2.5- and 5-Gbps time-delay self-homodyne interference differential phase-shift keying optical receiver for space-to-ground communication link

Abstract. The availability and reliability of the high data rate space-to-ground optical coherent communication links are critically challenged because the wave front of the signal beam is distorted and impaired when propagating through atmospheric turbulence. Based on the free-space interference of two successive data bits delayed by an unequal-arm-length Mach–Zehnder interferometer (MZI), a pupil-matching time-delay self-homodyne interference differential phase-shift keying (DPSK) optical receiver with 2  ×  4 90-deg optical hybrid is designed for the high data rate space-to-ground optical communication links due to its immunity from wave front impairment. The delayed optical path difference (OPD) in the unequal-arm-length MZI corresponds to the duration of one bit and can be stabilized to below 1000th of the wavelength by the closed-loop feedback control. The maximum system insertion loss is <1  dB. The measurement accuracy of OPD in unequal-arm-length MZI is 0.01 mm by the deramping method from a chirped laser. The double-bit-rate 2.5-  /  5-Gbps DPSK optical receiver has been presented. Parallel and separate atmospheric measurement along the optical communication link is also performed simultaneously. The 2.5-and 5-Gbps optical communication links have already been verified with a ϕ300-mm receiving telescope between two buildings in downtown over a distance of 2.4 km in the worst-case atmospheric conditions. The measured bit-error-rate is better than 10  −  6. Without wave front compensation of the adaptive optics, local oscillator, and optical phase-locked loops, the pupil-matching time-delay self-homodyne interference DPSK optical receiver has great significance for future developments of space-to-ground optical communication links.


Introduction
Free-space optical communications, including remote optical communications between deep space, geosynchronous orbit (GEO), medium earth orbit, low earth orbit (LEO) satellites, high-altitude platform, airborne platform, and optical ground station (OGS), can compose an integrated space-ground real-time high-speed optical communication network in the future. 1 With a larger communication capacity, larger aperture gains, and smaller diffraction losses at optical wavelengths as compared to radio frequency (RF) wavelengths, free-space optical communications can offer many potential benefits for future space missions by enabling higher data rate links with reduced size, weight, and power burden on a spacecraft and smaller ground terminals on the Earth, and break through the bottleneck of RF communication.
In 2001, the European Space Agency (ESA) successfully performed semi-conductor inter satellite link experiment, the world's first intersatellite 50-Mbps laser communication link experiment based on incoherent intensity-modulated/ direct-detection modulation between GEO and LEO satellites. 2 Since that time, a series of duplex optical link experiments between GEO/LEO satellites and OGS have been verified. [3][4][5][6][7][8][9] The homodyne binary phase-shift-keying (BPSK) systems can achieve the best receiver sensitivity and the longest transmission distance among all optical communication systems. 10 In 2008, an optical intersatellite communication link based on the homodyne BPSK transmitted data with a bidirectional data rate of 5.625 Gbps with a bit-error-rate (BER) better than 10 −9 between two LEO satellites. 11,12 The first demonstration of a BPSK laser communication link from LEO to OGS at 5.625 Gbps was reported in 2010. 13,14 These results demonstrate the applicability of free-space optical communication for not only high data rate intersatellite optical links, but also space-toground optical links. However, BPSK free-space optical communication is critically challenged by the effects of atmospheric turbulence in practical deployment. The refractive index inhomogeneity generated by atmospheric turbulence distorts the phase integrity of the propagating beam (shown in Fig. 1), which is vital for high data rate BPSK demodulation. The link performance will be seriously degraded because the interference contrast is deteriorated when a signal beam is mixed with a local oscillator (LO) laser. 15 Atmospheric turbulence has been well-studied and welldescribed using the Kolmogorov theory by Fried. 16 Several techniques have been performed to mitigate the effects of atmospheric turbulence to increase the heterodyne efficiency and signal to noise ratio (SNR) of space-to-ground optical communication links. Small apertures can be adopted when the wave front deformations are small within a certain diameter if the aperture is smaller than the Fried parameter.
