Letter | Published:

# Security and eavesdropping in terahertz wireless links

## Abstract

Resiliency against eavesdropping and other security threats has become one of the key design considerations for communication systems. As wireless systems become ubiquitous, there is an increasing need for security protocols at all levels, including software (such as encryption), hardware (such as trusted platform modules) and the physical layer (such as wave-front engineering)1,2,3,4,5. With the inevitable shift to higher carrier frequencies, especially in the terahertz range (above 100 gigahertz), an important consideration is the decreased angular divergence (that is, the increased directionality) of transmitted signals, owing to the reduced effects of diffraction on waves with shorter wavelengths. In recent years, research on wireless devices6,9,8 and systems9,12,11 that operate at terahertz frequencies has ramped up markedly. These high-frequency, narrow-angle broadcasts present a more challenging environment for eavesdroppers compared to the wide-area broadcasts used at lower frequencies12,13. However, despite the widespread assumption of improved security for high-frequency wireless data links14,15,16, the possibility of terahertz eavesdropping has not yet been characterized. A few recent studies have considered the issue at lower frequencies5,6,,12,13,17,18, but generally with the idea that the eavesdropper’s antenna must be located within the broadcast sector of the transmitting antenna, leading to the conclusion that eavesdropping becomes essentially impossible when the transmitted signal has sufficiently high directionality15. Here we demonstrate that, contrary to this expectation, an eavesdropper can intercept signals in line-of-sight transmissions, even when they are transmitted at high frequencies with narrow beams. The eavesdropper’s techniques are different from those for lower-frequency transmissions, as they involve placing an object in the path of the transmission to scatter radiation towards the eavesdropper. We also discuss one counter-measure for this eavesdropping technique, which involves characterizing the backscatter of the channel. We show that this counter-measure can be used to detect some, although not all, eavesdroppers. Our work highlights the importance of physical-layer security in terahertz wireless networks and the need for transceiver designs that incorporate new counter-measures.

## Main

Wireless networking is on the cusp of a revolution. For more than 100 years, wireless links have relied on wide-angle broadcasts, using transmit and receive antennas with gains that are relatively insensitive to the angle of emission or reception (and therefore with relatively low directivity). With the roll-out of 5G cellular mobile communications systems, this approach will soon change to an entirely new one, in which highly directional (and steerable) antennas provide links that are more like directed beams than like omnidirectional broadcasts2,19. This change is an unavoidable consequence of the move to higher carrier frequencies, a necessary step for increased bandwidth and higher rates of data transfer. There are numerous advantages to using more directional channels, including improved data security. Here, we focus on the new challenges faced by an eavesdropper when communication channels become directional5,18, with a beam divergence angle much smaller than that used by existing mobile networks, which often use 120° sectors20.

Security mechanisms are available at every layer of a network, and can be used jointly across layers for redundancy or in a subset of layers when resources are constrained. These mechanisms can take many forms, including encryption and authentication at the upper layers2,3, as well as physical-layer techniques such as wave-front engineering, near-field antenna modulation and polarization multiplexing4,5,21,22. Physical-layer approaches have some advantages: they do not require a shared private key, they use little or no additional computing resources23 and they do not rely on the assumption that the attacker has limited computational power. In the terahertz frequency range, numerous researchers have envisioned the need for physical-layer security14,15,16,24. Highly directional beams and increased atmospheric attenuation will confine unauthorized users to be on the same narrow path as the intended user if they wish to intercept the signal. As a result, it is often assumed that terahertz signals are more secure than lower-frequency signals: a more directional transmission sends energy to a smaller range of locations, so it is more difficult for an eavesdropper to place a receiver that detects the signal without blocking the intended recipient and thereby raising an alarm. The equipment needed to collect, demodulate and amplify terahertz signals is large (always larger than the aperture of the detector) and bulky, so blockage would always be a concern.

