Optimization of Voltage Requirements in Electro-Optic Polarization Controllers for High-Speed QKD Systems

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

In the submitted manuscript, the authors presented a framework to optimize the voltage range of Electro-optic Polarization Controllers (EPC) in polarization-based Quantum Key Distribution (QKD) subsystems. The simulation is also conducted, analyzing voltage requirements around bias points and around zero volts, as well as the maximum voltage swings for various EPC configurations. I find that the manuscript is well written, it can be recommended for publication in Photonics after the following questions are satisfactorily addressed.

1. For the application of the polarization control in the polarization - encoded QKD system, it mainly involves two parts: the encoders/decoders and channel polarization compensation. Different application scenarios have different limiting conditions for the practical and effective application of this algorithm. It is recommended that the author supplement the following relevant content. (1) Research progress on the stability of the QKD polarization encoders: Currently, the research on the stability of the QKD polarization encoders is of great significance. Supplementing this content can enable readers to have a more comprehensive understanding of the developments in this field, clarify the current research status of encoders stability, and its important influence on the entire polarization-encoded QKD system. Some references are recommended to be considered for citation. (Opt. Express., vol. 24, no. 8, pp. 8302–8309, Apr. 2016, doi: 10.1364/OE.24.008302;Optica, vol. 7, no. 4, pp. 284–290, Apr. 2020, doi: 10.1364/Optica.381013; IEEE PHOTONICS JOURNAL, VOL. 15, NO. 5, OCTOBER 2023; ……). (2) Analysis of the reasons for system performance degradation in practical applications for the proposed optimize method: For different application scenarios, such as the calibration of the QKD encoders and the polarization compensation of the optical fiber channel, based on the simulation results in this paper, it is necessary to deeply explore the reasons that may lead to system performance degradation in practical applications. This will help readers better evaluate the feasibility and potential problems of this method in practical applications.
2. There are many factors influencing the encoding/decoding rate and the polarization compensation rate for the entire QKD system. Merely optimizing the voltage does not necessarily increase the rate. It is recommended to appropriately summarize the reasons for increasing the rate in view of various technical requirements. The influence of other factors should also be properly mentioned.

Author Response

Q1.1: “Research progress on the stability of the QKD polarization encoders: Currently, the research on the stability of the QKD polarization encoders is of great significance. Supplementing this content can enable readers to have a more comprehensive understanding of the developments in this field, clarify the current research status of encoders stability, and its important influence on the entire polarization-encoded QKD system. Some references are recommended to be considered for citation. “

Response: We thank the reviewer for bringing attention to this important topic. Indeed, the stability of polarization encoders in QKD systems is critical for maintaining consistent key generation rates and overall system performance.

Following the reviewer’s suggestion, we have added the recommended citation by Wang et al. (2016) in line 39, next to the iPognac implementation citation, to provide additional context regarding inherently stable polarization-modulated units. We also evaluated the reference by van Rees et al. (2023) but decided not to include it, as it focuses primarily on frequency stabilization, which we consider slightly tangential to the discussion on polarization encoding stability.

Page 2 of the manuscript:

“The most widely used techniques employ phase modulators [6, 14, 15], whose key advantage is the potential to achieve encoding/decoding speeds in the tens of GHz range, far exceeding the physical limitations of current single-photon detector (SPD) technology [16,17].”

Q1.2: “Analysis of the reasons for system performance degradation in practical applications for the proposed optimize method: For different application scenarios, such as the calibration of the QKD encoders and the polarization compensation of the optical fiber channel, based on the simulation results in this paper, it is necessary to deeply explore the reasons that may lead to system performance degradation in practical applications. This will help readers better evaluate the feasibility and potential problems of this method in practical applications.”

 

Response: We thank the reviewer for this comment. The proposed optimization algorithm is designed to determine the best voltage combinations for each of the three SOP bases that minimize the overall voltage range required for operation. In the simulation and practical environment, the algorithm operates using only “alpha” and “delta” parameters, assuming a well-defined input polarization after an initial calibration stage.

 

In practical applications, several factors may lead to system performance degradation. Drifts in electrical and environmental conditions such as temperature fluctuations, mechanical strain, and electrical noise can affect the stability of the LiNbO3-based modulators [1], leading to polarization drift over time. These factors may introduce variations in the applied voltages, requiring periodic recalibration to maintain optimized performance

Additionally, the proposed method does not include an active polarization drift compensation mechanism, meaning that external polarization fluctuations, such as those caused by fiber birefringence changes over time, are not corrected within this optimization framework. Since polarization drift primarily affects the receiver side, additional compensation methods are required to maintain system stability in long-distance optical fiber channels. Without such compensation, uncontrolled polarization variations could degrade the quality of the received signal, impacting the overall performance of the QKD system.