In 2011, the ESA demonstrated 5.625-Gbps bidirectional laser communication between an LEO [near-field infrared experiment (NFIRE)] and a ground communication terminal with a 65-mm diameter aperture of TESAT OGS hosted in Tenerife. 13 However, using small apertures results in low received power and strong intensity fluctuations. The adaptive optics (AO) can measure the deformations of a received wave front using a Shack-Hartmann sensor and finish the corresponding corrections by applying inverse deformations with a deformable mirror. The AO schemes to mitigate the atmospheric turbulence have been investigated for satelliteto-ground laser communication in ESA OGS. 17,18 The mixing efficiency and the BER performances of the closed-loop AO integrated with the coherent optical communications were analyzed. [19][20][21] Different AO approaches for OGS were presented and compared. 22 Although the AO was proven to be a very promising technique that can largely improve the performances of the space-to-ground optical communication links, the complex systems and expensive costs limit its application. The orbital angular momentum (OAM) has the potential for enabling multiplexing of multiple data-carrying beams to increase the transmission capacity and spectral efficiency of a communication system. A phase correction method for OAM states was proposed to mitigate the turbulent aberrations more effectively. 23 The low-density parity-check coded OAM modulation scheme was proved to be suitable for the high data rate free-space optical communication link under strong atmospheric turbulence. 24 Increasing the spacing between the detected OAM modes was also shown to effectively mitigate the effects of atmospheric turbulence. 25 A hybrid input-output-algorithm-based AO system has been proposed to compensate the distorted OAM-based free-space optical communication systems. 26 However, the complexity and high cost of the schemes also limit their applications in the implementation of space-toground optical communication links.
Time-delay self-homodyne interference differential phase-shift keying (DPSK) is a promising alternative to AO and OAM for overcoming atmospheric turbulence. Timedelay self-homodyne interference DPSK is based on the interference of two successive data bits in an unequal-arm-length interferometer, where the delayed optical path difference (OPD) corresponds to the duration of one bit. The relative phases of two successive data bits can be converted into the intensity by the interference. The modulation contrast in an interferometer depends on the interference of successive wave fronts, not on the quality of the wave front itself. So, without wave front compensation of AO, timedelay self-homodyne interference DPSK can completely eliminate wave front distortions if the two interfering bits experience identical wave front distortions when passing the atmospheric turbulence. A 5.625-Gbps multimode Michelson-type DPSK interferometer was developed for the satellite-to-ground communication in the ESA's OGS. 27 The unequal-arm-length MZI-based coherent DPSK receiver has also been proposed. 28,29 In this paper, the 2.5-and 5-Gbps pupil-matching time-delayed self-homodyne interference DPSK optical receivers based on the free-space unequal-arm-length MZI and 2 × 4 90-deg optical hybrid 30 for space-to-ground optical communication link are designed and verified between two buildings in the downtown are over a distance of 2.4 km in the worst-case atmospheric conditions.
2 Theoretical Analysis 2.1 Time-Delay Self-Homodyne Interference DPSK Receiver DPSK modulation has already been successfully applied in high data rate optical fiber communications and has recently gained significant research attention. 31 Using differential decoding in the BPSK transmitter, bit 1 will be transmitted by a shift the phase of the modulated signal π relative to the previous phase of the modulated signal. Bit 0 will be transmitted without a shift of the phase of the modulated signal relative to the previous modulated signal. The encoding operation is described mathematically by where S n ðtÞ is the baseband signal, S 0 n ðtÞ is the DPSK signal, and S 0 n−1 ðt − τ B Þ is the adjacent DPSK signal at a time interval of τ B . τ B is the duration of one bit. The È denotes the Boolean XOR operator.