Although this argument is reasonable for conventional eavesdropping attacks, it does not consider alternative approaches that could circumvent the blockage problem and enable a successful attack. In our measurements, we consider a different approach for the eavesdropper (Eve). Rather than adopting the conventional assumption that Eve must place a large bulky receiver within the narrow beam path5,13,17,18, we instead consider the possibility that she can place a smaller passive object in the beam that will scatter some of the transmitted radiation towards her receiver, which is located elsewhere12. This set-up affords Eve considerable additional flexibility, and can enable successful eavesdropping even at high frequencies with very directional beams.

We assume a line-of-sight configuration connecting a single transmitter (Alice) and a single receiver (Bob), as is standard for a highly directional millimetre-wave or terahertz wireless link through the air25,26,27 (see Methods section ‘Radiation patterns for directional horn antennas’). In our scale model data link (Fig. 1), we position objects at various locations in the beam between Alice and Bob, and evaluate the signal strength and bit error rate detected by Eve at various receiver locations. Eve’s goal is to choose a scattering object, and its location, in such a way that the signal measured by Bob is not attenuated too much (otherwise, Bob might detect the attack) and the signal that she measures is large enough for her to intercept the communication. This corresponds to a successful eavesdropping configuration. To quantify these statements, we define the attenuation of Bob’s signal due to the scattering object, which we refer to as the blockage, as:

$$b=1-\frac{{{\rm{SNR}}}_{{\rm{Bob}}}^{{\rm{object}}}}{{{\rm{SNR}}}_{{\rm{Bob}}}^{{\rm{no\; object}}}}$$

where SNR represents the signal-to-noise ratio on a linear scale. A value of b = 0.5 would correspond to a situation in which Eve’s scattering object is blocking half of Bob’s signal. This arbitrary value may be considered a threshold beyond which Bob is certain to be aware of the change in the characteristics of his line-of-sight channel. Further, as a small modification to the conventional approach28, we define a normalized secrecy capacity, which relates the strength of Eve’s signal to Bob’s signal:

$${\bar{c}}_{{\rm{s}}}=\frac{{\rm{l}}{\rm{o}}{\rm{g}}(1+{{\rm{S}}{\rm{N}}{\rm{R}}}_{{\rm{B}}{\rm{o}}{\rm{b}}})-{\rm{l}}{\rm{o}}{\rm{g}}(1+{{\rm{S}}{\rm{N}}{\rm{R}}}_{{\rm{E}}{\rm{v}}{\rm{e}}})}{{\rm{l}}{\rm{o}}{\rm{g}}(1+{{\rm{S}}{\rm{N}}{\rm{R}}}_{{\rm{B}}{\rm{o}}{\rm{b}}})}$$

This quantity incorporates the particular modulation and coding methods used and characterizes the empirical limits of Bob’s and Eve’s reception capabilities. It is equal to unity if Eve receives no signal and to zero if Eve and Bob receive the same signal. In general, a threshold value for $${\bar{c}}_{{\rm{s}}}$$ is not easy to define, because Eve’s ability to decode a signal depends on additional factors, including the modulation scheme and the absolute power level. In an information-theoretic sense, secure transmission is possible under certain circumstances even if $${\bar{c}}_{{\rm{s}}} < 0$$29. Thus, this quantity is not a perfect metric for defining the security of a channel. However, it is reasonable to assume that networks would be designed to strive to maximize $${\bar{c}}_{{\rm{s}}}$$, to minimize the likelihood of a successful eavesdropping attack5. To frame our discussion, we use an illustrative value of $${\bar{c}}_{{\rm{s}}}=0.5$$ as an arbitrary threshold, below which we presume that eavesdropping is feasible. Both b and $${\bar{c}}_{{\rm{s}}}$$ depend on the size, shape, composition and location of the object placed in the beam path, and the carrier frequency. We find that for any frequency, Eve can always find a successful configuration that permits her to eavesdrop undetected, in the absence of any counter-measures.

