Quantum communication, and more specifically Quantum Key Distribution (QKD), enables the transmission of information in a theoretically secure way, guaranteed by the laws of quantum physics. Although fiber-based QKD has been readily available since several years ago, a global quantum communication network will require the development of space links, which remains to be demonstrated. NICT launched a LEO satellite in 2014 carrying a lasercom terminal (SOTA), designed for in-orbit technological demonstrations. In this paper, we present the results of the campaign to measure the polarization characteristics of the SOTA laser sources after propagating from LEO to ground. The most-widely used property for encoding information in free-space QKD is the polarization, and especially the linear polarization. Therefore, studying its behavior in a realistic link is a fundamental step for proving the feasibility of space quantum communications. The results of the polarization preservation of two highly-polarized lasers are presented here, including the first-time measurement of a linearly-polarized source at λ = 976 nm and a circularly-polarized source at λ = 1549 nm from space using a realistic QKD-like receiver, installed in the Optical Ground Station at the NICT Headquarters, in Tokyo, Japan.
© 2016 Optical Society of America
Quantum communication  has become one of the first mature applications to be developed out of the new field of quantum information. Generally, it relies on the transmission and detection of single photons, as its security is based on the impossibility to copy the quantum state of a single photon. Now it is clear that the secure-communications market will be transformed in the near future, in advance of the threat to the information security that the development of quantum computers will bring . The currently-used classic cryptography will be substituted by the quantum alternative in several critical applications. However, although optical-fiber implementations have rapidly widespread, until reliable quantum repeaters are developed, they are limited to distances below ~200 km .
Extending the operation of quantum-information processing to space will allow performing fundamental quantum physics experiments as well as applications related to quantum communications on a global scale . Among the most important ones will be the development of a global quantum communication network, in which a key step is the utilization of satellites capable of performing space-to-ground links . So far, a number of theoretical studies have been carried out proving the feasibility of quantum communication from space [6–9].
However, up to date no demonstrations have been carried out showing a proof-of-concept in an operational quantum-communication link from space. There have been several steps towards that direction though, like the reception of single photons in the ground after being reflected in passive retroreflector satellites [10,11], being the closest attempts a quantum-communication link from an airplane  and from an air balloon , replicating some characteristics of actual links from space. For these reasons, experiments validating different aspects of the fundamental principles of space quantum communications are very valuable for future developments.
This paper is organized as follows. Section 2 introduces the SOTA mission and the characteristics related to the polarization experiment. Section 3 describes the relation between polarization and quantum communications, presenting the polarization characterization of the SOTA lasers. In Section 4, we explain the effects on the polarization in a quantum communication link from space. Section 5 explains the peculiarities of measuring the polarization on the ground from a laser source in space. In Section 6, the ground receiving system is presented, along with a characterization for predicting the measurements from the space terminal. Section 7 shows the results of the polarization measurements from the laser sources in space, including examples of the satellite passes and a comparison with the expected results. Finally, Section 8 presents our conclusions.
2. SOTA: Small Optical TrAnsponder
Most of space communications still use radiofrequency links, in particular microwave links. However, free-space optical communication links outperform microwave links in a number of relevant parameters, such as the data rate, the power consumption and the size and mass of the space terminals. The technology is close to reaching the maturity stage, with more and more demonstrations by the different space agencies as well as the private industry.
The National Institute of Information and Communications Technology (NICT) in Japan has conducted research and development related to space optical communications for a number of years, with several relevant demonstrations so far [14,15]. The last one started in May 2014, when NICT launched the microsatellite SOCRATES (Space Optical Communications Research Advanced Technology Satellite), with the aim of demonstrating the operation of the SOTA (Small Optical TrAnsponder) lasercom terminal . SOTA, shown in Fig. 1, is a light (<6 kg) and small transponder designed to carry out a number of different lasercom experiments.
