Review Reports - 19 × 1 Photonic Lantern for Mode Conversion: Simulation and Adaptive Control for Enhanced Mode Output Quality

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

This manuscript entitled “19×1 photonic lantern for mode conversion: simulation and adaptive control for enhanced mode output quality” by P. F. Liu et al, theoretically demonstrated a 19×1 photonic lantern capable of outputting 19 linear polarization modes and 16 orbital angular momentum modes. A detailed simulation results for the mode evolution process and insertion loss are given from such photonic lantern component.

The analysis results are good and this manuscript is organized. Therefore, I would like to recommend this paper for publication.

In table 1, Does the physical parameters of the taper ratio and taper length will affect the output performance of such photonic lantern?
According to an actual situation, could the authors add other simulation results under different single-mode fibers? e.g., 10/125 μm.
From an experimental perspective, how to fine fabricate and control such 19×1 photonic lantern during a fused taper process? Could the authors further explain it?
If reduce the input port numbers of photonic lantern, e.g., 7×1, how about the output performance?

Author Response

Comment 1:[In table 1, Does the physical parameters of the taper ratio and taper length will affect the output performance of such photonic lantern?]

Response 1:[I am very grateful to your comments for the manuscript. According with your advice, we amended the relevant part in manuscript.

The taper ratio and taper length indeed affect the output performance of such photonic lantern. As for the taper ratio, it directly determines the core diameter of the multimode output end of the photonic lantern, and thus determines the number of modes that the output end can support. As the taper diameter gradually becomes finer, the single-mode fiber bundle turns into the core of a multimode fiber, while the jacket becomes the cladding of the multimode fiber. An excessively large taper ratio would result in the multimode output end supporting more than 19 modes, which would require more input arms to control the output modes. On the other hand, an overly small taper ratio would lead to the multimode output end supporting fewer than 19 modes. Taking the above into account, a taper ratio of 1/5 is a suitable value. While for the taper length affects energy loss during the process of mode coupling, adiabatic transition requires that the change in the transverse dimensions of the single-mode fiber bundle within the tapering region must occur sufficiently slowly. This ensures that the mode coupling is minimized and energy loss is reduced during the transition process. A taper region that is too short fails to satisfy the adiabatic approximation, causing some light to leak out of the photonic lantern and increasing its loss. Conversely, a longer taper region ensures excellent mode coupling and low loss but increases fabrication difficulty. Considering both the insertion loss and the practical constraints of the manufacturing process, a taper length of 3 cm is deemed appropriate. ]

Comment 2:[According to an actual situation, could the authors add other simulation results under different single-mode fibers? e.g., 10/125 μm.]

Response 2：[

I am very grateful to your comments for the manuscript. According with your advice, we amended the relevant part in manuscript.

Based on our preliminary research, we found that the ratio of the core diameter to the cladding diameter (core-to-cladding ratio) of the input single-mode fibers is a critical factor influencing the mode evolution results in photonic lanterns. Using 5×1 and 3×1 non-mode photonic lanterns as examples, we simulated the evolution process of the LP₀₁ mode and evaluated the mode purity using the beam quality M² factor. The results are shown in the figure below. From the figure, it is evident that a larger core-to-cladding ratio of the input single-mode fiber (with a core diameter of 10 μm) results in an M² factor closer to the ideal value of 1. To ensure that the better output beam, photonic lanterns with larger output core diameters require larger core-to-cladding ratios. Therefore, in designing this photonic lantern, we selected the commercially available single-mode fiber with the largest core-to-cladding ratio (8/80) and corroded its cladding to 30 μm, rather than using the commonly employed 10/125 single-mode fiber.

]

Comment 3:[From an experimental perspective, how to fine fabricate and control such 19×1 photonic lantern during a fused taper process? Could the authors further explain it?]

Response 3:[

I am very grateful to your comments for the manuscript. According with your advice, we amended the relevant part in manuscript.

The aim of this paper is the design and simulation of a 19×1 photonic lantern, with the goal of developing a device capable of outputting 19 LP modes and 16 OAM modes with low loss. It is important to note that the fabrication and control of such a 19×1 photonic lantern are not within the scope of this study. Although the fabrication and control of the 19×1 photonic lantern are not the primary focus of this study, we can briefly introduce some of the most critical aspects of the drawing process:

Cleanliness: Cleanliness of both the optical fibers and glass tubes is crucial throughout the entire process. Any dirt or liquid residue can cause intense burning during the fusion and tapering stages, significantly reducing the yield. Even tiny contaminants can become the primary source of loss in the final photonic lantern.

Bundling Techniques: Precise control over the inner diameter of the pre-tapered glass sleeve is essential to ensure that the fiber bundle is tightly and uniformly arranged. Custom molds may be required to achieve specific geometric arrangements, ensuring optimal packing density and alignment.

Fusion Tapering Parameters: The choice of flame size and tapering speed during the fusion process affects the collapse effect, which alters the size of air holes between single-mode fibers or determines whether the glass sleeve and fiber bundle shrink proportionally. Ensuring proportional shrinking is critical for the success of the splicing step. Therefore, selecting appropriate fusion tapering parameters is vital for achieving high-quality results.

]

Comment 4:[If reduce the input port numbers of photonic lantern, e.g., 7×1, how about the output performance?]

Response 4:[

I am very grateful to your comments for the manuscript. According with your advice, we amended the relevant part in manuscript.