To illustrate this point, we show in Fig. 2 the measured values of b and $${\bar{c}}_{{\rm{s}}}$$ for a set of scattering objects. These long cylindrical metal pipes of various diameters are inserted into the beam path along an axis parallel to the polarization direction of the beam. In Fig. 2a, the objects are situated on the line-of-sight transmission axis, so that they cast a direct shadow on the aperture of Bob’s receiver30. Unsurprisingly, the value of b therefore increases with the size of the scattering object. We also observe a roughly opposite trend for $${\bar{c}}_{{\rm{s}}}$$: a larger scattering object directs more signal to Eve, so the secrecy capacity decreases.

For any realistic line-of-sight millimetre-wave or terahertz data link, even if the beam is highly directional, it is likely that the size of the beam when it reaches the receiver will exceed the aperture of the receiver. This is necessary to provide some margin of error for beam steering and for channel fluctuations such as atmospheric scintillation31. Therefore, it is possible that a scattering object could intercept a portion of the beam but not cast a shadow on the receiver. To illustrate this point, we repeat the experiments of Fig. 2a, except that we move the cylindrical objects off the centre line to minimize blockage due to direct shadowing. As a result, the values of b are reduced considerably (Fig. 2b), so that Bob may not notice the effect of the object in some cases. For the lower frequencies, the secrecy capacity is also quite low for larger objects, because they still scatter a substantial amount of power. Eve is much more readily able to implement a successful attack (b < 0.5 and $${\bar{c}}_{{\rm{s}}} < 0.5$$) with an off-centre object.

From Eve’s point of view, cylindrical objects are advantageous because they scatter radiation over a wide range of angles. Whereas Fig. 2 depicts results for one particular location of Eve’s receiver, Fig. 3 shows Eve’s received power as a function of the angular location, for a few of the situations that satisfy both b < 0.5 and $${\bar{c}}_{{\rm{s}}} < 0.5$$. The dashed curves show a few examples of predicted values corresponding to the measured situations. These calculations use an ab initio model to compute the diffracted field from a perfectly smooth cylindrical conductor, under illumination from a point source32 (see Methods section ‘Diffracted field from a uniform metal cylinder’). As indicated by the measurements and calculations, Eve has the freedom to place her receiver in many different locations, without sacrificing signal strength.

The wide-angle scattering of a cylindrical object can also be used to develop a new type of physical-layer counter-measure2. A cylindrical object scatters some radiation at 180°, back towards the transmitter (Alice)33. It can also block radiation reflected from Bob’s receiver, which would otherwise have returned to Alice. If Alice can measure these incoming signals and distinguish them from the variable backscatter off mobile objects or the environment, then any change, either an increase or a decrease, would be a sign of a possible attack. The effectiveness of this strategy relies on the assumption that Alice has previously characterized the back-scatter of the channel before any attempted attack by Eve—a rather strong assumption. Nevertheless, when combined with other, more conventional eavesdropping counter-measures2,3,4,5, this approach can provide an additional level of security. We define the success of this counter-measure through a back-scatter parameter:

$${S}_{180}=\left|1-\frac{{{\rm{SNR}}}_{{\rm{Alice}}}^{{\rm{no\; object}}}}{{{\rm{SNR}}}_{{\rm{Alice}}}^{{\rm{object}}}}\right|$$

If the object placed in the beam by Eve causes no change in the back-scattered signal, then S180 = 0 and the counter-measure has failed. But a larger value, greater than some pre-determined threshold (for example, S180 > 0.5), could be regarded as a warning of possible eavesdropping. In the inset in Fig. 3a we show measured values of S180 for all of the conditions for which both b < 0.5 and $${\bar{c}}_{{\rm{s}}} < 0.5$$ (see Methods section ‘Measurement of S180’). These measurements indicate that some, but not all, of Eve’s attacks are detected by this counter-measure.