Until now, all the goals of the SOTA mission have already been accomplished, e.g. up-to-10 Mbit/s downlinks at different wavelengths, space-to-ground transmission of onboard-camera images, experiments on error-correcting codes and interoperability with other international optical ground stations. The extra-success phase of the mission comprises basic experiments to validate the feasibility of quantum communications from space . For these experiments, SOTA allows using four different laser sources in space, as shown in Fig. 2: Tx1 at 976 nm with an undefined polarization at the time of the SOTA development, 800-nm band Tx2 and Tx3 with mutually non-orthogonal linear polarizations and circularly-polarized Tx4 at 1549 nm.
3. Polarization-encoded QKD
Quantum cryptography, or more specifically Quantum Key Distribution (QKD) has become the most developed application in the field of quantum communication. A QKD protocol consists of two parts: firstly, during the quantum communication phase, two distant parties (Alice and Bob) share a sequence of random bits by means of transmitting and receiving single photons; secondly, a classical-communication phase takes place, in which both parties use the previously shared bits to generate a secret key out of them by using a public and authenticated channel.
The most extended free-space QKD protocols, like BB84  and B92 , use polarization for encoding the quantum information into the transmitted photons. Polarization-encoded QKD involves at least two non-orthogonal States of Polarization (SOP) to ensure its security by means of introducing uncertainty in the measurements of the single photons. Usually, linear polarizations are used for this matter, e.g. in B92, two linear polarizations with orientations separated by 45° are utilized.
If an eavesdropper tried to spy the quantum communication using the most basic attack, i.e. intercept and resend, a measurement basis had to be chosen (rectilinear or diagonal). Either of them would provide a random result for some fraction of the measurements, introducing errors in the communication that could be detected by the quantum protocol when evaluating the Quantum Bit Error Rate (QBER). The QBER is the fundamental metric in QKD protocols and is calculated by counting the photons received in the different polarizations. Therefore, differently from classical lasercom, the preservation of the polarization characteristics is a key issue in free-space quantum communications.
The polarization of the laser sources were characterized using a polarimeter in a clean room for measuring the Tx1 and Tx4 signals from the SOTA engineering model (a replica of the SOTA flight model). The result is shown in Fig. 3 in terms of the Stokes parameters represented in the Poincaré sphere, along with the numerical values. These measurements were transformed to show the point of view of the emitter from SOTA. At the time of the mission design, the polarization of Tx1 was not within any of the purposes of the mission, hence no specific polarization was defined. In spite of this, Tx1 is a highly-polarized laser source, which can be used for measuring a linear polarization (the one usually used for QKD) from space for the first time. Tx4 is also a circularly highly polarized source, which can be used for measuring the polarization of a 1549-nm laser from space for the first time.
4. Space QKD and polarization
Horizontal free-space quantum-communication links are usually static, which allows easily aligning and calibrating the polarizations transmitted from Alice and received in Bob. When Alice is implemented within a moving terminal in space (e.g. LEO), the satellite motion makes the reference frame change with time . This will affect the polarization measurements in two different ways: firstly, the reference-frame changes will induce basis misalignments ψRF between the two parties, as shown in Fig. 4(a), and secondly, the varying incident angle on Bob’s optics will introduce phase delays χPD between the orthogonal components of each polarization, making them generally elliptical, as shown in Fig. 4(b).
In a space-to-ground link, the polarization characteristics can be altered by the atmospheric turbulence. If a worst-case scenario among several existing theoretical models  is assumed for a linearly-polarized laser at λ = 1 µm, the root mean square polarization fluctuations would not be higher than ~25 nrad/km. When this figure is applied for the maximum SOTA distance (~1000 km), it would be translated into ~25 µrad, a negligible figure for all practical purposes.
Another effect that may alter the received polarization on the ground in a QKD link from space is the scattering of the laser by the aerosols and atmospheric molecules. Backscattering depolarization measurements by LIDAR systems is a well known technique, useful to discriminate between different kinds of particles in the atmosphere . However, their effect on the transmitted light has not been well characterized through experimental measurements.