Reducing the number of input ports in the photonic lantern can cause a mismatch between the number of input arms and the number of output modes. In such a scenario, the photonic lantern will only be able to support low-loss transmission for a limited set of lower-order modes. Higher-order modes, on the other hand, will experience substantial losses and contribute less to the final output. This imbalance leads to an inefficient utilization of the available modes and can limit the overall performance and versatility of the photonic lantern.

]

Reviewer 2 Report

Comments and Suggestions for Authors

The manuscript by Liu et al presents a numerical design of a 19×1 photonic lantern (PL) capable of supporting both high-order linear polarization (LP) modes and orbital angular momentum (OAM) beams. Mode control is implemented using an SPGD (Stochastic Parallel Gradient Descent) combined with a basin hopping optimization algorithm. The use of photonic lanterns for adaptive mode shaping and switching is a well-established and active area of research, with several related works available in the literature (including work from same authors on using SPGD-based mode control on a 6 mode PL (10.3390/mi15111342 )). The key novelty of this study appears to lie in the reported high coupling efficiency into the PL outputs (with insertion loss as low as 0.07 dB), as well as the integration of the SPGD + basin hopping approach for mode control. These aspects are promising, but further clarification on the insertion loss calculation methodology, simulation assumptions, and experimental feasibility of the adaptive control system is required.

The work could potentially be published in MDPI Photonics. I would like to see the response from the author before making a final recommendation.

See my comments below,

Page 5, line 186

Is there an alternative to basin hopping that can be used with the SPGD + basin hopping algorithm (e.g., Gerchberg–Saxton)? It would be ideal to show a comparison (e.g., a table) showcasing the computational enhancements of the SPGD+BH with other techniques.

Page 6, PL design

What is the NA of the PL MM end? what is the length of the SMFs used in the setup?

There is a mismatch in mode numbers, considering the 30um core of the PL MM end (unless the NA matches). The MM end supports > 50 modes (for a typical NA of 0.12), while the number of SM fibres are 19. If the refractive indices of core and cladding is taken from the given parameters from the table, 36 guided modes at 1064 nm (NA 0.095). Can you comment on this?

The mode mismatch loss is still high; however, this will be only evident when coupling light from MM end to input.

This reason for poor mode quality before applying the adaptive control “……..the output beam contains a substantial amount of unwanted modes”….. (page 11, line 309), is likely due to the mode mismatch, that the MM end supports more than 19 LP modes.

In line 226, it says taper length of 3 cm, which is 30,000µm? In the table below, taper length is 3000 µm.

Page 9, insertion loss measurements

I would like to see more detailed information regarding the insertion loss (IL) measurements. What specific steps were followed during the measurement process? Was the analysis based on the assumption that the splitter distributes light equally to all PL inputs? Additionally, what power level was used as the input source? It is also important to clarify whether Fresnel reflection losses at the input and output facets were taken into account. The IL values presented appear to reflect optimal or ideal conditions, but in practical implementations, Fresnel reflections can contribute to additional loss. If these were included in the measurement, this should be explicitly stated to avoid ambiguity.

Page 12, figure 9.

Is there a specific reason why the LP_21e mode was chosen as the target for the figure? Most existing adaptive mode-shaping approaches focus on targeting the fundamental mode (e.g., LP₀₁), due to transmission and processing benefits of signal.

Additionally, simulating the M² value as a function of process time for a given target mode (or a few selected modes) would strengthen the results and provide a more complete picture of mode evolution.

Page 12,13. Figures 10,11

I am particularly interested in the conversion efficiency and mode purity of the LP₀₁ mode, as well as other supported modes. Including a plot showing the mode purity or conversion efficiency for all LP modes would significantly strengthen the paper by offering a more comprehensive view of the modal evolution and performance.

Other minor comments.

Page 2. Line 35

“Photonic lanterns serve as spatial mode converters, seamlessly integrating single-mode signals from multiple individual waveguide cores into a single multimode waveguide” –

the individual waveguide cores can be few-moded or multimoded too.

Modify the sentence accordingly. Something like…., the simplest PL design consisting of multiple SM inputs with smaller core size and a multimode core with larger core size…

It would be also informative to include conditions for lossless PL systems in terms of mode numbers (Number of SM waveguides = Number of modes supported by MM core). Refer to Citation 16.

Page 6, figure1

Caption. Be consistent on capitalising (or not) first letters of the abbreviations full form.

Author Response

Comment 1: [Page 5, line 186

Response 1: [

I am very grateful to your comments for the manuscript. According with your advice, we amended the relevant part in manuscript.

The SPGD algorithm augmented with basin hopping is selected for adaptive control because of the SPGD algorithm’s advantages, including rapid convergence, strong scalability, and high tolerance to power fluctuations. Despite these benefits, the SPGD algorithm can easily become trapped in local optima. The basin hopping algorithm mitigates this issue by facilitating the escape from local optima and enhancing the search for the global optimum. By combining the SPGD algorithm with the basin hopping algorithm, the system can achieve both rapid convergence and robust global optimization.

I also studied the Gerchberg-Saxton (GS) algorithm, which is a widely used iterative phase retrieval algorithm in optics and signal processing. The GS algorithm is designed to solve the problem where the intensity distributions of a signal are known in two different domains (such as the spatial domain and the frequency domain), but the phase information is missing. The goal is to recover the complete phase information through an iterative method, thereby reconstructing the original signal.