A second disadvantage for Eve is that, because the scattered radiation is dispersed in all directions, the power collected in a small receiver aperture is relatively low. Consequently, for example, at 400 GHz even the largest of our metal cylinders does not scatter enough radiation to permit Eve to decode the signal ($${\bar{c}}_{{\rm{s}}}$$ is nearly unity for all measurements at this frequency). An alternative for Eve is to use objects that scatter more selectively. Although this limits Eve’s freedom of location considerably, it also increases her signal strength substantially. Instead of cylinders, we consider a set of square planar metal reflectors, which direct a portion of Alice’s transmitted beam at 90° to the original propagation direction by specular reflection. When placed on the transmission axis, these plates block a substantial portion of Bob’s signal. However, when moved off axis (Fig. 4a), the blockage drops to nearly zero, while the secrecy capacity remains low at all frequencies. Moreover, because these objects generate little back-scatter towards Alice, we find that S180 is small except for the largest plates, which block radiation reflected by Bob back towards Alice. For instance, for all of the measurements at 400 GHz, we find S180 < 0.2. As an extreme case, we imagine that Eve has the capacity to fabricate a lossless beam splitter, which is large enough to encompass the entire beam generated by Alice and which can be engineered to split off any desired fraction of the power in the transmitted beam. Figure 4b demonstrates that this type of attack is always effective if the transmittance of the beam splitter is chosen correctly (see Methods section ‘The beam-splitter attack’). Moreover, if Alice cannot measure back-scattered signals (and is therefore unable to use the associated counter-measure), then the beam-splitter attack becomes even more devastating: Eve can readily obtain a bit error rate that is nearly as good as Bob’s.

Our results demonstrate that a narrow pencil-like beam does not guarantee immunity from eavesdropping. Although this claim has often been cited as one of the advantages of using millimetre or terahertz waves, our analysis reveals that an agile eavesdropper can always succeed in implementing an undetected attack, unless counter-measures are used. Traditional counter-measures such as those that rely on beam forming5,21,24 or on more advanced multiplexing schemes22 may be less effective against these attacks, because the portion of the wave front that is sampled by Eve is almost coincident with that intended for Bob. On the other hand, our new counter-measure requires Alice to use a transceiver, not merely a transmitter. Thus, to incorporate security into a directional wireless link, systems will require new physical-layer components and new protocols for channel estimation. For our measurements, the transmitter-to-receiver distance is considerably less than what is often envisioned for line-of-sight terahertz links for backhaul applications. Nevertheless, our results apply equally well to communication over a longer range, by scaling the transmitter and receiver gain (see Methods section ‘Radiation patterns for directional horn antennas’). Our results also demonstrate the ease with which line-of-sight communications can be diverted, which could have implications other than for eavesdropping, such as for distributing signals to multiple receivers in a network.

## Methods

### Radiation patterns for directional horn antennas

Our transmitter consists of a waveguide-coupled horn antenna together with a dielectric lens (diameter, 5 cm; focal length, 7.5 cm). In Fig. 1, we show the measured radiation pattern for the 200-GHz transmitter (WR-5.1 conical horn) used in our measurements, as well as a computed antenna pattern using a finite-element solver. For this configuration, the measured far-field beam pattern has a directivity of 34 dBi, which corresponds to an angular full-width of about 4°. The radiation patterns for the 100-GHz and 400-GHz measurements are similar; these transmitters also use horn antennas (WR-8 conical horn and WR-2.2 diagonal horn, respectively), coupled to the same dielectric lens. The measured directivities are shown in Extended Data Table 1. As might be anticipated from simple diffraction arguments, higher frequencies produce narrower beams. Yet, even for our narrowest beam, the diameter of the beam when it reaches Bob is still twice as large as the aperture of his receiver.

Our detection system, which we use for Bob’s and Eve’s measurements, uses the same type of lens and horn as used for the transmitter and therefore has similar angular sensitivity. For bit error rate (BER) measurements, the transmitter signal is modulated with a pulse-pattern generator using amplitude shift keying, with a bit rate of 1 Gb s−1. The detected signal is amplified and analysed in real time using a BER tester. Other specifications of our transmitter and receiver equipment are given in Extended Data Table 1, with more details in ref. 10.

