1. Introduction

Quantum key distribution (QKD), a secure key sharing protocol [1,2], is vastly expanding in implementation, with development moving towards commercial realisation. To make this commercially possible, large-scale networks will need to be established. While fiber-based networks are being widely constructed [3–5], there are limitations to performance and practicality when scaling these up to global distances due to dispersion and losses over long distances. However, significant research into QKD protocols and low dispersion and loss optical fiber continue to push the achievable distance [6–8]. Free-space network links are seen as a complementary to optical fiber networking, able to achieve access to remote stations, mobile platforms [9–11], and a way to achieve practical global coverage using satellites [12,13].

Due to the multimodal nature of propagating through a free-space channel, receivers either implement adaptive optics to couple into single-mode fiber, [14], or utilize a free-space receiver and multi-mode fiber [15,16]. The choice between single or multimode coupling is driven by the choice of single-photon detector technology used [17], which also has a dependency on wavelength utilized for the implementation. Free-space coupled detectors are also an option, but typically suffer from high background noise, for example in daylight [18–20], or that have space envelope limitations between the receiver optics and the QKD receiver. The feasibility of utilizing free-space coupled 2D pixel arrays of single-photon detectors, which can post-process out spatial noise, has been demonstrated [21]. While promising, the 2D detector array technology is at a lower development stage in comparison to single-pixel detectors, specifically for QKD applications. Overall, coupling to optical fiber can reduce the complexity of the receiver system as well as ease optical alignment, as the incoming signal needs only be aligned to the receiver optics, which will be then carried through the system.

In free-space QKD, the use of single-photon avalanche diodes (SPAD) detectors is common due to their size, weight, and power consumption, particularly when the operational wavelength is in the visible or near-infrared where silicon SPAD technology can be utilized. Single-photon detectors are a big driver of the performance of QKD, with dependencies on parameters such as dark count rate, detection efficiency, after-pulsing, and timing response (typically presented as full-width at half-maximum (FWHM) time-jitter) [14].

The timing jitter for a SPAD is the delay between photon arrival time at the detector and the time it is absorbed by the detector and further read out by the electronics [22]. The timing jitter places a resolution limit of how fast a QKD system can be operated, as at high operational, or clock, frequencies, timing jitter limitations increase the probability that an incorrect photon will be recorded within a time-bin window [23,24]. The increase in cross talk between time-bins will increase the total quantum bit error rate (QBER) of the system. The QBER is a sum of several contributions which encompass the encoding, decoding, dark counts, and timing jitter [23]. This work is focusing on the timing jitter contribution and for clarity in this paper, any discussion of the timing jitter contribution to the QBER will be referred to as QBER_jitter.

The timing jitter response of SPADs have primarily focused on the use of single-mode fibers [23], due to the previous focus of the community on optical fiber implementations. It is known that the spot size and relative position of the spot on the active area of a SPAD impacts the timing jitter [25,26]. It is also understood that multimode fibers will allow for increased coupling efficiency, due to their bigger numerical apertures than single mode fibers. However, a quantitative study of the impact of multimode fibers on QBER of high repetition rate QKD has yet to be presented.

Another factor of fiber coupling is the effect of modal dispersion within the fiber [27], the larger the core, the more dispersion and the higher the losses occurred. The effects of modal dispersion can be improved by using fibers with graded-index profiles rather than step-index. Graded-index fibers have been used for telecommunications for this reason and we expect it to perform similarly at the single-photon level. The propagation for step and graded index fiber is highlighted in Fig. 1. A secondary property of graded-index fibers, the Kerr effect, has been recently studied, but still not completely understood [28,29]. Also known as spatial mode self-cleaning, this effect describes how over time or through induced stress on the fiber, higher order modes condense towards equilibrium, which results in a lower order mode profile that reduces the beam-profile and modal-dispersion of the signal through the fiber.

 

Fig. 1. The differences between a step-index fiber and a graded-index fiber. For the refractive index profile, n₀, n₁, n₂ are the refractive index of the fiber jacket, core and cladding respectively.

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It is unknown whether it is possible to compensate for the losses with using shorter fibers than standard, which will be part of the study here.