To address this point, additional discussion has been included in Section 5, Discussion and Conclusions:

 

“While the optimization method provides an efficient way to determine the best voltage settings, it does not incorporate an active polarization drift compensation mechanism. This means that external polarization fluctuations, such as those caused by fiber birefringence changes over time, are not corrected within this optimization framework. In long-distance optical fiber channels, these dynamic variations could impact system stability on the receiver side, requiring additional real-time compensation methods. Separately, long-term environmental fluctuations, including temperature variations, mechanical strain, and electrical noise can cause voltage drift in LiNbO3-based modulators [31], necessitating periodic recalibration. In practical applications, this issue must be addressed through active feedback control to maintain stable performance.”

 

[1] - J. Salvestrini, L. Guilbert, M. Fontana, M. Abarkan, and S. Gille, "Analysis and Control of the DC Drift in LiNbO3-Based Mach–Zehnder Modulators," J. Lightwave Technol.  29, 1522-1534 (2011).

 

Q2: “There are many factors influencing the encoding/decoding rate and the polarization compensation rate for the entire QKD system. Merely optimizing the voltage does not necessarily increase the rate. It is recommended to appropriately summarize the reasons for increasing the rate in view of various technical requirements. The influence of other factors should also be properly mentioned.”

Response: We thank the reviewer for highlighting this important point. While optimizing the voltage range for each wave-plate can help improve the system’s efficiency, it does not necessarily translate to a higher encoding/decoding rate. The QKD system performance is influenced by multiple factors beyond voltage optimization, including synchronization constraints, attenuation, polarization drift compensation, and SPD configurations. These elements define the system’s practical limits and must be considered when evaluating potential improvements.

To address this topic and improve the manuscript, we have added the following clarification in Section 3, page 4:

“This distribution reduces the voltage range required for each wave-plate, leading to smaller voltage swings between the target SOPs. Lower voltage swings reduce electronic power demands and allow for faster response times, potentially enabling higher encoding and decoding frequencies for the polarization-based QKD subsystem. However, the maximum achievable rate is also influenced by other system factors such as synchronization constraints, attenuation, polarization drift compensation, and SPD configurations, which together define the overall system performance”

Author Response File: Author Response.pdf

Reviewer 2 Report

Comments and Suggestions for Authors

This manuscript demonstrates the ability to minimize voltage ranges and swings while maintaining high state of polarization accuracy by using multi-objective optimization algorithms and reducing the number of variables through waveplate splitting. The results presented are of great significance for the development of Quantum Key Distribution system.  They have the potential to increase the modulation speed, reduce hardware complexity, and improve system efficiency. The conclusion is clear and can be well supported by the simulation results. Here are some questions listed as below:

Q1: According to the product specification (https://www.eospace.com/polarization-controller), EOSPACE polarization controller has eight wave-plates, is it possible to further decrease the absolute maximum voltage deviation or swing if more wave-plates are used?

Q2: Multi-objective optimization methods have high computational complexity and are time-consuming. How long does it typically take for different tasks in the work? What is the convergence behavior of the model?

Q3: The results in the manuscript were obtained through simulation. Is it possible to fully or partially validate the simulated conclusion via experiments?

Q4: There are still some typos, for example the error in Line 389. Could the author go through the whole manuscript to remove them?

Comments on the Quality of English Language

The English is overall good. But there is still room to improve, in terms of typos and logic.

Author Response

(view uploaded pdf)
Q1: “According to the product specification (https://www.eospace.com/polarization-controller), EOSPACE polarization controller has eight wave-plates, is it possible to further decrease the absolute maximum voltage deviation or swing if more wave-plates are used?”

Response: We thank the reviewer for this question. According to the manufacturer’s datasheet, Vπ is the voltage required for a 180-degree phase shift between two orthogonal polarization modes in a single wave-plate, whereas V0​ provides a complete transfer of optical power from one mode to the other. Under the current design, the ratios 2xV0/n and Vπ/n, where n is the number of wave-plates, ​​ remain close to 10, regardless of the number of wave-plates. Consequently, increasing the number of wave-plates while maintaining these proportions does not lead to any additional reduction in the overall absolute voltage deviation or swing. Thus, under the current design, adding more wave-plates alone does not minimize the encoder/decoder voltages further.