DPSK modulation for free-space optical communication through an atmospheric channel is based on the interference of two successive data bits in an unequal-arm-length interferometer (depicted by Fig. 2). It is the time-delay selfhomodyne interference without independent LO and optical phase-locked loop (OPLL). The relations between the baseband signal SðtÞ, DPSK signal S 0 ðtÞ, DPSK phase signal φðtÞ, delayed DPSK phase signal φðt − τ d Þ, phase of the self-homodyne interference φðtÞ − φðt − τ d Þ, and recovered signal S r ðtÞ are shown in Fig. 3. The one-to-one match between the baseband signal and received interference pattern should be underlined. The delayed time τ d and the corresponding delayed OPD in air ΔL OPD can be determined by E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 0 2 ; 3 2 6 ; 1 3 8 where B is the transmission bandwidth and c is the velocity of light. The delayed OPD corresponds to the duration of one bit: at a data rate of 2.5 Gbps, the duration of one bit is 0.4 ns, equivalent to an OPD of 120-mm in air; at a data rate of 5 Gbps, the duration of one bit is 0.2 ns, equivalent to an OPD of 60-mm in air. The coherence length of the light needs to be larger than the OPD in the interferometer, and the delayed OPD has to be stabilized to a fraction of the wavelength. Then the interferometer will differentiate the data from two successive bits and convert the relative phase into intensity modulation: if two successive bits of data are identical, the output will be constructive interference; if two successive bits of data are different, the output will be destructive interference. The received bit sequence will recover the original bit sequence as transmitted by Fig. 2 The schematic diagram of time-delay self-homodyne interference DPSK receiver-based on the interference of two successive data bits in an unequal-arm-length interferometer. the homodyne BPSK. Because the unequal-arm-length interferometer superimposes the identical wave fronts, the optimum modulation contrast is obtained despite a distorted input wave front. This is the physical basis of time-delay self-homodyne interference DPSK systems. In this way, the time-delay self-homodyne interference DPSK receiver can be immune from wave front distortion when passing through the atmospheric turbulence and is suited for high data rate space-to-ground coherent optical communication links. In this paper, the free-space unequal-arm-length MZI is used as the time-delay self-homodyne interference DPSK optical receiver.

Free-Space Unequal-Arm-Length MZI-DPSK Receiver with 2 × 4 90-deg Optical Hybrid
The schematic diagram of an unequal-arm-length MZI-DPSK receiver with 2 × 4 90-deg optical hybrid is presented in Fig. 4. The Jones' polarization matrix calculus is employed to describe the light wave field. The input signal with the unknown polarization can be given by E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 0 3 ; 6 3 ; 5 2 7 where E S is the amplitude of the light wave. k 1 and k 2 are the components of the x and y coordinates, respectively. φðtÞ is the modulated phase function of the signal beam. ψ N ðtÞ is the noise phase caused by atmospheric turbulence and is the difficult problem to be resolved in this research. Note that the time factor of the light wave has been ignored. The matrix of half-waveplate HWP1 is Λ HWP1 ¼ cos 2α 1 sin 2α 1 sin 2α 1 − cos 2α 1 , where α 1 is the angle of the principal axes tilted to the x coordinate. Thus the passed field E ⇀ 0 S is given by E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 0 4 ; 6 3 ; 3 5 7 8 > > < > > : where η and κ are the component of the x and y coordinates, respectively. PBS1 and PBS2 act as the splitter and combiner in the unequal-arm-length MZI, respectively. PBS1 is considered to be a nonideal polarization beam splitter. The reflection and transmission matrix of PBS1 are given by E Q -T A R G E T ; t e m p : i n t r a l i n k -; s e c 2 . 2 ; 6 3 ; 2 3 4 where ρ v_1 is the phase shift of the reflection matrix; τ p_1 is the induced phase shift of the transmission matrix; r v_1 is the amplitude of the reflection matrix; t p_1 is the amplitude of the transmission matrix. After passing the splitter PBS1, the two divided beams in the unequal-arm-length MZI can be given by E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 0 5 ; 3 2 6 ; 4 8 3