5. Measuring the polarization from space
To date, the only polarization measurement from a laser source in space was made by our group in NICT in 2009 , measuring on the ground the circularly-polarized laser source transmitted from the LUCE (Laser Utilizing Communications Equipment) terminal onboard the LEO satellite OICETS (Optical Inter-orbit Communications Engineering Test Satellite). In that experiment, Stokes parameters and the degree of polarization were measured, proving for the first time the polarization preservation of a highly-polarized laser source after going through a slant atmospheric path.
However, LUCE and SOTA are very different lasercom terminals, as they were designed with different goals. The main purpose of SOTA is to demonstrate different optical technologies for microsatellites. In consequence, SOTA lasers have less power and are emitted through a smaller aperture (thus with a bigger divergence). For example, as can be seen in Fig. 5, the main transmitting aperture of LUCE had a 26-cm diameter, and the biggest aperture in SOTA (Tx4) is over 5 times smaller, with a divergence angle in SOTA being 40‒160 times higher (depending on the beam waist and the wavelength) than LUCE’s. As a result, the power received on the ground is much lower in the case of SOTA due to the design constraints of the microsatellite platform, accounting for a difference of about 30 dB.
The consequence of the lower received power on the ground is that a much bigger aperture is required. The accuracy of a polarization measurement, characterized by the mean and standard deviation of the DOP (Degree of Polarization), depends on the received optical power. Figure 6 shows the dependence of the DOP using our polarization measuring system when a highly-polarized linear (horizontal) state is received at different optical-power levels. In the OICETS experiments, a 1.5-cm diameter lens was enough to detect a high-enough power to measure the polarization correctly. However, if the same lens was used to measure the SOTA signals, a very-low power would be received, resulting in a deterioration of the polarization characteristics caused by the measuring system.
6. Ground receiving system
A big telescope is required in order to receive enough power to measure the polarization of the SOTA lasers on the ground. For this purpose, the 1.5-m diameter Optical Ground Station (OGS) at NICT Headquarters (Koganei, Tokyo) was utilized for these measurements, increasing the gain in 20 dB in relation to the 1.5-cm lens used in the OICETS measurements. With this receiving gain, more accurate polarization measurements can be made. Besides, using a big telescope is a much more realistic scenario, since quantum communications will require this kind of apertures to close the links due to the low photon flux . Furthermore, when using only a lens for measuring a circular polarization, none of the two effects shown in Fig. 4 happen. Conversely, using a big telescope involves additional optics that will affect the polarization in the same way it will happen in actual quantum communication links from space.
Figure 7 shows the setup used for the SOTA polarization measurements, which is similar to the basic setup used for the SOTA communication links, hence as realistic as it can be. The main tracking and receiving system is a 1.5-m diameter Cassegrain telescope  with parabolic primary and secondary mirrors and one flat reflector to redirect the radiation towards the Nasmyth bench, where the measuring system was located. The first element in the Nasmyth bench is a collimating lens followed by a × 10 beam reducer, designed to provide an output beam with a diameter of 10 mm at λ = 1.55 µm. After the beam reducer, a 50/50 beamsplitter (BS) is used for deflecting part of the beam towards an infrared (IR) camera for tracking purposes and finally towards the polarimeter, where the measurements were taken. The BS before the polarimeter is also a representative element of any QKD receiver, so the polarization measurements were taken just where the single photon detector of an equivalent QKD receiver would be.