In this work, we have taken empirical measurements (1 MHz repetition rate) and simulated (1 GHz repetition rate) expected responses for a range of multimode fibers, both in core diameter and in length, to understand the impact on the timing jitter and QBER contribution of a single pixel, 180 µm SPAD. The results are generic for QKD protocols implementing single-photon detectors, and values can be used to seed many QKD protocol implementations.

2. Experimental setup

Understanding the impact of the timing jitter of the SPAD depending on the core diameter of multimode fibers is important to moving forward within the QKD field. By taking a selection of empirical measurements and using these to simulate responses and estimate the timing jitter for the SPAD, it will enable us to draw conclusions on the feasibility of multimode fiber-based coupling for QKD.

The light source for the study was a free-space Picoquant laser that emitted at a wavelength of 850 nm. The laser driver was electronically triggered by a pulse generator, at a repetition rate of 1 MHz, which then generated an electrical pulse to obtain an optical pulse with a FWHM pulse-width of <70 ps. The pulse generator also provided a synchronization pulse for a time-correlated single-photon counter (TCSPC). The laser was coupled to single-mode optical fiber to provide a common and easy to use interface for the experiments. After fiber coupling, an in-line optical fiber attenuator was used to control the incident photon count rate on the single-photon detector, down to the single-photon level. Following the attenuator, the various test optical fibers were then be connected to the photon source and single-photon detector. The set-up described here is outlined in Fig. 2.

 

Fig. 2. Set up for testing fibers with the SPAD. The laser pulse is generated using a pulse generator with a 10 MHz reference to the time correlated single-photon counter (TCSPC). The laser output is attenuated using a single mode (SM) fiber attenuator and passed through the test fibers (orange) that are coupled into the SPAD. The readout from the SPAD is measured with the time-tagger.

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The multimode optical fibers (step-index) tested in this experiment were 20, 50, 100 and 150 cm in length with corresponding core diameters of 10, 50, 105, 200 and 400 µm, respectively. Two graded index fibers of core diameter 50 µm and 62.5 µm were also tested in the four lengths described. For a comparison, four matching lengths of single mode fiber were tested. Due to the non-standard lengths of the fibers included in the study, they were custom made. In addition, a selection of commercial off the shelf (COTS) fibers of 1 m in length with 50 and 100/105 µm core diameters were tested to provide comparison to assess the performance of the custom fibers used in this work.

The single-photon detector used for these measurements was a single pixel fiber-coupled silicon SPAD (Excelitas SPCM-AQRH-12). The detector had a circular active area with a diameter of 180 µm, a dark count rate of 1000 cps, a detection efficiency at 850 nm of 65%, and a stated FWHM timing jitter of 350 ps at 825 nm. The module had a fiber connector port for coupling optical fibers to the detector, which contained a graded index interfacing lens.

A TCSPC module with a time-bin resolution of 1 ps was used to record the photon arrival times. This 1 ps is much smaller than the timing jitter of the SPAD itself. To set a reference point for measurements, the total summated count levels were attenuated to 10⁵ counts per second. Measurements were recorded for 30 seconds to take a statistical average.

To study the impact of the timing jitter change with the multimode optical fibers, we utilized methodology from [23], which maps the shape of the time-response, to take the 1 MHz data and simulate the impact at 1 GHz by mapping the data to that repetition rate. This method allowed analysis of the detector itself by ensuring consecutive pulses are not overlapping, which would create unreliable jitter results, without relying on narrow pulse-width GHz clock frequency capable equipment for the laser source. The methodology can be expanded to any repetition rate, but 1 GHz was chosen as a good representative of an achievable repetition rate for a commercial QKD system when utilizing silicon SPAD technology.

The analysis code was created in MATLAB, which incorporated the tail of the pulse as well as FWHM of the pulse, Fig. 3. The full timing responses were post-processed in MATLAB, which was used to quantify the QBER_jitter and total loss with assigned time-gating of the optical pulse. The QBER_jitter was estimated from the ratio of photons from pulses lying in the incorrect time bin and the photons from the pulse expected to be in the time bin of a defined gate width. Meanwhile the time-gating loss was the number of measured events that lie outside the defined gate width, which will be discarded during post-processing. These time-gating loss will reduce with a larger gate width, but it will also increase the QBER_jitter as it will include more of the overlapping pulses. This loss will be an important factor to when considering the trade-offs in performance when implemented in a full QKD system. The method is described in further detail in Section 6 – Appendix.