Assuming that the half-waveplate HWP2 has a principal axis tilted by π 8 to the x coordinate, the matrix can be given by E Q -T A R G E T ; t e m p : i n t r a l i n k -; s e c 2 . 2 ; 3 2 6 ; 3 3 2 Assuming that the QWP has a principal axis tilted by π 4 þ ε to the x coordinate, the matrix can be given by E Q -T A R G E T ; t e m p : i n t r a l i n k -; s e c 2 . 2 ; 3 2 6 ; 2 6 6 where ε ≪ π 2 and the small angle approximation can be performed. Taking the delay time τ d and the random noise phase σðtÞ between the two optical paths into consideration, the passed fields before the combiner PBS2 can be presented as E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 0 6 ; 6 3 ; 1 6 4 8 > > > > > < > > > > > : PBS2 is also considered to be a nonideal polarization beam splitter. The reflection and transmission matrices of PBS2 are given by E Q -T A R G E T ; t e m p : i n t r a l i n k -; s e c 2 . 2 ; 6 3 ; 7 2 8 After passing the combiner PBS2, the divided two beams can be given by E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 0 7 ; 6 3 ; 6 2 3 Assuming that the half-waveplates HWP3 and HWP4 have the principal axes tilted by α 3 and α 4 to the x coordinate, respectively then the matrix can be given by where both PBS3 and PBS4 are considered to be nonideal polarization beam splitters. The matrices of PBS3 and PBS4 are given by E Q -T A R G E T ; t e m p : i n t r a l i n k -; s e c 2 . 2 ; 3 2 6 ; 5 2 0 The four output fields are given by Thus, the four electrical signal outputs can be given by E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 0 9 ; 6 3 ; 2 8 5 The corresponding DC coefficients can be presented as  I dc_in;1 ¼ 1 2 ½r v_1 t p_2 t p_3 κ cos 2α 3 E S 2 k in þ 1 2 ½t p_1 r v_2 t p_3 η sin 2α 3 E S 2 k in I dc_in;2 ¼ 1 2 ½r v_1 t p_2 r v_3 κ sin 2α 3 E S 2 k in þ 1 2 ½t p_1 r v_2 r v_3 η cos 2α 3 E S 2 k in I dc_qu;1 ¼ 1 2 ½t p_1 t p_2 t p_4 η cos 2α 4 E S 2 k qu þ 1 2 ½r v_1 r v_2 t p_4 κ sin 2α 4 E S 2 k qu I dc_qu;2 ¼ 1 2 ½t p_1 t p_2 r v_4 η sin 2α 4 E S 2 k qu þ h 1 2 r v_1 r v_2 r v_4 κ cos 2α 4 E S i 2 k qu ; (10) where k in and k qu are the photodetector responsivities of the inphase and quadrature-phase channels, respectively. Two pairs of 180-deg phase-shifted outputs are obtained: one pair (I in;1 and I in;2 ) is the in-phase branch, and the other pair (I qu;1 and I qu;2 ) is the quadrature-phase branch.
In order to realize balanced detection, the DC components must be eliminated. The tilted α 3 and α 4 of HWP3 and HWP4 can be satisfied by E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 1 1 ; 3 2 6 ; 6 6 6 8 < : The in-phase and quadrature-phase output can be given by E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 1 2 ; 6 3 ; 5 6 6 8 > > > > > < > > > > > : If the small angle ε can be satisfied by E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 1 3 ; 6 3 ; 4 6 6 ε ¼ ρ v_2 − τ p_2 : The residual phase difference induced by PBS2 can be removed. So, rotating the quarter waveplate QWP by a small angle ε, the accurate phase difference π 2 between the in-phase and quadrature channels can be guaranteed despite the polarization behavior of beam combiner PBS2.
The noise phase ψ N ðtÞ caused by atmospheric turbulence is considered to be slow varying compared with the high-speed data rate. So ψ N ðtÞ can be removed in the unequalarm-length MZI-DPSK receiver by E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 1 4 ; 3 2 6 ; 4 5 5 The mathematical derivation manifests the feasibility to eliminate the noise phase caused by atmospheric turbulence by use of the unequal-arm-length MZI-DPSK receiver.