In the OICETS measurements campaign, the received circular polarizations were registered as they were transmitted, without any alteration  because the only receiving optical system was a lens. However, in this case the polarization is expected to be affected due to the use of other optical elements before the measuring system. In general, the reflection of a linear polarization on a metallic mirror will turn the linearity into ellipticity, depending on the polarization angle, the incidence angle and the complex refractive index of the reflective material . In practice, a Cassegrain telescope, like the one used in this experiment, should have a very small effect on the polarization if the Cassegrain focus was used. However, when using the big telescopes required for space QKD, it is impractical to install the receiving system in this location because it would move along with the telescope. Normally, the receiving system is placed in the Nasmyth bench, which is stationary. A tertiary mirror is required for redirecting the light, which steers the beam 45° towards the Nasmyth bench. The complex refractive index of the metal surface introduces a phase shift between the s-polarized and p-polarized components, which is maximum when both components are equal (polarization angle equals to 45°), and minimum when one of the components is zero. Therefore, a 45°-incidence on a metallic mirror preserves the linear polarization only for polarization angles equals to 0° or 90°, and for other cases the linear polarization turns into elliptical.
The figure below shows the result of the experimental characterization of the receiving system of Fig. 7, as well as a simulation of the same system to predict the behavior of the SOTA signals in the actual measurements. For the simulation, the reflective material of the 1.5-m telescope mirrors was considered to be Aluminum  and linear SOPs with different azimuth orientations were used as the input light. The apparatus used for the experimental characterization is shown in Fig. 8, consisting of the same receiving system shown in Fig. 7. In the figure, instead of SOTA, a replica of the SOTA laser sources was employed for generating the input SOPs. A rotating system was built to emulate the motion of the satellite. The effect of this motion is a change in the polarization angle, or equivalently, a rotation of the SOP around the S3 axis in the Poincaré sphere. This system rotates the light source (including a fiber collimator and a linear polarizer) around the optical axis with steps of 15° between consecutive SOPs. A half-wave plate is not adequate for this purpose, since the retardance of the plate changes with the wavelength, the incident angle and the temperature. As a consequence, the actual retardance can be shifted several degrees from 180°, meaning that the effect of rotating the half-wave plate is not only that of modifying the polarization angle as desired, but also of turning the linear polarization into elliptical before entering the telescope. For example, a 10° shift away from the theoretical 180° retardance of the half-wave plate has the same effect turning the linear polarization into elliptical as the worst-case of Fig. 9 (an input SOP corresponding with a 45° linear polarization). The error bars in the experimental data of Fig. 9 are smaller than the data points, not allowing its visualization.
As can be seen in Fig. 9, the linear polarization transmitted from SOTA will be modified by the receiving system. This perturbation will always be present since quantum protocols require transmitting non-orthogonal polarization basis: two polarizations in case of B92, with 0° and 45° polarization angles, or four in the case of BB84, with 0°, 90°, 45° and −45° polarization angles. Depending on the orientation of the reference frame between the satellite and the OGS, there will be two extreme situations: when both reference frames are aligned, then one polarization basis (0° in B92, or 0° and 90° in BB84) will be received with no distortion, and the other one (45° in B92, or 45° and −45° in BB84) will be transformed into the worst-case elliptical SOPs; and when both reference frames are not aligned, all the received polarizations will be generally elliptical.
7. SOTA polarization measurements
The experiments with SOTA were carried out from January to March 2016 at nighttime, as SOTA was specifically designed for this kind of operation. This is representative of a QKD link from space as to date it seems unlikely that daytime single-photon quantum communications can be carried out . In these experiments, the polarization of the Tx1 and Tx4 lasers were measured. They are communication signals modulated with On-Off Keying (OOK) at 10 Mbit/s with a Non-Return-to-Zero (NRZ) format and a ~50% duty cycle.
One experiment can be conducted each night, with the following sequence of events: firstly, the parameters of the experiment are sent to SOTA by RF; then, when the SOTA pass approaches, as can be seen in Fig. 10, the OGS starts emitting a high-power beacon towards the predicted position of the satellite according to its orbital information; when SOTA acquires the beacon signal, the downlink starts while SOTA keeps tracking the beacon; after several minutes, the line of sight is lost and the communication ends.