However, the GS algorithm is not suitable for the adaptive control of photonic lanterns. This is because both the amplitude and phase of the input arms in a photonic lantern vary with environmental disturbances. Simply controlling the phase cannot achieve high-quality mode output. Additionally, compared to the SPGD algorithm, the GS algorithm is slower in solving high-dimensional problems and is prone to getting trapped in local optima.

The key to effective control lies in the ability to quickly find the global optimum rather than getting trapped in a local optimum. Here, we present simulation results using only the SPGD algorithm, as shown in the figure below. The results indicate that the controlled output of the LP_21e mode content reached only 72%, which is significantly lower compared to the >95% mode content achieved in Figure 9. This discrepancy is due to the SPGD algorithm converging to a local optimum rather than the global solution.

]

Comment 2: [What is the NA of the PL MM end? what is the length of the SMFs used in the setup?

The mode mismatch loss is still high; however, this will be only evident when coupling light from MM end to input.

Response 2: [

I am very grateful to your comments for the manuscript. According with your advice, we amended the relevant part in manuscript.

The NA of the PL MM end is 0.095. The refractive index of the MM core is 1.4504, and the refractive index of the cladding is 1.4471. During the tapering process, the lantern cladding gradually evolves into the core of the multimode fiber, while the jacket transforms into the cladding of the multimode fiber. The core diameter of the multimode (MM) fiber is 30 μm, resulting in a normalized cutoff frequency V of 8.5295. This configuration supports only 19 guided modes, specifically: LP₀₁, LP₀₂, LP₀₃, LP_11e, LP_11o, LP_12e, LP_12o, LP_21e, LP_21o, LP_22e, LP_22o, LP_31e, LP_31o, LP_32e, LP_32o, LP_41e, LP_41o, LP_51e, and LP_51o. Therefore, there is no mismatch between the number of modes supported by the photonic lantern's (PL) MM end and the number of SMFs. What I mean in (page 11, line 309) is that the environmental disturbances cause the optical parameters of the photonic lantern's input arms to deviate from their optimal settings. As a result, the output beam contains a substantial amount of unwanted modes, which is not due to mode mismatch. We are currently investigating the impact of mismatches between the number of SMFs and the mode numbers in MMFs on the performance of photonic lanterns, which will be detailed in an upcoming paper. Besides, the length of the SMFs used in the setup is about 0.6 m。

]

Comment 3: [In line 226, it says taper length of 3 cm, which is 30,000µm? In the table below, taper length is 3000 µm.]

Response 3: [

I am very grateful to your comments for the manuscript. There was a typographical error in the original manuscript regarding the taper length. The correct taper length is 30,000 μm (30 mm). This has been updated in the new version of the paper to reflect the accurate value.

]

Comment 4: [Page 9, insertion loss measurements

Response 4: [

I am very grateful to your comments for the manuscript. The detailed information regarding the IL measurements has been added in the reversed version. Before calculating the insertion loss for various output modes of the photonic lantern, it is first necessary to obtain its transmission matrix. Using this matrix, the required optical parameters (including amplitude and phase) at the input arms for generating the corresponding output modes can be deduced. The detailed steps are as follows: First, inject laser light individually into each single-mode (SM) input port. Next, perform mode decomposition at the multimode (MM) output port to measure the normalized mode decomposition coefficients for each SM input. Using these coefficients, construct the transmission matrix MM of the photonic lantern, which is

Once the transmission matrix M is determined, apply the corresponding input vectors by injecting laser light with the appropriate intensity and phase differences into the SM input ports. For example, to generate the fundamental mode at the output, the input power for each SM fiber should be proportional to the square of the modulus of the corresponding element in the first row of the transmission matrix M , and the phase difference should match the phase of the corresponding element in the first row. Finally, measure the total output power at the MM output port to validate the performance. ,

where P_out is the power of the corresponding output mode， and P^{i}_{in} is the power at each input arm. Since this work focuses on the design and simulation of a 19×1 photonic lantern, the calculated insertion losses (IL) represent ideal conditions. These losses only account for the leakage of the optical field during the mode evolution process. Other practical losses, such as Fresnel reflection losses introduced during the actual fabrication process, have not been considered in these calculations.

]

Comment 5: [

Page 12, figure 9.

]

Response 5: [

I am very grateful to your comments for the manuscript. We arbitrarily selected the LP_21e mode as the control target for Fig. 9. The fundamental mode was not chosen for control because we had already conducted extensive simulations and experimental studies on fundamental mode control in our previous work. In response to your comment, we have added simulations of fundamental mode control and included the M² value as a function of process time in the revised version of our manuscript.

Figure (a) illustrates the mode content evolution and the final output beam generated by the 19×1 photonic lantern during the adaptive control process, specifically targeting the fundamental mode output. Figure (b) shows how the M² value varies with process time for the fundamental mode. These figures clearly demonstrate that the application of the active control algorithm results in a gradual increase in the fundamental mode content and a progressive decrease in the M² factor. Ultimately, the output fundamental mode content surpasses 99%, achieving an M² value better than 1.07, indicating excellent beam quality.