For the high-speed balanced detectors, the in-phase channel and quadrature-phase channel photocurrents are given by E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 1 5 ; 6 3 ; 3 4 6 I in ¼ I in;1 − I in;2 ¼ 1 2 r v_1 t p_1 r v_2 t p_2 κη sin 4α 3 k in E 2 S cos½φðtÞ − φðt − τ d Þ − σðtÞ I qu ¼ I qu;1 − I qu;2 ¼ 1 2 r v_1 t p_1 r v_2 t p_2 κη sin 4α 4 k qu E 2 S sin½φðtÞ − φðt − τ d Þ − σðtÞ : The phase can be extracted by multiplying both the inphase and quadrature-phase signals. Finally, the high-speed phase signal φðtÞ − φðt − τ d Þ can be extracted from the obtained phase after being high-pass filtered. The lowspeed random phase noise σðtÞ produced between the two optical paths in the unequal-arm-length can be extracted from the obtained phase after being low-pass filtered. The closed-loop feedback control can be realized by compensating the low-speed random phase noise in order to stabilize the OPD and eventually optimize the receiver performance.

Design of High Data Rate Optical Communication Link
The block diagram of the high data rate space-to-ground optical communication link based on a free-space unequalarm-length MZI-DPSK receiver is shown in Fig. 5. The optical communication link includes the transmitting module, receiving module, and atmospheric channel. The transmitting module is composed of a 1550-nm laser source, digital source, differential encoder, high-speed phase modulator, erbium-doped fiber amplifier, and transmitting telescope. Taking the high-speed relative motion between the satellite and the ground terminal into consideration, only the circularly polarized light can be fit for the future satellite-ground optical communication link. So the circularly polarized light is used and transmitted from the ϕ50-mm transmitting telescope in this work. The nonreturn-to-zero DPSK method is used for laser beam modulation, and the bit rate is up to 2.5 and 5 Gbps, respectively. The receiving module is composed of a receiving telescope, tracking control loop for intensity stabilization, unequal-arm-length MZI-DPSK receiver, OPD control loop for OPD stabilization, and a signal processing unit. The near-diffraction-limited beam is propagated from the transmitting telescope to a ϕ300-mm receiving telescope over an atmospheric channel with a distance of 2.4 km between two buildings in downtown.

Design of Receiving Module Based on
Free-Space Unequal-Arm-Length MZI-DPSK Figure 6 depicts the schematic of the receiving module based on a free-space unequal-arm-length MZI-DPSK. The circularly polarized input beam light from the receiving telescope is collimated, and the tip/tilt correction is performed at the internal pupil using a fast steering mirror (FSM). After being reflected by the FSM, the circularly polarized beam passes a QWP, which converts it into linearly polarized light. The linearly polarized signal beam can be adjusted by the HWP in front of PBS1. PBS1 acts as a splitter for beam position detection and unequal-arm-length MZI-DPSK. The beam position detection is performed by Xenics infrared CCD camera. The infrared CCD camera, FSM, image acquisition card (IAC), digital-to-analog converter (DAC), and control computer compose the tracking control module in order to inhibit the influence of beam wandering, 32 angle-of-arrival fluctuation, and scintillation. 33 The double 4-f confocal lens groups (L1-L4) work as the pupil matching in order to eliminate the wave front aberration resulting from the diffraction in the unequal-arm-length MZI for optimized coherent detection. 34 The maximum system insertion loss is measured to be <1 dB.