Figures 11 and 12 show examples of the reception of, respectively, the Tx1 (λ = 976 nm) and Tx4 (λ = 1549 nm) signals from SOTA in the NICT OGS using the receiving system of Fig. 7. The variation of the Stokes parameters, the degree of polarization and the received power with time during one whole SOCRATES pass can be observed. The information of the OGS elevation and azimuth angle is included as a reference of the SOTA tracking by the 1.5-m telescope. The interruption in Fig. 11 is due to a SOTA high-elevation pass, since the ground telescope has to rotate with the azimuth to find the position of the satellite again when the maximum allowed elevation is reached. Only the Stokes parameters corresponding with powers above −55 dBm are shown, since low received powers do not allow measuring the polarization correctly, as can be seen in the DOP plot of Fig. 6.
Figure 13 shows the polarization measurements of the Tx1 and Tx4 signals received in the NICT OGS from the two SOTA passes presented in Figs. 11 and 12. The experimental results are compared with their predicted values, explained in Section 6, showing a good agreement between the expected and the measured behavior. The dots color (starting from red to green), as well as the red arrows, indicates how the received signals evolve in time according to the satellite motion. Both polarizations suffer from the two effects explained in Section 4: first, the rotation of the reference frame between the satellite and the OGS has the same effect of the apparatus used in Section 4 for characterizing the OGS, i.e. a rotation around the S3 axis in the Poincaré sphere, or equivalently, a rotation of the polarization angle around the optical axis. Second, as explained in Section 4, the variation of the incidence of the polarization angle in the optics of the receiving system, especially in the 45° reflections, causes a phase shift between the s-polarization and the p-polarization that turns the linear polarization into elliptical.
For the case of Tx1, the first effect can be seen in Fig. 11, where the S3 component (circular component) starts with a non-zero value and it tends to zero according to the satellite motion, when the reference frames are aligned. The second effect can be seen in Fig. 13(a), where the polarization is rotated around the S3 axis according to the prediction. This result is consistent with the simulation and characterization presented in Section 4, when the input polarization is linear, which proves that the linear polarization is preserved during the atmospheric propagation.
In the case of the circular polarization of Tx4, the cause of the polarization modification is different. A circular polarization can be seen as made up by two 90° phase shifted orthogonal linear components of the same amplitude. Therefore, even when using a receiving system like the SOTA one presented in Fig. 7, in which each component reach a 45°-mirror at different polarization angles, if the polarization is perfectly circular, the ellipticity of both components is mutually compensated and the circularity is preserved. However, if the incoming polarization is not perfectly circular (as was shown in Fig. 3 for Tx 4), or equivalently, if the transmitter’s and receiver’s planes orthogonal to the propagation are not perfectly parallel, then the compensation is not perfect and some ellipticity is observed. This is the case in the Tx4 received signal shown in Fig. 13(b), as it was validated with the simulation shown in the upper right corner of the same figure, using the model explained in Section 6.
If signals like the ones shown above were received by a real QKD system, these polarization disturbances should be compensated before their detection in order to maintain a low QBER. As was explained in Section 3, this is critical in QKD systems, where the QBER is the fundamental metric for assessing the security of the communications and detect a possible eavesdropping. Therefore, once the effect of the atmosphere is ignored, as this paper shows that will be the case, each particular receiving system has to be analyzed to understand how the polarization is modified and what kind of correction should be used. There are several techniques for compensating the effect of the optical system in the polarization, such as using a half-wave plate to align the polarization basis with the receiving system, using the Cassegrain focus to avoid the 45° reflection, using a reflecting material that helps to preserve the linear polarization after the 45° reflections, or using a set of quarter-wave plates to compensate the ellipticity of each DOP.