]

Comment 6: [Page 12,13. Figures 10,11

Response 6: [

I am very grateful to your comments for the manuscript. In the revised version, we have added a table to show the mode purity or conversion efficiency for all LP modes and OAM modes. The results are summarized as follows:

]

Comment 7: [Page 2. Line 35

“Photonic lanterns serve as spatial mode converters, seamlessly integrating single-mode signals from multiple individual waveguide cores into a single multimode waveguide” –

the individual waveguide cores can be few-moded or multimoded too.

Modify the sentence accordingly. Something like…., the simplest PL design consisting of multiple SM inputs with smaller core size and a multimode core with larger core size…

It would be also informative to include conditions for lossless PL systems in terms of mode numbers (Number of SM waveguides = Number of modes supported by MM core). Refer to Citation 16.]

Response 7: [

I am very grateful to your comments for the manuscript. The sentences have been modified in the new version of our paper. “The simplest photonic lanterns serve as spatial mode converters, seamlessly integrating single-mode signals from multiple individual waveguide cores into a single multimode waveguide. To achieve minimal loss operation, it is essential that the number of input single-mode (SM) fibers should equal the number of modes supported by the multimode (MM) fiber.”

]

Comment 8: [Page 6, figure1

Caption. Be consistent on capitalising (or not) first letters of the abbreviations full form.]

Response 8: [

I am very grateful to your comments for the manuscript. The capitalization consistency of the abbreviations' first letters has been modified.

]

Reviewer 3 Report

Comments and Suggestions for Authors

Review of Manuscript:

This manuscript presents a simulation-based study of a device designed to generate Linear Polarized (LP) and Orbital Angular Momentum (OAM) modes at the output of a multimode fiber. The proposed system combines a 19x1 photonic lantern (PL) with phase modulators and pre-amplifiers, whose control voltages are adjusted to select desired output modes. To compensate for perturbations and environmental instabilities, the authors implement an adaptive control loop based on feedback from output mode measurements, using a gradient descent algorithm. The PL’s transmission matrix is simulated using a finite-difference beam propagation method.

The topic—combining all-fiber systems and photonic lanterns to generate high-order modes—is of high interest in photonics. However, I believe the manuscript requires major revisions prior to publication. Key concerns relate to the novelty of the work, insufficient methodological detail, and a lack of practical considerations or experimental validation. This review is devided in major and minor concerns.

MAJOR CONCERNS

Novelty and Contribution

While photonic lanterns have been already used for mode conversion—including for generating OAM modes—this work does not appear to provide new insights or innovations in PL design while the use of adaptive control via gradient descent is not well introduced and justified (cf. next comment). The manuscript lacks a clear distinction from existing literature, especially given that the work is entirely simulation-based. Without experimental validation or a novel theoretical contribution, the manuscript’s originality is difficult to establish.

Adaptive Control: Lack of Clarity

The rationale for implementing adaptive control is unclear due to insufficient details regarding the perturbations the system aims to correct and the lack of quantitative analysis. Key questions include:

What specific perturbations are introduced? What are their temporal and spatial characteristics?
Where in the optical system are they applied?
At what rate must the system respond to these perturbations?
Is it assumed that the transmission matrix of the PL is known a priori?
Is a mode decomposition device required during operation, or only for calibration? If removed after initialization, are perturbations assumed to be static?
the claim that "resemblance indicates a high-level of purity" lacks quantitative support. What level of mode purity is considered acceptable or targeted? These metrics are not discussed.

Moreover, since the system appears linear in the optical field, why is a gradient descent algorithm needed? Could a direct inversion of the PL transmission matrix, combined with output mode measurement, be used?

PL Design Justification

The design of the photonic lantern is not justified. Were the structural parameters selected based on specific criteria or optimized in any way? While the internal modal transformations are discussed qualitatively, the rationale behind the chosen geometry, tapering, or other PL parameters is not provided. Similarly, simulation parameters used in lightbeam for calculating the PL transmission matrix are not described.

Practical Considerations and Experimental Outlook

The conclusion refers to the “practical utility” of the method, yet no experimental validation is provided, nor are any plans or challenges for implementation discussed. Missing points include:

Required precision/resolution for phase modulators and amplifiers to reach the desired beam quality.
Impact of deviations between simulated and fabricated PLs.
Possible mitigation strategies for such discrepancies.
A proposed experimental setup to validate the approach.

Without these considerations, the claim of practical relevance remains unsubstantiated.

MINOR COMMENTS

Introduction:

Maintain consistent terminology for LP modes (use either a/b or o/e throughout).
Avoid repeating the abstract at the end of the introduction; instead, briefly outline the structure of the paper.

Section 2.1:

Equation 1: Define all variables, including r and z.
Clarify the statement about “one-to-one correspondence.” If the PL acts as a linear operator, say so explicitly.

Equation 9: Unclear. The goal is to have a matrix that describe the transmission of the PL from: input basis being the amplitude and phase of the fundamental mode in the i-th single mode fiber to the OAM basis (let’s call this basis S). Therefore I believe that the matrix of interest is

M = M_change × M_PL × S_to_LP

where S_to_LP converts from SMF inputs to LP modes. By inverting this one, one can compute what command to send on phase modulator and pre-amplifier to produce a given OAM mode.

Section 2.2:

Typo: “an mode decomposition” → “a mode decomposition.”
Clarify notation for μ upper index: does 1 represent phase modulators and 2 pre-amplifiers?
Figure 2: Text inside the figure is too small. Replace full sentences with symbols or equation references and move detailed explanations into the main text.