The self-developed 10-GHz bandwidth avalanche photodiode balanced detectors with double aspheric lenses are used for the high-speed balanced photodetection. The fieldprogrammable gate array (FPGA) is used for high-speed data acquisition, phase demodulation, and corresponding filtering. The random phase noise is extracted, and the feedback control signal is generated for the finely motorized platform, which works as an OPD compensation. The high-dynamics, very stable piezo-nano-positioner is applied as the finely motorized platform to optimize and stabilize the OPD. The positioning resolution is 0.05 nm and the pitch is AE5 μrad. The high-response bandwidth of the hardware is up to 10 kHz. In this work, the OPD can be stabilized to below one thousandth of the wavelength. Figure 7 shows the 3D design of a free-space unequalarm-length MZI-DPSK receiver for a 5-Gbps optical Fig. 6 The schematic of the receiving module based on unequal-arm-length MZI-DPSK receiver. PBS, polarization beam splitter; λ∕2, half-waveplate; λ∕4, quarter-waveplate; FSM, fast steering mirror; L, lens; AL, aspheric lens; RAP, right-angle prism (protected gold coating); FPGA, field-programmable gate array; ADC, analog to digital converter; DAC, digital-to-analog converter; and IAC, image acquisition card. communication link. Note that the input collimator is just for system debugging and will be removed during the link experiment. Figure 8(a) shows the picture of the high data-rate free-space optical communication link based on free-space unequal-arm-length MZI-DPSK receiver. Figure 8(b) shows the picture of a free-space 2 × 4 90-deg optical hybrid with a compact structure. Figure 8(c) shows the picture of an orthogonal receiving module including the free-space 2 × 4 90-deg optical hybrid and the balanced photodetectors. Figure 8(d) shows the working electronic equipment, including the BER Tester, digital oscilloscope, electro-optic phase modulator, and power supply, during the indoor link testing.

OPD Measurement in Free-Space
Unequal-Arm-Length MZI The data rate of 2.5 Gbps is equivalent to an OPD of 120 mm in air, and the data rate of 5 Gbps is equivalent to an OPD of 60 mm in air. From Eq. (2), the error of the bit rate can be determined by E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 1 6 ; 3 2 6 ; 5 0 2 where B is the transmission bandwidth, c is the velocity of light, and ΔL OPD is the delayed OPD in air. Thus, the measurement accuracy of OPD in a free-space unequal-armlength MZI is very critical for a high data-rate optical communication link. However, accurate OPD measurement in unequal-arm-length MZI is difficult to achieve by the general geometry method. Here, a scheme based on the deramping method from a chirped laser is presented and shown in Fig. 9. The chirped frequency for the OPD measurement can be given by E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 1 7 ; 3 2 6 ; 3 4 8  where f 0 is the initial frequency and f is the chirp rate of the frequency. The generated beat frequency F b can be given by E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 1 8 ; 6 3 ; where λ is the chirp rate of the wavelength. So, the OPD in an unequal-arm-length MZI can be calculated by measuring the beat frequency. In this work, the chirp light source is a Newfocus TLB-6800 single-mode tunable laser, and the chirp rate of the wavelength is 250 nm∕s. The spectral resolution of the spectrum analyzer is 1 nm. In this way, the measurement accuracy of the OPD in an unequal-armlength MZI is 0.01 mm.

Double-bit-rate 2.5/5-Gbps Free-Space Unequal-Arm-Length MZI-DPSK Optical Receiver
Because of the one-to-one correspondence between the bit rate and OPD, the bit rate of the optical communication link can only be determined by the OPD in an unequal-armlength MZI. In order to realize the variable bit rate optical communication link between LEO/GEO satellite and OGS, a double-bit-rate 2.5∕5-Gbps free-space unequal-arm-length MZI-DPSK optical receiver is proposed in Fig. 10. The unequal-arm-length MZI for 2.5 Gbps is composed of PBS2, PBS3, PBS8, PBS9, L1, L2, L5, and L6. The unequal-armlength MZI for 5 Gbps is composed of PBS2, PBS3, PBS6, PBS7, L1, L2, L3, and L4. By the gate actions of the HWP and PBS, the 2.5-and 5-Gbps bit rates can be adopted, respectively. The minimum crosstalk between the two channels is larger than 30 dB.