 

Fig. 3. Increasing time resolution as a function of the fiber core diameter – the diffusion tail widens with core diameter. (a) Time resolution for all fibers. Note the secondary pulses for the core diameters of 50 µm, these are back reflection artefacts within the experimental setup for fibers above 1 m in length. These have been disregarded as part of the study, as they do not impact the timing jitter significantly. (b) The time resolution for smaller core diameters that have been used including a single mode fiber (SM).

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3. Results

The results are presented in the following order. Firstly, the relationship between the length of the fiber and the timing jitter at the detector and the contribution to the QBER_jitter are discussed. Following this, the responses at the detector for the various fiber core diameters are reported, starting with the timing jitter of the full response and its QBER_jitter contribution and finishing with a comparison discussion between the fibers custom built for the work and commercial off-the-shelf (COTS) optical fiber patch cables. In all QBER results reported, these are the contribution to the QBER_jitter from the timing jitter alone, not factoring in dark counts or other parameters.

It was found that the timing jitter response at each length of fiber cable is similar for a given core diameter, seen in Fig. 4(a). This result shows the time response in this QKD application is independent of fiber length – which is to say that increasing the length of the fiber will not impact the response on the detector when the interest is having the single-photon detector displaced a short distance (less than a few meters) from the main free-space optical receiver. This is equivalent for both the timing jitter response, and the QBER_jitter contribution as seen in Fig. 4(b) and summarized for all core diameters in Table 1. Noticeably, the responses for the graded fibers have a shorter timing jitter and lower QBER_jitter contribution than was expected for their given core diameters.

 

Fig. 4. (a) Timing jitter for all core diameters across all four lengths of fiber. The red circle encircling the purple diamond notes the response from the 100 cm 62.5 µm graded fiber which had issues during the build process and currently lies as an outlier to the other results. The result set gives an indication that with a successful built, this fiber would follow the same trend. (b) Quantum bit error rate jitter (QBER_jitter) response with changing fiber length up to a length of 1.5 m. Across all core diameters, the length of the fiber remains independent of the response on the detector. Errors are based on the statistical average of three data points with each data point encapsulating 30 seconds of single photon detections.

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Table 1. QBER_jitter (%) contribution for all core diameters with increasing length of fiber. All errors were based on statistical average of 3 data points with each data point encapsulating 30 seconds of single photon detections, giving a value of below 1% for all table entries. The contribution at a given core size is independent of the length – this is visually represented in Fig. 4

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Due to the independency of the fiber length on the timing jitter contribution – only one length of fiber will be reported in this paper for clarity of the results. In comparison, the core-size and time-response have a direct dependence on the results, as a larger core typically leads to an increase in QBER_jitter.

Since the fibers have been manufactured in-house, measurements have been carried out using both ends of the fiber. This was to check for any unknown defects occurring during the build process. An indication of a poor build is noted in Fig. 4(a) and Table 1 with the 62.5 µm fiber of 100 cm length, as the response is significantly worse (red circle around purple diamond). The performance of the fibers using either end of the fiber is similar, so just one set of results will be reported in this paper. This is an average of the results measured from the selected connector.

The measured insertion loss for these fibers range between 0.2-2.7 dB with the average insertion loss of 1 dB. These losses are due to the construction of the fiber, but for the experiment, the input intensity has been attenuated so that the count rate on the detector was the same in all cases. As the diameter of the detector area of the Excelitas SPAD is 180 µm, the response for the larger core sizes (200 and 400 µm) may not be truly indicative of the QBER_jitter contribution as they are overfilling the detector, which also causes optical loss, separate from the insertion loss. A larger area detector would allow the response to be measured as it would no longer be overfilled.

Figure 5 shows time-gated QBER_jitter for a fixed length of 50 cm. The time-gate was centered around the peak of the time-response, capturing the majority of the time-response. The QBER_jitter does increase with time-gate width due to the increase in background noise, as more of the diffusion tail is captured and the cross talk of the time-responses at 1 GHz. Again, it is clear there is a dependence of QBER_jitter on increasing core size, seen in Fig. 5(a). However, the graded-index fibers produce a significantly lower contribution considering they are the largest core size suitable for the detector area, Fig. 5(b). The increased modal bandwidth provided by the graded-index core structure in comparison to the step-index equivalent will have an influence on this. This highlights a huge benefit in utilizing graded-index fibers for high repetition rate free-space QKD implementations.