Results and Discussion:

Figure 3: The 3D diagram is not to scale. This should be explicitly stated.
The decomposition of OAM modes as a combination of LP modes is well known; this need not be emphasized.
Figure 8: Arrows are hard to distinguish. Consider increasing their thickness and using contrasting colors.

Conclusion:

This work explores a promising direction in mode control using photonic lanterns and adaptive feedback, but several critical elements are missing. Specifically:

The novelty relative to existing literature is not clearly established.
The description of the control scheme and perturbation modeling lacks rigor.
Design choices and simulation assumptions are insufficiently justified.
There is no experimental plan or discussion of real-world implementation.

I recommend substantial revisions before this manuscript can be considered for publication.

Author Response

Comment 1: [Novelty and Contribution

Response 1: [

I am very grateful to your comments for the manuscript. This manuscript is a simulation study designed to theoretically validate the feasibility of generating high-quality higher-order LP and OAM modes, as well as achieving mode switching. Traditional control systems using bucket power as the evaluation function can only stabilize a few low-order modes like the fundamental mode and LP_11e/LP_11o. We have improved the adaptive control system by integrating a high-speed mode decomposition system (using optical correlation filtering) at the output. The real-time decomposed target mode content serves as the evaluation function, enabling effective mode distinction and enhancing system performance. By switching different mode components, the desired modes can be output. Using this method, the system is capable of generating 19 LP modes and 16 OAM modes. The work specifically targets the challenge of low target mode content observed in previous theoretical and experimental studies. Previous work used the SPGD algorithm for adaptive control, which tends to get trapped in local optima. This problem worsens with an increase in the number of input arms, leading to higher control complexity. The figure below shows the control results for the LP_21e mode using only the SPGD algorithm, where performance converges at about 72%, with significant unwanted modes present. The algorithm cannot escape this local optimum to find a better solution. By introducing the new SPGD + basin hopping control algorithm, this study successfully increases the mode content to over 95%. These findings provide essential theoretical guidance for subsequent experimental efforts, paving the way for more effective high-order mode control and manipulation.

]

Comment 2: [Lack of Clarity

What specific perturbations are introduced? What are their temporal and spatial characteristics?

Where in the optical system are they applied?

At what rate must the system respond to these perturbations?

Is it assumed that the transmission matrix of the PL is known a priori?

Is a mode decomposition device required during operation, or only for calibration? If removed after initialization, are perturbations assumed to be static?

the claim that "resemblance indicates a high-level of purity" lacks quantitative support. What level of mode purity is considered acceptable or targeted? These metrics are not discussed.

Response 2: [

I am very grateful to your comments for the manuscript. The details regarding the perturbations in the systems and the quantitative analysis will be included in the revised version. Here is a brief overview:

In this paper, both amplitude and phase perturbations are introduced due to environmental disturbances. Environmental disturbances, such as air flow and base vibrations causing fiber jitter, can affect the amplitudes and phases of the input arms of the PL. The fiber output is highly sensitive to these disturbances, impacting signal stability. Additionally, instabilities in optical components, such as laser seeds and splitters, also contribute to these perturbations. Our preliminary measurements in the laboratory indicate that phase jitter occurs at frequencies ranging from approximately 1 kHz to 10 kHz, while amplitude jitter occurs at frequencies ranging from about 1 Hz to 10 Hz. To simulate these disturbances, we modeled all jitter as a combination of a 10 kHz random phase jitter and a 10 Hz random amplitude jitter applied to the input arms of the photonic lantern. The controller bandwidth was designed to be 10 MHz to effectively correct these environmental perturbations. In this simulation, the transmission matrix of the PL needs to be determined in advance because we rely on it to obtain the mode information at the multimode output of the PL. However, in actual experiments, it is not necessary to know the transmission matrix beforehand. The SPGD algorithm can adaptively search for the optimal solution as long as there is a one-to-one correspondence between the input and output, without requiring explicit knowledge of their specific relationship.

During operation, the mode decomposition device is continuously required because perturbations are not static. Real-time adaptive control is necessary based on the results of mode decomposition to maintain optimal performance.

In the revised version, we have added a table to show the mode purity or conversion efficiency for all LP modes and OAM modes. The results are summarized as follows:

Although the system appears linear in the optical field, directly inverting the PL transmission matrix combined with output mode measurements is challenging. Firstly, direct inversion of the PL transmission matrix requires high-speed and precise measurement of the amplitude and phase information at the input arms of the photonic lantern. Secondly, accurately obtaining the transmission matrix of a photonic lantern is extremely difficult due to manufacturing defects, which cause the actual transmission matrix to deviate from the designed one. Lastly, in high-power applications, we typically connect a fiber amplifier after the photonic lantern. Mode decomposition for control is performed after the amplifier, where the SPGD adaptive control algorithm can also account for mode competition within the fiber amplifier.

Given these challenges, the SPGD combined with the basin hopping algorithm is a more suitable control strategy for managing the modes in photonic lanterns.

]

Comment 3: [PL Design Justification

The design of the photonic lantern is not justified. Were the structural parameters selected based on specific criteria or optimized in any way? While the internal modal transformations are discussed qualitatively, the rationale behind the chosen geometry, tapering, or other PL parameters is not provided. Similarly, simulation parameters used in lightbeam for calculating the PL transmission matrix are not described.]

Response 3: [

I am very grateful to your comments for the manuscript.

