the favorable results.\n4. The authors have explained my questions about both the different dimensions of MIMO-DSP and the b2b modal crosstalk level. Based on the responses, I believe that the statement in the Abstract of “A 4-linearly polarized mode multiplexer ,……, achieves over 90% mode purity and mode crosstalk less than -17 dB” is misleading. There are both strongly-coupled and weakly-coupled mode-division-multiplexing schemes for FMF transmission systems. The authors’ claim of high-purity MPLC device (only the multiplexer instead of both multiplexer/demultiplexer) may mislead the readers that it achieves so excellent performance that weakly-coupled systems could be supported by the MPLC devices, which is much more difficult than those for typical strongly-coupled systems. However, the b2b modal crosstalk is high and inter-mode MIMO-DSP for strongly-coupled transmission has to be adopted. So, I suggest modifications to this statement.\nBased on the above concerns, I’m sorry to say that I still think the two innovations of the manuscript including the MPLC design and the transmission experiment are not significant enough for the publication of NC."}, {"round": 3, "reviewer_comments": {"reviewer_1": "The authors have replied all the comments. But I do not think that all the modifications are satisfying.\n1. The authors have explained some details of the “wide-spectrum dispersion compensation mechanism”, in which the key technique point is to “establish a multi-parameter space incorporating wavelength, mode, spatial coordinates, free-space transfer functions, and the number of diffractive layers”. However, all these factors are widely considered in previous MPLC designs. So, it confirms my concerns for the innovations of the MPLC design.\n2. The authors state that the main difference between the manuscript and the Ref. A (another MPLC design covering beyond-400 nm spectrum) is that they have different design objectives, specifically, two fixed wavelengths V.S. multiple wavelengths. But I do not think that this kind of optimization could be a significant innovation.\n3. The authors have explained the reception method of degenerate modes. It’s said that “During transmission, unintentional coupling between LP21a and LP21b may occur due to fiber imperfections and perturbations, potentially causing partial energy leakage. To mitigate this, we inserted a polarization controller before the demultiplexer to suppress coupling and retain more energy in the LP21a component.”\nHowever, I disagree this experimental method of applying a PC before the receiver, only detecting on of a pair of degenerate modes, and choosing the favorable results.\n4. The authors have explained my questions about both the different dimensions of MIMO-DSP and the b2b modal crosstalk level. Based on the responses, I believe that the statement in the Abstract of “A 4-linearly polarized mode multiplexer ,……, achieves over 90% mode purity and mode crosstalk less than -17 dB” is misleading. There are both strongly-coupled and weakly-coupled mode-division-multiplexing schemes for FMF transmission systems. The authors’ claim of high-purity MPLC device (only the multiplexer instead of both multiplexer/demultiplexer) may mislead the readers that it achieves so excellent performance that weakly-coupled systems could be supported by the MPLC devices, which is much more difficult than those for typical strongly-coupled systems. However, the b2b modal crosstalk is high and inter-mode MIMO-DSP for strongly-coupled transmission has to be adopted. So, I suggest modifications to this statement.\nBased on the above concerns, I’m sorry to say that I still think the two innovations of the manuscript including the MPLC design and the transmission experiment are not significant enough for the publication of NC."}, "author_response": "15 November 2025\nResponse to the reviewers’ comments (Nature Communications, Research Article, No. NCOMMS-24-56122C)\nDear Reviewers,\nWe would like to express our sincere gratitude for your valuable time and consideration of our manuscript. Your efforts are essential in upholding the journal's high standards.\nThe author proposes a wide-spectrum, dispersioncompensated, multi-layer diffractive network design method to create ultra-wideband optical components, based on which an ultra-wideband, 4-LP-mode MPLC mode multiplexer is designed and fabricated for proof-of-concept demonstration. The MPLC component can achieve high efficiency over a very broad bandwidth ranging from 1260 nm to 1680 nm. However, I do not recommend this paper to be published by Nature Communications for the following reasons:\nultra-wideband MPLC-based devices and transmission experiments have been experimentally demonstrated in previous works. Its novelty, progress, and practical significance must be clarified and highlighted before it be considered for publication. Here are my comments:\n1. The authors claim a wide-spectrum device design method for multi-layer diffraction network by compensating the spatial dispersion. Please clarify the innovations of your approach by comparing it with previous work in reference [1], in which an ultra-wideband MPLC with a bandwidth over 900 nm is achieved by optimizing phase planes at various wavelengths. Mathematical and simulation descriptions must be added to highlight the practical significance of this work.\n2. The ultra-wideband transmission covering OESCLU bands has been demonstrated in reference [2]. Please clarify the novelty of your experimental transmission work.\n3. The descriptions of this algorithm should be clarified for it’s a main innovation of this work, such as:\na. Which phase plane is the spatial dispersion compensation algorithm applied to?\nb. How does the algorithm be applied? Is it just in the judgment step after calculating the correlation?\nc. The significance of this algorithm in reducing the number of phase planes and decreasing computational complexity?\nd. I suggest that the detailed mathematical algorithm should be described like that in the reference [2].\n4. With the same number of phase planes, the optimization of bandwidth will cause a deterioration in assembly tolerance such as the error planes or incident angle. Please add mathematical and simulation descriptions for them.\n5. Please provide some details for the transmission experiment:\na. Are both LP21a and LP21b modes demultiplexed at the receiver?