Our measurement set-up is a scale model of a typically envisioned terahertz wireless link—our transmitter–receiver distance is only 2 m. Our work is not intended to emulate a real system, which would require a transmitter–receiver distance of tens or hundreds of metres. In part, this is due to the limitations of our measurement apparatus. A system designed for longer distances would probably use a higher-power transmitter, higher-gain antennas and/or a more sensitive receiver. For example, a system operating at 220 GHz was used previously34 to achieve a data transmission rate of 3.5 Gb s−1 using a transmitter antenna with a gain exceeding 50 dBi. This enhanced link budget enables a scaling of the broadcast distance to 200 m. If we assume that Eve’s detector sensitivity scales in the same way as Bob’s, then this scaling would not only enhance Bob’s ability to detect signals at larger distances but also equally enhance Eve’s ability. In more complex wireless links, such as those that use multiple inputs and multiple outputs (MIMO), the issue of scaling is more subtle and requires a more detailed consideration35. However, for the single-transmitter, single-receiver link discussed here, a scale-model approach is valid, on the basis of the linearity of the Friis equation. Our results should apply equally well to any line-of-sight link at any range.

### Measurement of S180

The data shown in Fig. 2 can be used to select a limited number of configurations for further investigation. Using the criteria that b and $${\bar{c}}_{{\rm{s}}}$$ must both be less than 0.5, we down-select from the 42 different measurements displayed in Fig. 2. The remaining ten points are a representative set of configurations that illustrate successful eavesdropping situations. Eight of these are at the lowest frequency of 100 GHz and the remaining two are at 200 GHz. We measured the back-scatter parameter S180 for each of these ten situations; the results are shown in the inset of Fig. 3a.

An additional measurement challenge in our experiment is that our transmitter (Alice) is not able to also detect received signals. Therefore, we cannot measure signals in exactly the back-scattered direction (180°). Instead, we place our receiver as close as possible to this direction, limited by the size of the equipment. The measured values of S180 reported in Fig. 3 were obtained at an angle of about 175° and are therefore only approximate values. Nonetheless, they illustrate the key points of our discussion.

### Diffracted field from a uniform metal cylinder

The three dotted curves shown in Fig. 3 were computed using an ab initio model for diffraction from a uniform cylinder made of a perfectly conducting metal28. In brief, if Eve is not located in the geometric shadow of the cylinder, then the power received by Eve (in units of decibels) is

$${P}_{{\rm{Eve}}}={P}_{{\rm{Alice}}}+{G}_{{\rm{tx}}}+{G}_{{\rm{rx}}}+20log\left|{R}_{{\rm{D}}}\right|-20log\left[\frac{4{\rm{\pi }}({d}_{{\rm{i}}}+{d}_{{\rm{r}}})}{\lambda }\right]$$

where PAlice is the power transmitted by Alice, Gtx and Grx are the gains of the transmitter and receiver, respectively, di and dr are the propagation distances from the transmitter to the specular surface reflection point on the cylinder and from that point to Eve’s receiver, respectively (so that their sum is the total propagation distance for a ray detected by Eve), and λ is the wavelength of the radiation. The factor RD accounts for the strength of the diffracted signal. In a coordinate system centred on the central axis of the cylinder, it is given by

$${R}_{{\rm{D}}}=-{\rm{\exp }}\left(-i\frac{{\rm{\pi }}}{4}\right)\sqrt{\frac{-4}{\xi }}{\rm{\exp }}\left(-i\frac{{\xi }^{3}}{12}\right)\left[\frac{-F(X)}{2\xi \sqrt{{\rm{\pi }}}}+p(\xi )\right]$$

where ξ = −2(ka/2)X1/3, X = 2kdicos2(ϕ), k is the free-space wavevector and a is the radius of the cylinder. The angle ϕ is the difference between the angle at which Eve’s receiver is located and the angle of the point on the surface of the cylinder that corresponds to the specular surface reflection, both measured relative to the line from the transmitter through the centre of the cylinder. The function F is defined as

$$F(x)=2i\sqrt{x}{{\rm{e}}}^{ix}\underset{\sqrt{x}}{\overset{\infty }{\int }}{{\rm{e}}}^{j{u}^{2}}{\rm{d}}u$$

and the function p is the Fock scattering function, which is defined in appendix F of ref. 36.