 

Fig. 5. (a) Quantum bit error rate jitter (QBER_jitter) contribution with increasing time-gate for all fiber core sizes over a fixed fiber length of 50 cm. (b) QBER contribution for the smaller core diameters including the graded-index fibers (5-62.5 µm).

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The full timing jitter response of the detector has an equivalent behavior to that seen with the QBER_jitter contribution. The timing jitter across all the core diameters is summarized numerically in Table 2. To cover the full response, including the diffusion tail, the full-width at 10^th-maximum (FW10M) and the full-width at 100^th-maximum (FW100M) are reported alongside the characteristic full-width half-maximum. It can be seen that as the core diameter increases, so to do the parameters. However, while the FWHM for graded fibers does broaden, the FW10M and FW100M are narrower than the equivalent step index fiber, which highlights why the QBER_jitter contribution is less for those fibers, due to less optical cross talk between adjacent time-bins.

Table 2. Timing jitter for all core diameters. All errors were based on statistical average of 3 points with each data point encapsulating 30 seconds of single photon detections, giving a value of below 50 ps for all table entries

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A secondary check of standard for the performance of the fibers that have been produced for the work is to directly compare them against COTS equivalents. Ideally, all graded-index core diameters would be compared with the step-index cores, however 62.5 µm core diameters are only available in graded-index. This has meant that only the 50 µm and 100 µm can be compared in this case. In Table 3, the QBER_jitter contributions are seen to be similar.

Table 3. QBER_jitter contribution and timing jitter at detector for fibers with 50 µm and 100 µm core diameters. Unless stated as graded, the fibers are step-index. *This fiber suffered from fabrication issues which prevented the results from being directly compared in the wider results

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A variation on the QBER_jitter contribution and timing jitter is to be expected between fibers of the same core diameter, due to coupling losses, manufacturing processes and potential impurities in the material. However, a marked improvement on performance is noted when switching from step-index to graded-index for the 50 µm core as outlined in Table 3.

The graded-index 100 µm core from Coherent would be expected to be better than the step-index cores, however, due to a mismatch of fiber cladding diameter and ferrule bore size during construction, the core lies away from the centre of the connector ferrule increasing coupling losses and spot position on the detector. A visual inspection of the mismatch is shown in Fig. 6. The broader response from the spatially offset 100 µm core matches up with previous literature that highlights the timing jitter has dependence on the spot position [30].

 

Fig. 6. Evidence of core diameter and ferrule bore size mismatch, which will increase the coupling losses significantly for this fiber, negating performance of the 100 µm graded-index fiber. The depreciation in the performance is noticeable in Table 3.

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4. Discussion

The larger core diameter is a concern for the timing jitter of the SPAD as the long diffusion tail of the recorded pulse becomes larger as the diameter of the core increases due to modal dispersion and the resultant spot size on the SPAD active area. The use of graded-index cores, due to their larger modal bandwidth, lower modal dispersion, results in a significant increase in performance in comparison to their step-index core diameter equivalent. The unique property of the Kerr effect in graded-index fibers will also contribute towards the increase in performance, but to what degree is unknown. Graded-index fibers are used in telecommunication networks as they provide higher data rates than step-index fibers. These results show that using this fiber technology in a single-photon setting can remove limitations associated with multimode fibers in high repetition rate QKD settings. Specific to free space implementation, utilizing larger core optical fibers will also allow for greater coupling efficiency than can be achieved with single mode fibers due to the larger apertures so a larger acceptance angle into the fiber. The multimode fibers also reduce the reliance of high specification adaptive optics, creating a more affordable solution.

5. Conclusion

To achieve sufficient secret key rates, operating at high operational frequencies are increasingly important for many QKD scenarios, particularly in free-space applications where the channel loss is highly variable and communication windows are often time limited. In addition, to make QKD commercially viable, more cost-effective design solutions are needed in system design to improve or maintain performance.

Single-photon detectors play a significant role in the overall performance of QKD. The optical interface between the quantum receiver and the single-photon detectors, while studied for optical fiber based QKD, has yet to be analyzed for free-space applications, where multimode fibers will typically be used.