When designing photonic lanterns, it is crucial to arrange single-mode fibers (SMFs) as compactly as possible to ensure efficient coupling of light fields, and the geometric arrangement of the SMF cores should match the modal structure supported by the multimode end because theoretical studies have shown that matching these structures can effectively reduce device loss[Advances in Optics and Photonics, 2015, 7:107, Optics Express, 2012, 20:27123]; for step-index fibers, the morphology of LP modes follows a specific pattern where for an LP_mn mode, the light field splits into 2m lobes angularly and n-1 layers radially, with two degenerate states when m > 0, therefore the corresponding arrangement of single-mode fiber cores in the photonic lantern should be angularly distributed in a regular polygon with 2m+1 cores and radially distributed across n-1 layers of cores. Following these design principles, the optimal configuration for a 19×1 photonic lantern is arranged in a three-layer circularly symmetric configuration: the innermost layer contains 1 SMF, the middle layer contains 6 SMFs, and the outermost layer contains 12 SMFs, ensuring efficient light coupling and minimizing losses by matching the modal structures between the single-mode and multimode ends.

To maximize the confinement of energy within the guided modes of single-mode beams and minimize energy dissipation in the cladding due to radiative modes, it is essential to design the taper region of the single-mode fiber bundle in photonic lanterns to satisfy the adiabatic transition condition, that is, satisfying the condition expressed by the equation . Here, and are the propagation constants of the two modes within the tapered single-mode fiber bundle that could potentially couple energy. Theoretically, the longer the taper region, the better it is for ensuring adiabatic transition and maximizing energy confinement within the guided modes. However, considering the limitations of photonic lantern fabrication processes, excessively long taper regions are difficult to fabrication. Therefore, it is necessary to select an appropriate taper length that balances the benefits of adiabatic transition with the practical constraints of manufacturing, ensuring both efficient energy confinement and feasible fabrication.

The photonic lantern is designed with a single-mode fiber at the input end, featuring a core radius of 4 µm. After etching, the fibers have a cladding diameter of 30 µm and are bundled into an envelope with a radius of 75 µm, forming the "inner cladding." The single-mode fibers used have a core refractive index of 1.4539 and a cladding refractive index of 1.4504. A low-index glass jacket with a refractive index of 1.4471 surrounds the bundle. Following a taper length of 30 mm, the diameter of the tapered region is reduced to one-fifth of its original size, ensuring efficient transition from the input fibers to the output multimode structure.

]

Comment 4: [Practical Considerations and Experimental Outlook

The conclusion refers to the “practical utility” of the method, yet no experimental validation is provided, nor are any plans or challenges for implementation discussed. Missing points include:

Required precision/resolution for phase modulators and amplifiers to reach the desired beam quality.

Impact of deviations between simulated and fabricated PLs.

Possible mitigation strategies for such discrepancies.

A proposed experimental setup to validate the approach.

Without these considerations, the claim of practical relevance remains unsubstantiated.]

Response 4: [

I am very grateful to your comments for the manuscript.

The simulations in this paper have theoretically demonstrated that the developed 19×1 photonic lantern adaptive control system can produce the desired 19 LP modes and 16 OAM modes with high quality, achieving over 90% mode content. Next, we will design an experimental system to achieve this functionality, with the experimental setup shown in the figure below. The mode decomposition system, based on the optical correlation filter method, can achieve a decomposition rate of over 10MHz. Previously, we have implemented adaptive control systems for 3×1, 5×1, and 6×1 photonic lanterns. On this basis, we will expand the control channels to 19 and introduce a high-speed mode decomposition system into the control. In the previous work, we used PMs (phase modulators) which are lithium niobate phase modulators with a 10GHz bandwidth and control accuracy of 0.1mrad, and Pre-AMP with an output of the 10W level and control accuracy in mW, whose precision has far exceeded the required control accuracy of 10^-3.

During the fused tapering process of PLs, factors such as flame stability, uniformity of flame temperature, and taper speed stability can lead to discrepancies between the actual fabricated PLs and simulated PLs. These inconsistencies primarily manifest in two ways: 1) transmission matrix mismatch; 2) increased insertion loss. The transmission matrix mismatch does not affect control performance, as the SPGD algorithm can adaptively search for the optimal solution as long as there is a one-to-one correspondence between the input and output, without requiring explicit knowledge of their specific relationship. While for increased insertion loss, it exacerbates the thermal effect, which leads to reduced mode coupling efficiency and changes in modal field shape. To mitigate the thermal effect, first, select PLs with low insertion loss, and second, place the PLs in a water-cooled environment to minimize the adverse effects caused by thermal effects.

]

Comment 5: [Introduction:

Maintain consistent terminology for LP modes (use either a/b or o/e throughout).]

Response 5: [

I am very grateful to your comments for the manuscript.

The consistent terminology for LP modes has been changed to be uniform throughout the document.

]

Comment 6: [Avoid repeating the abstract at the end of the introduction; instead, briefly outline the structure of the paper.]

Response 6: [

I am very grateful to your comments for the manuscript.

I am very grateful for your comments on the manuscript. I have revised the corresponding sections based on your suggestions.

]

Comment 7: [Equation 1: Define all variables, including r and z.

Clarify the statement about “one-to-one correspondence.” If the PL acts as a linear operator, say so explicitly.]