Q2: “Multi-objective optimization methods have high computational complexity and are time-consuming. How long does it typically take for different tasks in the work? What is the convergence behaviour of the model?”

Response: We thank the reviewer for this question. The computational complexity of the multi-objective optimization (MOO) process is directly influenced by parameters such as the number of independent variables, which ranges from 2 for a single wave-plate to 16 for an eight-wave-plate setup. The use of NSGA-II as the optimization algorithm is motivated by the multi-objective nature of the problem and allows us to compare different independent wave-plate configurations efficiently.

To evaluate the performance of the optimization process, we conducted simulations using the following hardware setup:

• CPU: Intel Core i7-6800K @ 3.4 GHz
• RAM: 32 GB
• GPU: Nvidia GeForce GTX 1050

For equivalent starting points and proportional models, we used a population size of 200 for configurations with four or fewer wave-plates and 400 for setups with more than four wave-plates. The convergence behavior was monitored based on the number of generations required to reach stable solutions, with an Objective Space Tolerance (ftol) from pymoo defining the breakpoint at < 0.01.

The table below summarizes the average computational time to obtain each of the six SOP targets, with an input SOP defined as  for different wave-plate configurations:

 

Wave-plates (n)	Time Taken (s)	Generations	Evaluations	Non-Dominated Solutions
1	15.73	86.67	17133	200
2	38.82	390.33	77867	120.5
3	59.57	564	112600	72
4	94.58	863.67	172533	48.67
6	171.1	379.17	151267	26.33

From these results, we observe that:

• Computation time increases with the number of wave-plates, as expected due to the increase in independent variables.
• The convergence behavior varies depending on the SOP target and wave-plate configuration. Cases with higher numbers of variables typically require more generations and function evaluations to reach an optimized solution.
• The number of non-dominated solutions (NDS) is generally lower for higher-stage models, as more constraints impact the search space.

To address this topic and improve the manuscript, we have added the following information in Section 3.2, Algorithm Implementation:

“The population size is set to 200 if the current calculation only involves 4 wave-plates or lower, and 400 if it involves a higher number of wave-plates. This approach provides a reasonable trade-off between exploration and computational efficiency while accounting for the increased number of optimization variables as wave-plate count increases. As a result, computational time scales accordingly, with the optimization for 6 wave-plates requiring an average of 171.1 seconds, while 4 and 2 wave-plate configurations require 94.58 and 38.82 seconds, respectively, when running on an Intel Core i7-6800K @ 3.4 GHz CPU, 32 GB of RAM, and an Nvidia GeForce GTX 1050 GPU. The constraints in NSGA-II, as well as in ‘pymoo‘, are handled using a constraint dominance approach [26].”

 

Q3: “ The results in the manuscript were obtained through simulation. Is it possible to fully or partially validate the simulated conclusion via experiments?”

 

Response: We thank the reviewer for this comment. We have acquired both a six-wave-plate and an eight-wave-plate polarization controller, and are finalizing the development of a driver capable of generating voltages within a ±70 V range. This setup will allow us to perform experimental tests to validate the main conclusions from our simulations, at least in part, under controlled laboratory conditions.

 

Q4: There are still some typos, for example the error in Line 389. Could the author go through the whole manuscript to remove them?

Response: We thank the reviewer for the opportunity to correct the grammatical errors. We have carefully reviewed the entire manuscript to eliminate any remaining errors, including the one on Line 389.

Author Response File: Author Response.pdf

Reviewer 3 Report

Comments and Suggestions for Authors

This paper presents an outstanding contribution to the field of quantum key distribution (QKD) by offering a well-structured and innovative approach to optimizing the voltage range requirements of electro-optic polarization controllers (EPCs). The authors provide a thorough and technically rigorous framework for reducing voltage swings and electronic demands while maintaining precise polarization state control, which is crucial for high-speed QKD implementations.

One of the key strengths of this work is its comprehensive analysis of multi-wave-plate EPCs and the systematic application of multi-objective optimization (MOO) techniques, particularly the Non-dominated Sorting Genetic Algorithm (NSGA-II). The authors effectively demonstrate how voltage ranges can be minimized while ensuring reliable state-of-polarization (SOP) transformations, significantly improving the efficiency of polarization-based QKD systems. The use of wave-plate splitting techniques is another noteworthy innovation, allowing for a reduction in independent variables while maintaining high optimization accuracy.