Figure 14 shows the variation of the degree of polarization with the received power from SOTA signals (Tx1 and Tx4) in the NICT OGS. The DOP stabilizes around ~100% (100.69% in the case of Tx1 and 100.32% in the case of Tx4) when the received power is over −55 dBm, proving that the polarization is preserved through the space-to-ground path. The average DOP values which are slightly higher than 100% are associated to artifacts of the measuring system, e.g. a dependence of the DOP with the wavelength was identified, being up to 0.25% per nm. The measured DOPs present a small difference with the original DOPs of the SOTA laser sources (0.53% for Tx1 and 0.12% for Tx4, see Section 3). This is the first time the degree of polarization is measured in a linear-polarization laser source from space and also the first time to measure the polarization characteristics of 1550-nm wavelength, proving the feasibility of polarization-based quantum communication from space.
We have presented the first measurement of a linearly-polarized laser source in space using a QKD-like receiver on the ground installed in the NICT 1.5-m diameter telescope. To date, no space quantum-communication link has been demonstrated, thus experiments validating different aspects of the fundamental principles of space quantum communications are valuable for future in-orbit validations. In this paper, a realistic receiver was utilized for measuring the polarization of the SOTA lasercom terminal in LEO-to-ground links. Two different highly-polarized laser sources were measured, including a linear polarization at λ = 976 nm and a circular polarization at λ = 1549 nm. The linear polarization is the basic property for encoding information in free-space QKD links, and it had not been measured before from a source in space, as well as the polarization of a source at λ = 1549 nm. In our experiments, the measured data agreed well with the predicted values and the degree of polarization was preserved in both sources, proving the feasibility of polarization-based QKD from space.
References and links
1. N. Gisin, G. Ribordy, W. Tittel, and H. Zbinden, “Quantum cryptography,” Rev. Mod. Phys. 74(1), 145–195 (2002). [CrossRef]
2. P. W. Shor, “Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer,” SIAM J. Comput. 26(5), 1484–1509 (1997). [CrossRef]
3. V. Scarani, H. Bechmann-Pasquinucci, N. J. Cerf, M. Dusek, N. Lutkenhaus, and M. Peev, “The security of practical quantum key distribution,” Rev. Mod. Phys. 81(3), 1301–1350 (2009). [CrossRef]
5. M. Aspelmeyer, T. Jennewein, M. Pfennigbauer, W. R. Leeb, and A. Zeilinger, “Long distance quantum communication with entangled photons using satellites,” IEEE J. Sel. Top. Quantum Electron. 9(6), 1541–1551 (2003). [CrossRef]
6. C. Bonato, A. Tomaello, V. Da Deppo, G. Naletto, and P. Villoresi, “Feasibility of satellite quantum key distribution,” New J. Phys. 11(4), 045017 (2009). [CrossRef]
7. J.-P. Bourgoin, E. Meyer-Scott, B. L. Higgins, B. Helou, C. Erven, H. Huebel, B. Kumar, D. Hudson, I. D’Souza, R. Girard, R. Laflamme, and T. Jennewein, “A comprehensive design and performance analysis of LEO satellite quantum communication,” New J. Phys. 15(2), 023006 (2013). [CrossRef]
8. P. Villoresi, T. Jennewein, F. Tamburini, M. Aspelmeyer, C. Bonato, R. Ursin, C. Pernechele, V. Luceri, G. Bianco, A. Zeilinger, and C. Barbieri, “Experimental verification of the feasibility of a quantum channel between space and Earth,” New J. Phys. 10(3), 033038 (2008). [CrossRef]