Response 7: [

I am very grateful to your comments for the manuscript.

I have revised the corresponding sections based on your suggestions.

]

Comment 8: [Equation 9: Unclear. The goal is to have a matrix that describe the transmission of the PL from: input basis being the amplitude and phase of the fundamental mode in the i-th single mode fiber to the OAM basis (let’s call this basis S). Therefore I believe that the matrix of interest is

M = M_change × M_PL × S_to_LP

where S_to_LP converts from SMF inputs to LP modes. By inverting this one, one can compute what command to send on phase modulator and pre-amplifier to produce a given OAM mode.]

Response 8: [

I am very grateful to your comments for the manuscript.

Eq. 9 describes the relationship between the transmission matrix of the photonic lantern in the OAM mode basis and the transmission matrix of the photonic lantern in the LP mode basis. The instructions that should be sent to the phase modulator and pre-amplifier to generate a specific OAM mode should be calculated using Formula 1, which is

]

Comment 9: [Typo: “an mode decomposition” → “a mode decomposition.”

Clarify notation for μ upper index: does 1 represent phase modulators and 2 pre-amplifiers?]

Response 9: [

I am very grateful to your comments for the manuscript.

I have revised the corresponding sections based on your suggestions.

]

Comment 10: [Figure 2: Text inside the figure is too small. Replace full sentences with symbols or equation references and move detailed explanations into the main text.]

Response 10: [

I am very grateful to your comments for the manuscript.

I have revised the corresponding sections based on your suggestions.

]

Comment 11: [Results and Discussion:

Figure 3: The 3D diagram is not to scale. This should be explicitly stated.

The decomposition of OAM modes as a combination of LP modes is well known; this need not be emphasized.]

Response 11: [

I am very grateful to your comments for the manuscript.

I have revised the corresponding sections based on your suggestions.

]

Comment 12: [Figure 8: Arrows are hard to distinguish. Consider increasing their thickness and using contrasting colors.]

Response 12: [

I am very grateful to your comments for the manuscript.

I have revised the corresponding sections based on your suggestions.

]

Round 2

Reviewer 3 Report

Comments and Suggestions for Authors

This is the second review I have conducted on this manuscript. I would like to thank the authors for taking the time to revise their work and to consider the comments I raised in my initial review. Despite these efforts, I remain uncertain about several points in their responses, and I believe the manuscript may still not be ready for publication.

In my first review, I expressed concerns regarding the originality of the work. The authors responded by emphasizing that the novelty lies in their use of a high-speed mode decomposition system combined with their adaptive optics control algorithm. However, the literature indicates that the authors have already published studies based on the combination of photonic lanterns, mode decomposition, and the SPGD algorithm. As I understand it, the current manuscript claims novelty mainly in the number of modes controlled and the higher level of purity achieved thanks to modifications in the control algorithm. Nevertheless, the manuscript itself does not convincingly convey this: the algorithm is summarized in a single paragraph with one figure, while the mode decomposition system is not described in sufficient detail. As I noted in my previous review, the originality of the contribution remains unclear.

Since the authors insist that the novelty lies primarily in the adaptive control strategy, I would like to highlight the following major points that remain unclear to me:

I am not convinced why the SPGD + basin hopping control algorithm should outperform other possible approaches. With a high-speed mode decomposition system providing direct access to the output, the problem seems much simpler. For instance, one could in principle measure the transmission matrix of the system and apply a straightforward matrix inversion for control—would this not work?
Regarding perturbations: the manuscript provides some temporal values for the evolution of perturbations, but no amplitudes are given. Nor is there any description of their temporal spectrum (e.g., white noise?). The origin of these temporal values is unclear: the authors refer to "preliminary measurements" but without citation, which is confusing. What measurements are being referred to exactly?
Concerning controller bandwidth: the manuscript states that a 10 MHz bandwidth is designed to reject perturbations. Why must it be so high? Do the authors actually mean the framerate of the controller rather than its bandwidth? Moreover, how can such a rate be achieved when the referenced mode decomposition system is reported to reach only a few kHz? Am I missing something here? Also : is the SPGD + basin hopping algorithm really expected to run at 10 MHz?
The authors have made an effort to better describe the design choices for the photonic lantern. However, the explanations remain largely qualitative (e.g., citing "preliminary research" without specifying which). Furthermore, the manuscript acknowledges that the actual transmission matrix might deviate from the expected one. If so, how can we be confident that a one-to-one correspondence between modes is preserved? What if it is not the case? Would this pose a risk, and if so, how is it mitigated?

Once again, I would like to once again thank the authors for their effort in responding to many of the points raised in my first review through their dedicated answers and modification in the manuscript. However, I believe that the issues outlined above must be adequately addressed before this manuscript can be considered suitable for publication.

Author Response

Comment1: [ I am not convinced why the SPGD + basin hopping control algorithm should outperform other possible approaches. With a high-speed mode decomposition system providing direct access to the output, the problem seems much simpler. For instance, one could in principle measure the transmission matrix of the system and apply a straightforward matrix inversion for control—would this not work?

Concerning controller bandwidth: the manuscript states that a 10 MHz bandwidth is designed to reject perturbations. Why must it be so high? Do the authors actually mean the framerate of the controller rather than its bandwidth? Moreover, how can such a rate be achieved when the referenced mode decomposition system is reported to reach only a few kHz? Am I missing something here? Also : is the SPGD + basin hopping algorithm really expected to run at 10 MHz?