1. It is best to revise the introduction to include recent advancements in the larger field of quantum networks to appeal to the broader community, such as:
   
   -  "The quantum internet: A synergy of quantum information technologies and 6G networks." IET Quantum Communication 4.4 (2023): 147-166.

• "Towards real‐world quantum networks: a review." Laser & Photonics Reviews 16.3 (2022): 2100219.

2. Can you give a full description of the physical details in the caption of Figure. 1?
3. The analysis indicates that for a 6 SOP implementation, (HL) or (DL) are optimal input polarization states, while for a 4 SOP setup, a circular basis state provides the best results. Could you elaborate on the underlying physical or mathematical reasoning why these specific input SOPs minimize voltage deviations compared to others?
4. In page 8., the section discusses how unused wave-plates, when left unconnected or connected to ground, introduce additional polarization modulation, leading to variations in the relative magnitudes of each input SOP across different wave-plate configurations. This effect contrasts with cases where unused wave-plates exhibit zero birefringence. Would it be possible to develop a systematic compensation method to counteract the unintended polarization modulation caused by unused wave-plates? If so, how would such a compensation scheme impact the optimization process and the required voltage ranges? It would be interesting to discuss this in this section.
5. Consider adding a brief statement on potential future research directions in the conclusion.

 

Author Response

Q1: “It is best to revise the introduction to include recent advancements in the larger field of quantum networks to appeal to the broader community, such as:”

Response: We thank the reviewer for this valuable suggestion. In order to provide a broader context for our work, we have revised the first paragraph of the Introduction to include the second suggested reference “Towards Real-World Quantum Networks: A Review”:

“As advances in quantum computing pose future risks to an important part of currently employed encryption algorithms [1], Quantum Key Distribution (QKD), together with quantum networks [2], has emerged as a promising solution for secure key distribution.”

Q2: “Can you give a full description of the physical details in the caption of Figure. 1?

Response: We thank the reviewer for this request. We have revised the caption of Figure 1 to include further physical details about the polarization controller:

“Figure 1. Schematic representation of the controller waveguide, x-cut on LiNbO₃ substrate with a titanium-diffused channel waveguide in the z-direction, featuring an independent electrode configuration (VA, VB, VC).”

Q3: The analysis indicates that for a 6 SOP implementation, (HL) or (DL) are optimal input polarization states, while for a 4 SOP setup, a circular basis state provides the best results. Could you elaborate on the underlying physical or mathematical reasoning why these specific input SOPs minimize voltage deviations compared to others?

Response: We thank the reviewer for this insightful question. The observed behaviour is closely linked to the nonlinear nature of polarization modulation, described by the Müller matrix formalism, and the voltage equations, used to connect the wave-plate characteristics with VA and VC​, which exhibit sinusoidal behaviour.

For cases where V0 and Vπ​ maintain the proportion 2xV0=Vπ, the peaks of the sinusoidal variations in VA and VC occur at π/2 intervals, with an initial shift of π/4. This leads to distinct results when comparing implementations with different wave-plate counts.

For example, in a one-wave-plate implementation starting from (H):

• Achieving (D) or (A) requires δ=π with α=π/4 and α=π/4+3π/2, both of which align with the peaks of the VA and VC ​ functions.
• In contrast, achieving (V) requires δ=π and α=3π/2, which does not align with a peak.

This means that for a six-SOP implementation, the transitions to (D),(A) represent the limiting factor in a one-independent wave-plate system. However, when an additional independent wave-plate is introduced, more parameter combinations become available. In this case, a two-independent wave-plate configuration allows for:

• δ1=0.46π, δ2=0.44π (with their sum remaining close to π)
• α1=1.36 rad, α2=0.17 rad

This alternative α-combination helps avoiding sinusoidal peaks, equalizing the voltage range required to obtain (D),(A) with that of (V), leading to a 28% improvement. Notably, this improvement would go unnoticed in four-SOP or three-SOP implementations, as the problematic transitions are not included. This 28% gain does not appear when the input SOP is (HD), the case presented in the paper. Instead, using the (HD) SOP input results in a 38% improvement when transitioning from one independent wave-plate to two or more independent wave-plates for three-SOP combinations. This occurs because the optimization algorithm finds better α values for (HD)→(H),(D) transitions, when using two-independent wave-plates compared to one.