### The beam-splitter attack

In the context of Fig. 4, we envision that Eve has the freedom to create an ideal lossless thin-film beam splitter and place it in the transmission channel at a location where the beam is small enough so that it does not illuminate the edge of the beam splitter (that is, close to Alice), as shown in the inset in Fig. 4a. This placement eliminates any edge-scattering effects. Therefore, the only effect of the beam splitter is to redirect a portion of the transmitted power towards Eve, with the remaining power continuing to Bob, undistorted. We further imagine that Eve has the freedom to choose the power-transmission coefficient of the beam splitter T to be any value between 0 and 1. In this case, we can compute directly the three relevant parameters discussed in the text. First, the blockage is given by b = 1 – T. From this result, we conclude that Eve must choose T > 0.5, or else the blockage would be too high. Second, the back-scatter parameter S180 is found by noting that the signal-to-noise ratio of a back-reflected signal measured by Alice is reduced by a factor of T2 when the beam splitter is inserted (because such signals must pass through the beam splitter twice to reach Alice, once in each direction). Thus, S180 = |1 – 1/T2|. Third, to compute the normalized secrecy capacity, we note that Eve’s signal-to-noise ratio (SNREve) is given by the product of 1 – T and the transmitter power, or

$${{\rm{SNR}}}_{{\rm{Eve}}}=\frac{1-T}{T}{{\rm{SNR}}}_{{\rm{Bob}}}$$
(1)

where SNRBob is the signal-to-noise ratio measured by Bob with the beam splitter in place. The normalized secrecy capacity is therefore

$${\bar{c}}_{{\rm{s}}}=1-\frac{{\rm{l}}{\rm{o}}{\rm{g}}[1+{{\rm{S}}{\rm{N}}{\rm{R}}}_{{\rm{B}}{\rm{o}}{\rm{b}}}(1-T)/T]}{{\rm{l}}{\rm{o}}{\rm{g}}(1+{{\rm{S}}{\rm{N}}{\rm{R}}}_{{\rm{B}}{\rm{o}}{\rm{b}}})}$$

To proceed with this analysis and compute the value of $${\bar{c}}_{{\rm{s}}}$$ that results when Eve uses a beam splitter with a particular value of T, we must make an assumption about Bob’s signal-to-noise ratio. For the purpose of illustration, we assume that Bob is receiving sufficient signal (with the beam splitter in place) to be able to achieve a BER of 10−9 (that is, an error-free signal). For amplitude-shift-keying modulation, the BER is related to the signal-to-noise ratio (SNR) by

$${\rm{BER}}=\frac{1}{4}{{\rm{e}}}^{-{\rm{SNR}}/4}{\rm{erfc}}\left(\sqrt{\frac{{\rm{SNR}}}{2}}\right)$$
(2)

where erfc(x) is the complimentary error function and the signal-to-noise ratio is expressed as a linear ratio (not in decibels). To achieve BERBob = 10−9, a signal-to-noise ratio of about 23.3 is required.

Using this assumed value for SNRBob, we can compute $${\bar{c}}_{{\rm{s}}}$$ and S180 for T between 0.5 and 1 (Fig. 4b). These calculations show that there is a narrow range of T for which all three criteria are satisfied: b < 0.5, $${\bar{c}}_{{\rm{s}}} < 0.5$$ and S180 < 0.5. In particular, the criterion on S180 is satisfied for $$T > \sqrt{2/3}\approx 0.82$$ and the criterion on $${\bar{c}}_{{\rm{s}}}$$ is satisfied for

$$T < \frac{{{\rm{SNR}}}_{{\rm{Bob}}}}{\sqrt{1+{{\rm{SNR}}}_{{\rm{Bob}}}}+{{\rm{SNR}}}_{{\rm{Bob}}}-1}\approx 0.86$$