Atmospheric Measurement
The parallel and separate atmospheric measurement along the optical communication link is also performed simultaneously by the scintillometer between two buildings in downtown over a distance of 2.4 km. The typical 24-h (from 7:30 am, breeze, clear weather) atmospheric measurement results including Fried coherence parameter r 0 and turbulence structure profile C 2 n are shown in Figs. 11(a) and 11(b). Figures 11(c) and 11(d) show the pictures of the transmitter (TX) and receiver (RX) modules of the Scintec scintillometer, respectively. From Fig. 10(a) and 10(b), as the representatives of atmospheric turbulence, the nonlinear variations of r 0 and C 2 n have an approximately inverse relation. In the morning, the atmospheric turbulence changes violently because of the combined action of sunshine and wind. The typical r 0 is within 10 mm, and the typical C 2 n is larger than 5 × 10 −13 m −2∕3 . In this strong turbulent environment, the typical C 2 n is even larger than 10 −12 m −2∕3 . In the afternoon, the atmospheric turbulence tends to weaken as the temperature decreases. The r 0 varies from about 10 to 30 mm, and the C 2 n varies from 10 −13 to 10 −14 m −2∕3 in this relatively moderate turbulent environment. In the night, the typical r 0 varies from about 20 to 40 mm, and the typical C 2 n is 10 −14 m −2∕3 in this relatively weak turbulent environment. Most of our optical communication link Fig. 10 The schematic of the double-bit-rate 2.5∕5-Gbps free-space unequal-arm-length MZI-DPSK optical receiver. PBS, polarization beam splitter; λ∕2, half-waveplate; λ∕4, quarter-waveplate; and L, lens. The 2 × 4 90-deg optical hybrid is indicated by the yellow dashed box.  experiments were performed within the relatively weak turbulent environment in the night.

Link Experiments
The 2.5-and 5-Gbps optical communication links have been verified between two buildings in downtown over a distance of 2.4 km. The relatively stable eye-diagrams acquired by the free-space unequal-arm-length MZI-DPSK receiver are shown in Figs. 12(a) and 12(b). The pseudorandom codes with finite-length (1 Mbit), which are differentially encoded, are cyclically transmitted for measuring the BER. Figure 12(c) shows the oscilloscope screenshot of the acquired pseudorandom binary signal. The acquisition time is usually set as 2 to 5 min. Then the BER can be measured by the BER tester or FPGA and is shown in Fig. 12(d). The experiments prove that the obtained BER is better than 10 −6 . The co-measured r 0 varies from 20 to 37 mm, which is shown in Fig. 12(e).

Discussions
Although DPSK modulation can only reduce the temporal phase errors in optical fiber communication, 31 it is proved that both temporal and spatial phase errors can be eliminated by DPSK modulation and a pupil-matching time-delay selfhomodyne interference DPSK optical receiver in a free-space optical communication link. In this work, the pupil-matching time-delayed self-homodyne interference DPSK optical receivers based on the free-space unequal-arm-length MZI and 2 × 4 90-deg optical hybrid are designed and verified for a space-to-ground optical communication link. To maintain identical wave fronts, the entrance pupil is imaged in both arms onto the same exit pupil. Because the wave fronts within the duration of one bit cannot change because atmospheric turbulence effects are much slower, this ensures that the unequal-arm-length MZI can superimpose identical wave fronts, and optimum modulation contrast is obtained despite a distorted input wave front. Because only free-space optics is used in the time-delay self-homodyne interference DPSK optical receiver, the effects of atmospheric turbulence on the fiber-coupled DPSK system can be neglected where the coupling efficiency of spatial light to single-mode fiber is limited and fluctuates randomly in a satellite-to-ground downlink. 35 The maximum system insertion loss is measured to be <1 dB, which can offer support for SNR. Optical communication links through the atmosphere can be challenging depending on the characteristics of the turbulence for the location, time, and the elevation angle of the transmitted beam. During the optical communication link experiments, the co-measured r 0 varies from 20 to 35 mm. Compared with the published atmospheric value measured by ESA, 13 the atmospheric conditions can be considered to be worst case. So the free-space unequal-arm-length MZI-DPSK optical receiver can overcome the worst-case atmospheric conditions. This work is a solid foundation for the future satellite-to-ground optical communication link between OGS, which is located at an elevation of 2080 m residing at Urumchi, China, and has better atmospheric conditions, and the coherent laser communication terminal on the Chinese Mozi satellite.