 

In summary, the optimization of input SOPs and waveplate configurations plays a crucial role in minimizing voltage deviations. Our findings indicate that using two or more independent waveplates provides sufficient flexibility to optimize transitions while maintaining a balanced voltage range. While specific input SOPs may offer advantages for certain transitions, under certain wave-plate counts, their impact diminishes when we require complete SOP basis. Additionally, the sinusoidal dependence of ?? and ?? on the ratio ??/?0 suggests that modifying this proportion shifts peak α  values, potentially enhancing performance for different input SOPs. The interplay between input SOP selection, waveplate count, and nonlinear voltage behavior underscores the need for a holistic optimization approach to achieve minimal voltage swings across all required SOP transitions.

To address this, the following paragraph has been added to Section 5, Discussion and Conclusion:

 

“The input SOP plays a crucial role in determining the overall voltage range required for modulation. In this work, we assume a worst-case scenario for the input SOP, how- ever, better-aligned starting points can reduce the required voltage range by up to 60%, depending on the SOP combination. The SOP dependency does not arise from the physical characteristics of the device but is influenced by them. In particular, the ratio between V0 and Vπ affects the sinusoidal behavior of VA and VC, shifting the corresponding α values at function peaks and thereby influencing the preferable voltage settings for a given α, δ pair. Consequently, the preferred input SOP varies based on the number of SOPs required and the number of independent wave-plates included in the implementation. Furthermore, the impact of input SOP selection is more pronounced when fewer wave-plates are available, as the system has fewer degrees of freedom to adjust voltage settings. Our optimization frame- work aims to find the best voltage combinations for each of the three SOP bases to minimize the required voltage range. Specifically, our results indicate that achieving all six SOPs requires a voltage range of ±6 V when centering operations around bias voltage points and ±9.5 V when centering around zero volts. Additionally, the maximum voltage swing between SOP combinations was evaluated, yielding a value of 6.8 V. Another key result is that reducing the number of independent wave-plates to two (four variables) achieves performance comparable to models with higher independent wave-plate counts. This reinforces the effectiveness of optimizing control parameters while maintaining minimal complexity, ensuring efficient system performance. “

 

Q4: “In page 8., the section discusses how unused wave-plates, when left unconnected or connected to ground, introduce additional polarization modulation, leading to variations in the relative magnitudes of each input SOP across different wave-plate configurations. This effect contrasts with cases where unused wave-plates exhibit zero birefringence. Would it be possible to develop a systematic compensation method to counteract the unintended polarization modulation caused by unused wave-plates?

If so, how would such a compensation scheme impact the optimization process and the required voltage ranges? It would be interesting to discuss this in this section..”

Response: We thank the reviewer for this insightful question. The unintended polarization modulation caused by unused wave-plates can be effectively mitigated by setting them to their associated bias values, which are determined through a calibration procedure [2]. This ensures that the wave-plates remain at their zero-birefringence points, preventing unwanted modulation effects.

Implementing such a systematic compensation method does not significantly impact the optimization process itself, as the bias values are pre-determined and incorporated into the voltage settings before the optimization begins. The reason for including these cases is to illustrate how previous optimizations might no longer be valid when wave-plates are left floating or grounded, as this alters the system's response.

To address this, the following has been added to Section 5, Discussion and Conclusion:

“The bias-centered approach reduces the required modulation voltage but requires the driver to operate around offset voltages, which may be impractical in some hardware configurations. In contrast, the zero-centered approach ensures that unused wave-plates remain at zero voltage but introduces unwanted modulation effects. A potential solution for this issue involves performing a calibration step to identify the exact zero-birefringence points of the unused wave-plates, thereby reducing unintended modulation shifts.  “

[2] - Xi, L.; Zhang, X.; Tian, F.; Tang, X.; Weng, X.; Zhang, G.; Li, X.; Xiong, Q. Optimizing the operation of LiNbO3-based multistage polarization controllers through an adaptive algorithm. IEEE Photonics J. 2010, 2, 195–202.  

 

Q5: “Consider adding a brief statement on potential future research directions in the conclusion.”

Response: We thank the reviewer for this suggestion. In the final paragraph of the conclusion, we now highlight our intention to experimentally validate this framework through a polarization-based feedback loop. This addition outlines our commitment to translating the theoretical findings into practical implementations.

To address this topic and improve the manuscript, we have added the following information to the end of Section 5, Discussion and Conclusion:

“These findings will contribute to the widespread adoption of efficient polarization control in polarization-based QKD subsystems. Future research will focus on implementing the optimization algorithm in a real-time polarization feedback loop and assessing its performance in practical QKD systems.”

Author Response File: Author Response.pdf