This paper presents an experimental analysis on the impact of utilizing multimode optical fibers for free-space QKD in high repetition rate (∼GHz) scenarios. The results focused on the contribution of the timing jitter full response to the QBER.

In our application of utilizing a short length (<2 m) of multimode fiber, it was found that the QBER_jitter contribution increased with core diameter but was relatively independent of length of fiber. Notably, the graded-index fibers have a lower QBER_jitter contribution than their step-index counterpart. The graded-index fiber performance is also reflected in the timing jitter response at the detector, which tracks along the tail of the pulse, encompassing the full response. Across COTS fiber cables, the response is similarly presented with an indication that a larger 100 µm graded-index fiber could, in theory, present a better response than its step-index equivalent and provide a larger core diameter for coupling. However, it must be considered that these larger cores will also couple more background noise.

The results highlight that there is an opportunity to utilize larger multimode fiber core diameters to significantly reduce the costs of QKD receivers by reducing requirements on adaptive optics and pointing and tracking. In high repetition rate scenarios, the use of graded-index cores is critical in the system design. This understanding will help to develop fiber-coupled based alternatives at the receiver level which can improve alignment and provide simplifications to the design.

The next step for the work is to gain an understanding of the multimode fiber impact on larger area detectors and the contributions to the QBER given misalignments of the input beam. This will aid the development of technologies to improve the link efficiency of a QKD receiver by specifying requirements for pointing and tracking systems for optical ground stations. The fiber-coupled alternative can allow for developments within daylight QKD and in the future, allow for implementation in commercial networks.

6. Appendix

The data manipulation noted in section 2 is described in more detail here. It is based on the methodology used in [23].

Firstly, the measured histogram for the pulse is converted to have a suitable time base. With a 1 ns pulse, the time base must have at least 10 ns to incorporate 10 pulses – the time base has been chosen to be 15 ns to incorporate all core diameters, as the pulse widths will vary with core diameter.

Once the time base is defined, 10 detections (pulses) are read into the function and each consecutive pulse is delayed with respect to the preceding pulse by a factor of the reciprocal of the frequency that the user is taking measurements at (a 1 GHz frequency, 1 ns delay between pulses). The center pulse is selected with a defined gate width, seen in Fig. 7 as the black lines.

 

Fig. 7. Application of the gate width (black lines) to the pulse readout. This gives an indication of the contribution from the other overlapping time bins (only 1 plotted here) within the gate width.

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The jitter contribution to the QBER and the associated losses are calculated by working out the sum of everything that falls within the defined gate width and considered whether they are correct or incorrect detections. A correct detection is anything contributed from the time bin that lands in the center of the gate width (i.e., the orange pulse), as it is considered the peak of interest and anything else lying in the gate width which is a contribution from other time bins will be considered incorrect (the yellow pulse).

The losses in the gate width and the jitter contribution to the QBER are estimated using the following functions.

(1)$$Losse{s_{gate\; width}} = 1 - \frac{{detections\; in\; gate\; width}}{{\sum all\; detections\; }}$$

(2)$$QBE{R_{jitter}} = \; \frac{{wrong\; detections}}{{\sum all\; detections\; in\; gate\; width}} \times 0.5$$

The QBER_jitter estimation includes a factor of 0.5 since for a noisy count in QKD, there is a 50% chance of it being correct [23]. The losses and the time-jitter contribution to the QBER have been estimated across all core sizes with an increasing gate width.

These results will then give an indication of the ideal gate width when the QBER_jitter contribution is suitably sized, and the losses are around 3 dB. This is an important trade-off to consider for implementation into a QKD system. Figure 8 highlights that as the gate width decreases, the QBER_jitter will decrease, however the losses from the gate width will increase, impacting the systems loss budget.

 

Fig. 8. Losses and QBER_jitter for a system running at 1 GHz repetition rate. Setting the gate width will affect bot the loss and the QBER_jitter.

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Funding

Engineering and Physical Sciences Research Council (EP/L01596X/1); Innovate UK (TS/S009353/1); Royal Academy of Engineering (RF\201718\1746).

Acknowledgments

We would like to acknowledge the EPSRC Centre for Doctoral Training in Applied Photonics.

Disclosures

The authors declare no conflicts of interest

Data availability

Data underlying the results presented in this paper are available in Ref. [31].

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