\nb. How to deal with the rotation, coupling, differential mode gain/loss in the degenerated LP21 mode in the DSP algorithm?\nc. Is 4X4 or 8X8 MIMO-DSP applied at the receiver?\nd. Are the transmission data absent for the S and U bands? If they are, the claim of “first OESCLU MDM transmission” will be inappropriate.\ne. I suggest the authors provide the typical back-to-back mode crosstalk at C-band, O-band, U-band.\n[1]. Y. Zhang et al., \"An Ultra-Broadband Polarization-Insensitive Optical Hybrid Using Multiplane Light Conversion,\" in Journal of Lightwave Technology, vol. 38, no. 22, pp. 6286-6291, 15 Nov.15, 2020.\n[2]. B. J. Puttnam et al., \"402 Tb/s GMI data-rate OESCLU-band Transmission,\" 2024 Optical Fiber Communications Conference and Exhibition (OFC), San Diego, CA, USA, 2024, pp. 1-3.\n[3]. N. K. Fontaine, R. Ryf, H. Chen, D. Neilson and J. Carpenter, \"Design of High Order Mode-Multiplexers using Multiplane Light Conversion,\" 2017 European Conference on Optical Communication (ECOC), Gothenburg, Sweden, 2017, pp. 1-3.\nI believe that the revised manuscript has been greatly improved. However, I still have some question as below:\n1. Although the authors have replied most of the comments, I notice that not all the responses have been updated to the manuscript. I suggest that more revision should be made to improve the quality of the manuscript.\n2. I think the authors has not clearly explained the novelty for the design of MPLC device. The author has compared the method in the manuscript to that in the reference, and said “Our research utilizes multi-layer diffraction network technology to advance combined WDM+MDM multiplexing technology, enabling the parallel transmission of multiple wavelengths and spatial modes within a single optical fiber.” However, supporting both WDM and MDM may be a well-known design requirement instead of new concept for MPLC. For example, the function of multi-wavelength multiplexer has been proposed in Ref. A. If it’s a whole new design for WDM+MDM, please state the novelty clearly with detailed processing methods, simulation algorithms, etc.\nRef. A. G. Labroille, N. Barre, P. Jian, and J. Morizur, \"Multi-Wavelength Multiplexer with Independent Mode Control Based on Multi-Plane Light Conversion,\" in ECOC 2016.\n3. The authors have not fully replied my previous question about degenerate mode reception in Comment 5(b). For example, the authors may have excited the LP21a mode, and only received LP21a or LP21b mode at the receiver for coherent detection. However, the rotation and coupling between degenerate modes is inevitable during FMF transmission, and joint MIMO-DSP for both should be carried out to combat the frequency-selective fading, otherwise the performance will be greatly degraded. If the joint reception is not performed limited by available experimental equipment, the authors should describe it as it is.\n4. “The isolation between the LP01, LP11, and LP21 modes is sufficiently small” should be “sufficiently large”.\n5. The authors have provided more information for MIMO-DSP. Please explain your considerations for the different dimensions of MIMO-DSP such as 8x8, 10x10, 3x3.\n6. I notice that only 20-km transmission reach could be achieved (above the FEC limit) in the experiment using 8x8 MIMO-DSP. It’s much less than a typical strongly-coupled MDM transmission reach. please analyze it. Dos the limit come from the MPLC device?\n7. In the response to Comment 5(e), the authors claim that the b2b modal crosstalk for a pair of mode MUX/DEMUX is -8.87 dB. However, I think there is contradiction between it and the claim “achieves over 90% mode purity and mode crosstalk less than -17 dB” in the Abstract. Please clarify it.\n8. and mode crosstalk less than -17 dB” parameter may be\nThe authors have replied all the comments. But I do not think that all the modifications are satisfying.\n1. The authors have explained some details of the “wide-spectrum dispersion compensation mechanism”, in which the key technique point is to “establish a multi-parameter space incorporating wavelength, mode, spatial coordinates, free-space transfer functions, and the number of diffractive layers”. However, all these factors are widely considered in previous MPLC designs. So, it confirms my concerns for the innovations of the MPLC design.\n2. The authors state that the main difference between the manuscript and the Ref. A (another MPLC design covering beyond-400 nm spectrum) is that they have different design objectives, specifically, two fixed wavelengths V.S. multiple wavelengths. But I do not think that this kind of optimization could be a significant innovation.\n3. The authors have explained the reception method of degenerate modes. It’s said that “During transmission, unintentional coupling between LP21a and LP21b may occur due to fiber imperfections and perturbations, potentially causing partial energy leakage. To mitigate this, we inserted a polarization controller before the demultiplexer to suppress coupling and retain more energy in the LP21a component.”\nHowever, I disagree this experimental method of applying a PC before the receiver, only detecting on of a pair of degenerate modes, and choosing the favorable results.\n4. The authors have explained my questions about both the different dimensions of MIMO-DSP and the b2b modal crosstalk level. Based on the responses, I believe that the statement in the Abstract of “A 4-linearly polarized mode multiplexer ,……, achieves over 90% mode purity and mode crosstalk less than -17 dB” is misleading. There are both strongly-coupled and weakly-coupled mode-division-multiplexing schemes for FMF transmission systems. The authors’ claim of high-purity MPLC device (only the multiplexer instead of both multiplexer/demultiplexer) may mislead the readers that it achieves so excellent performance that weakly-coupled systems could be supported by the MPLC devices, which is much more difficult than those for typical strongly-coupled systems. However, the b2b modal crosstalk is high and inter-mode MIMO-DSP for strongly-coupled transmission has to be adopted. So, I suggest modifications to this statement.\nBased on the above concerns, I’m sorry to say that I still think the two innovations of the manuscript including the MPLC design and the transmission experiment are not significant enough for the publication of NC."}]

Decision letter