The smaller aperture compared with the Fried coherence parameter has been adopted for the satellite-to-ground BPSK communication links in order to weaken the impairment of atmospheric turbulence because no AO and other mechanisms can be used at that moment. 13,14 However, both low receiver power and low receiver power stability will limit the SNR of satellite-to-ground BPSK laser communication links. Moreover, the strong intensity fluctuations cannot be compensated by aperture averaging. Therefore, the small-aperture BPSK receivers were only attempted at excellent astronomical locations (e.g., mountain tops), where the atmospheric conditions are consistent with the link requirements. On the one hand, a time-delay self-homodyne interference DPSK receiver can overcome the atmospheric turbulence. On the other hand, the sensitivity of the time-delay self-homodyne interference DPSK receiver is considerably reduced compared to an ideal homodyne BPSK receiver, which employs a LO to ensure the shotnoise limited performance. The large aperture antenna can receive the most transmitted signal and the SNR can be improved accordingly. A larger aperture receiver in the OGS allows the transmitter power and aperture size of the coherent laser communication terminal on the spacecraft to be reduced significantly. So the ϕ300-mm receiving telescope, which is much larger than the Fried coherence parameter, is used in this work. The ϕ1.2-m larger-aperture receiving telescope and the free-space unequal-arm-length MZI-DPSK optical receiver in the OGS will be applied to the next step satellite-to-ground optical communication downlink.
While the free-space unequal-arm-length MZI-DPSK receiver can effectively eliminate the noise phase caused by atmospheric turbulence, it cannot effectively inhibit the intensity fluctuation caused by beam wander, 32 arrival-ofangle fluctuation, and scintillation. 33 The present experimental results confirm that the stability of the eye-diagram for the MZI-DPSK receiver is deteriorated when the turbulence structure profile C 2 n exceeds 10 −12 m −2∕3 in the worst-case atmospheric conditions. The tracking control loop with tip/tilt correction by an FSM can deal with the problem to some extent. Now the high-dynamic range optical automatic gain controller is in development. The multiaperture DPSK receivers with inherent superposition are also under research.
In this work, the data rate of 2.5 Gbps is equivalent to an OPD of 120 mm in air, and the data rate of 5 Gbps is equivalent to an OPD of 60 mm in air. Compared with the traditional BPSK optical communication link, the maximum transmission bit rate of time-delay self-homodyne interference DPSK receiver is limited by the geometry of freespace unequal-arm-length interferometer. The higher datarate DPSK receiver with an integrated Michelson interferometer structure is under development. Due the high-speed relative motion between the satellite and the ground terminal, the terminals on the OGS receive a high Doppler frequency shift over the satellite-ground channel. The amount of Doppler shift is estimated to be more than gigahertz, which depends on the altitude of orbit of the LEO/GEO-ground downlink. In future work, the influence of the Doppler shift on high data-rate optical communication link will also be investigated carefully.

Conclusions
In conclusion, a pupil-matching time-delay self-homodyne interference DPSK receiver based on a free-space unequalarm-length MZI and 2 × 4 90-deg optical hybrid for a high data-rate space-to-ground optical communication link is presented. The free-space unequal-arm-length MZI splits the signal into two paths and recombines them. The delayed OPD is equivalent to the duration of one bit. This method, based on the free-space difference of two successive wave fronts, requires neither AO nor OPLL. In addition to detailed theoretical analysis, the relatively stable 2.5-and 5-Gbps optical communication link has already been performed with a ϕ300-mm receiving telescope between two buildings in downtown over a distance of 2.4 km. The measured BER is better than 10 −6 in the worst-case atmospheric conditions, which is monitored by atmospheric measurement. Both theoretical and experimental results can manifest that the time-delay self-homodyne interference DPSK receiver can overcome the distorted wave front and eliminate the noise phase caused by the atmospheric turbulence. The time-delay self-homodyne interference DPSK receiver has great significance for the future developments of satellite-to-ground optical communication link.