The authors have made an effort to better describe the design choices for the photonic lantern. However, the explanations remain largely qualitative (e.g., citing "preliminary research" without specifying which). Furthermore, the manuscript acknowledges that the actual transmission matrix might deviate from the expected one. If so, how can we be confident that a one-to-one correspondence between modes is preserved? What if it is not the case? Would this pose a risk, and if so, how is it mitigated?]

Response1: [

I would like to thank the reviewer again for reviewing this paper and providing valuable feedback and suggestions for improvement. I feel so sorry for not having explained your comments clearly last time. Now, I would like to provide a mode detailed clarification.

Previously, our description of the adaptive system was not sufficiently clear, especially regarding the mode decomposition system and its role. In the following section, we will provide a detailed explanation of the adaptive control system, particularly how the mode decomposition system is used for mode selection and output.

The schematic diagram of a switchable output mode adaptive control system based on a 19×1 photonic lantern is shown in the figure below[Frontiers in Physics 2023, 11, 1146208]. The light emerging from the pigtail of the photonic lantern first passes through a polarizing beam splitter (PBS) to achieve linear polarization. It is then divided into two beams by a non-polarizing beam splitter (NPBS). One beam is reflected towards a CCD to record the near-field intensity pattern, while the other beam is directed to a spatial light modulator (SLM). The SLM is loaded with the complex conjugate transmission function of the desired mode. After modulation by the SLM, the optical field passes through the NPBS and a lens, focusing at the lens's focal plane. The complex conjugate transmission function is loaded onto specific spatial frequencies, and an aperture is placed at the centroid of the +1 diffraction order spot. A photodetector (PD) is positioned after the aperture. The photodetector collects optical data, which serves as the evaluation function fed back to the controller. Through the adaptive control system, the amplitudes and phases of the input arms of the photonic lantern are adjusted to maximize the power detected by the PD. By changing the transmission function on the SLM, different modes can be switched.

Although we employed a mode decomposition system based on the optical correlation filter method, we did not perform a complete mode decomposition of the output intensity pattern. Instead, we only collected power data for the desired modes. Therefore, in principle, the method of simply measuring the transmission matrix of the system and applying a straightforward matrix inversion for control would not work. This aspect was not clearly explained in previous versions of the paper, which led to confusion among the reviewers. In this section, we aim to clarify this point by providing a detailed explanation. The speed of the mode decomposition system based on the optical correlation method mainly depends on the frame rate of the far-field camera. In this work, we replaced the far-field camera with a photodetector (PD). For commercial PDs, bandwidths exceeding 10 MHz can be readily achieved, thus significantly increasing the system's speed. In the experimental system, we designed a controller driven by an FPGA, which can achieve speeds of over 100 MHz. This system allows for frame rates exceeding 10 MHz, enabling it to correct for disturbances occurring at kHz frequencies. Furthermore, although the actual transmission matrix may deviate from the expected one, the photonic lantern device remains a linear optical device, and the one-to-one correspondence can still be preserved.

]

Comment2: [Regarding perturbations: the manuscript provides some temporal values for the evolution of perturbations, but no amplitudes are given. Nor is there any description of their temporal spectrum (e.g., white noise?). The origin of these temporal values is unclear: the authors refer to "preliminary measurements" but without citation, which is confusing. What measurements are being referred to exactly?]

Response2: [

Regarding the evolution of perturbations, we have already provided partial explanations in the paper. Perturbations originate from environmental white noise, such as air flow and base vibrations causing fiber jitter, which can affect the amplitudes and phases of the input arms of the photonic lantern. Additionally, instabilities in optical components, such as laser seeds and splitters, also contribute to these perturbations. Preliminary measurements in the laboratory indicate that phase jitter occurs at frequencies ranging from approximately 1 kHz to 10 kHz, while amplitude jitter occurs at frequencies ranging from about 1 Hz to 10 Hz. In simulations, we modeled amplitude perturbations with a variation of up to 20%, and phase perturbations with variations up to ±π radians. The preliminary measurements were tests conducted prior to the design of our experimental system. These tests were aimed at determining the design of the control system, and these test results have not been published.

]

Round 3

Reviewer 3 Report

Comments and Suggestions for Authors

I would like to thank the authors for their detailed response and clarifications.

Most of the concerns I raised during the review have been satisfactorily addressed. The mode-decomposition system, which is a central component of their approach, is now more clearly explained.

I have one final comment: the fact that this system provides only the value of the projected desired mode should be introduced earlier in the manuscript. It would be more appropriate to state this explicitly in the methodology section, as it directly justifies the use of the SPGD + hopping algorithm.

Author Response

Comment 1: [I have one final comment: the fact that this system provides only the value of the projected desired mode should be introduced earlier in the manuscript. It would be more appropriate to state this explicitly in the methodology section, as it directly justifies the use of the SPGD + hopping algorithm.]

Response 1:

[

Thank you very much for your final comment and for highlighting the importance of clarifying the system's capability to provide the value of the projected desired mode. We agree that this information is crucial and should be introduced earlier in the manuscript to better justify the use of the SPGD + hopping algorithm.

In response to your suggestion, we have revised the methodology section to explicitly state this aspect, which can be found in section 2.2. Thank you once again for your valuable feedback, which has helped improve the clarity and completeness of our manuscript.

]