9. L. Moli-Sanchez, A. Rodriguez-Alonso, and G. Seco-Granados, “Performance analysis of quantum cryptography protocols in optical earth-satellite and intersatellite links,” IEEE J. Sel. Areas Comm. 27(9), 1582–1590 (2009). [CrossRef]
10. J. Yin, Y. Cao, S.-B. Liu, G.-S. Pan, J.-H. Wang, T. Yang, Z.-P. Zhang, F.-M. Yang, Y.-A. Chen, C.-Z. Peng, and J.-W. Pan, “Experimental quasi-single-photon transmission from satellite to earth,” Opt. Express 21(17), 20032–20040 (2013). [CrossRef] [PubMed]
12. S. Nauerth, F. Moll, M. Rau, C. Fuchs, J. Horwath, S. Frick, and H. Weinfurter, “Air-to-ground quantum communication,” Nat. Photonics 7(5), 382–386 (2013). [CrossRef]
13. J. Wang, B. Yang, S. Liao, L. Zhang, Q. Shen, X. Hu, J. Wu, S. Yang, H. Jiang, Y. Tang, B. Zhong, H. Liang, W. Liu, Y. Hu, Y. Huang, B. Qi, J. Ren, G. Pan, J. Yin, J. Jia, Y. Chen, K. Chen, C. Peng, and J. Pan, “Direct and full-scale experimental verifications towards ground-satellite quantum key distribution,” Nat. Photonics 7(5), 387–393 (2013). [CrossRef]
14. K. Araki, Y. Arimoto, M. Shikatani, M. Toyoda, M. Toyoshima, T. Takahashi, S. Kanda, and K. Shiratama, “Performance evaluation of laser communication equipment onboard the ETS-VI satellite,” Proc. SPIE 2699, 52–59 (1996). [CrossRef]
15. T. Jono, Y. Takayama, N. Kura, K. Ohinata, Y. Koyama, K. Shiratama, Z. Sodnik, B. Demelenne, A. Bird, and K. Arai, “OICETS on-orbit laser communication experiments,” Proc. SPIE 6105, 610503 (2006). [CrossRef]
16. Y. Takayama, Y. Koyama, H. Takenaka, Y. Munemasa, M. Akioka, and M. Toyoshima, “Preparative demonstration of optical link establishment by using Small Optical Transponder,” in Proceedings of International Conference on Space Optical Systems and Applications (ICSOS, 2014), pp. 1–4.
17. M. Toyoshima, T. Sasaki, H. Takenaka, Y. Takayama, Y. Koyama, M. Fujiwara, and M. Sasaki, “Verification plan of basic quantum key distribution experiments by using a small satellite,” Trans. JSASS Aerospace Tech. Jpn. 10(28), 9–15 (2012).
18. C. H. Bennett and G. Brassard, “Quantum cryptography: public key distribution and coin tossing,” in Proc. of IEEE Int. Conf. on Computers, Systems and Signal Processing (IEEE, 1984), pp. 175–179.
20. C. Bonato, M. Aspelmeyer, T. Jennewein, C. Pernechele, P. Villoresi, and A. Zeilinger, “Influence of satellite motion on polarization qubits in a space-Earth quantum communication link,” Opt. Express 14(21), 10050–10059 (2006). [CrossRef] [PubMed]
22. R. M. Schotland, K. Sassen, and R. Stone, “Observation by lidar depolarization ratio for hydrometeors,” J. Appl. Meteorol. 23(5), 1011–1017 (1971). [CrossRef]
23. M. Toyoshima, H. Takenaka, Y. Shoji, Y. Takayama, Y. Koyama, and H. Kunimori, “Polarization measurements through space-to-ground atmospheric propagation paths by using a highly polarized laser source in space,” Opt. Express 17(25), 22333–22340 (2009). [CrossRef] [PubMed]
24. JAXA Optical Inter-orbit Communications Engineering Test Satellite, “Kirari” (OICETS) official website, http://www.satnavi.jaxa.jp/project/oicets/.
25. M. Er-long, H. Zheng-fu, G. Shun-sheng, Z. Tao, D. Da-sheng, and G. Guang-can, “Background noise of satellite-to-ground quantum key distribution,” New J. Phys. 7(1), 215 (2005). [CrossRef]
26. H. Kunimori, M. Toyoshima, and Y. Takayama, “Overview of optical ground station with 1.5 m diameter,” Special Issue on the Optical Inter-orbit Communications Engineering Test Satellite 59(1–2), 43–52 (2012).
27. M. Born and E. Wolf, “Refraction and reflection at a metal surface,” in Principles of Optics. Electromagnetic Theory of Propagation, Interference and Diffraction of Light (Cambridge University, 1999).