This quantity is not a very strong function of SNRBob. To reduce it from 0.86 to $$\sqrt{2/3}$$, SNRBob would need to be diminished considerably, from about 23.3 to about 10.9. This would eliminate the range of T for which eavesdropping is possible, but at the expense of increasing BERBob by nearly four orders of magnitude. Thus, under the assumptions that we have made for parameter thresholds, Alice and Bob can prevent a successful beam-splitter attack only by operating at greatly reduced BER and by using both blockage and back-scatter as detection counter-measures.

With the same assumption of BERBob = 10−9, we can use equations (1) and (2) to compute the BER that Eve can achieve (BEREve) when she uses a beam-splitter attack (Fig. 4b inset, solid red line). Doing so indicates that BEREve improves continuously as T decreases. If Eve is restricted to the range of T mentioned above, then the optimal value of BEREve is achieved for $$T=\sqrt{2/3}$$ and is 1.5 × 10−3. Although this value may be sufficient to decode information, Eve can do much better with a smaller value of T. If the back-scatter counter-measure proposed here is not used, then Eve can use any beam splitter with T > 0.5, enabling her to implement an attack with a secrecy capacity arbitrarily close to zero and thus achieve a BER essentially equal to that of Bob.

The above analysis assumes that the transmitter produces zero side lobes. Although small, the side lobes cannot be exactly zero. Therefore, in the beam-splitter configuration depicted in the inset of Fig. 4a, Eve would measure not only the signal from the beam splitter but also a small contribution from side lobes, which would effectively degrade her BER through added interference. The inset of Fig. 4b also contains a few measured data points (open squares), which quantify this effect. The data reproduce the predicted trend in BEREve, but with a slightly worse BER than predicted. This is presumably due to the effect of side-lobe interference, which amounts to an extra roughly 3 dB of noise.

To make these measurements, we fabricated a few large-aperture beam splitters (no such devices are commercially available for these frequencies). To avoid etalon effects, which could introduce substantial phase distortion that would artificially decrease the BER, we fabricated the beam splitters on very thin low-loss polyethylene substrates37. These substrates were stretched across a circular metal frame with a large enough diameter to encompass the beam at the output of Alice’s transmitter without much scattering from the frame. The polyethylene films were then coated with a thin metal layer, using a metallic spray paint. The paint adhered well to the surface and coated it uniformly. By varying the thickness of this metal layer, we fabricated beam splitters with different values of T, which were determined experimentally.

## Data availability

The data that support the findings of this study are available from the corresponding author on reasonable request.

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## Acknowledgements

This work was funded in part by the US National Science Foundation, the US Army Research Office and the W. M. Keck Foundation.

### Reviewer information

Nature thanks K.-Y. Lam and the other anonymous reviewer(s) for their contribution to the peer review of this work.

## Author information

### Affiliations

1. #### School of Engineering, Brown University, Providence, RI, USA

• Jianjun Ma
• , Rabi Shrestha
•  & Daniel M. Mittleman
2. #### Department of Electrical and Computer Engineering, Rice University, Houston, TX, USA

• Chia-Yi Yeh
•  & Edward Knightly
3. #### Department of Electrical Engineering, University at Buffalo, The State University of New York, Buffalo, NY, USA

• Zahed Hossain
•  & Josep Miquel Jornet

### Contributions

D.M.M. and E.K. conceived the experiments. J.M., R.S. and J.A. performed the measurements. Z.H. and J.M.J. performed the computations shown as dotted lines in Fig. 3. D.M.M., E.K. and C.-Y.Y. analysed and interpreted the data. All authors contributed to the writing of the manuscript.

### Competing interests

The authors declare no competing interests.

### Corresponding author

Correspondence to Daniel M. Mittleman.

## Extended data figures and tables

### DOI

https://doi.org/10.1038/s41586-018-0609-x