A novel Stokes parameters coding scheme for free-space coherent optical communication

Abstract

This paper proposes a novel continuous variable coherent optical communication mode. In this mode, two quadrature Stokes parameters are regarded as observed physical quantity, and single linearly polarized component is used as carrier wave. At the sending end, electro-optical amplitude modulator (EOM) of 45° azimuth is used to indirectly complete the linear modulation of S₂ component, and S₃ component is changed by continuously rotating a half-wave plate (HWP). The receiving end adopts the mode of Q-Q-H wave plate are rotated to select the component of measured S₂ or S₃. The circuit of balance homodyne detection is designed, and the detection system is built by combination with LabVIEW to complete signal demodulation. New optical path scheme is verified by both theory and experiment.

Key words： continuous variable coherent optical; Stokes parameters; electro-optical modulator; balance homodyne detection

Introduction

Quantum cryptography communication is a promising technology using the quantum natures of light to achieve the security in telecommunication [1]. Like single photon as quantum key distribution (QKD), continuous variable coherent optical beam can be used to quantum cryptography communication. Compared with the single photon QKD scheme, continuous variable coherent schemes are more attractive because of their high efficiency and compatibility [2]. The continuous variable quantum cryptography setups can easily encode and transmit. Gaussian cryptographic protocols for coherent states have shown its potential on the applications of quantum communication. These protocols are robust due to it can overcome the quantum channel noise.

In the continuous variables coherent optical communication systems, the signal light was injected into a M-Z interferometer. In the Mach-Zehnder (M-Z) interferometer, one beam linearly polarized light was divided into two quadrature coherent components and propagate along the two branches [3]. One of them can be used as the local oscillator (LO), and the amplitude and phase of the other one is Gaussian modulated to carry information. In the output of the M-Z interferometer, two components are coupled for detection. According to the Heisenberg uncertainty principle and quantum no-cloning theorem [4], the two components of coherent light can guarantee the security of signal transmission. In the M-Z interferometer based free-space transmission schemes, the optical coupling and phase synchronization is unstable in the receiver of the systems. Elser et al. [5] proposed a single-mode spatial optical signal transmission scheme to simplify the setup of the continuous variables coherent optical communication scheme. But this scheme cannot achieve the base selection for the Stokes components {\widehat{S}}_{\mathrm{2}} and {\widehat{S}}_{\mathrm{3}} at the receiver end, which is required by the applications of quantum cryptography communication.

In this paper, we demonstrated a scheme using Stokes parameter coding for the continuous variable coherent optical communication scheme to stabilize optical coupling and phase synchronization. The corresponding experimental setup is also demonstrated, which uses SU(2) converter in the detector setup to achieve the base selection.

Theoretical analyses

The state of polarization (SOP) of the input light can be presented by Jones matrix as follows [6-8]:

(1)E={\left(\begin{array}{ll}{E}_x& E_y\end{array}\right)}^{\mathrm{T}}.

When passing through the polarizer and two polarized beam splitters (PBS), the light is transformed to a pure horizontal polarization. The horizontal polarized light is presented as

(2){\mathit{E}}_{\mathrm{1}}={\left(\begin{array}{ll}{E}_x& \mathrm{0}\end{array}\right)}^{\mathrm{T}}.

In this scheme, an electro-optical amplitude modulator (EOM) is used to adjust the phase delay of the horizontal polarized light. Here, electro-optical crystal acts as a phase delay adjustable wave plate, which can be written as follows [9]:

(3){\mathit{E}}_{\mathrm{e}\mathrm{o}\mathrm{m}}=\left(\begin{array}{cc}{\mathrm{e}}^{-\mathrm{i}\tau /\mathrm{2}}{\cos }^{\mathrm{2}}\phi +{\mathrm{e}}^{\mathrm{i}\tau /\mathrm{2}}{\sin }^{\mathrm{2}}\phi & -\mathrm{i}\sin (\tau /\mathrm{2})\sin (\mathrm{2}\phi )\\ -\mathrm{i}\sin (\tau /\mathrm{2})\sin (\mathrm{2}\phi )& {\mathrm{e}}^{-\mathrm{i}(\tau /\mathrm{2})}{\sin }^{\mathrm{2}}\phi +{\mathrm{e}}^{\mathrm{i}(\tau /\mathrm{2})}{\cos }^{\mathrm{2}}\phi \end{array}\right),

where \phi is the angle between the crystal axis and the horizontal direction; \tau is the phase delay of the crystal. After passing through the EOM, the Jones matrix of the signal light becomes

(4){\mathit{E}}_{\mathrm{o}\mathrm{u}\mathrm{t}}=E_x\left(\begin{array}{c}{\mathrm{e}}^{-\mathrm{i}(\tau /\mathrm{2})}{\cos }^{\mathrm{2}}\phi +e^{\mathrm{i}(\tau /\mathrm{2})}{\sin }^{\mathrm{2}}\phi \\ -\mathrm{i}\sin (\tau /\mathrm{2})\sin (\mathrm{2}\phi )\end{array}\right).

As we know,, the SOP of light can be represented by Stokes parameters, which can be mapped to a Poincaré sphere face as shown in Fig. 1.

Fig.1 Poincaré sphere

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So that one point on the sphere face present one polarization states. As shown in Fig. 1 [10], the following formulas can be written.

(5)\begin{array}{c}{S}_{\mathrm{0}}=\sqrt{S_{\mathrm{1}}^{\mathrm{2}}+S_{\mathrm{2}}^{\mathrm{2}}+S_{\mathrm{3}}^{\mathrm{2}}},\\ S_{\mathrm{1}}=S_{\mathrm{0}}\cos (\mathrm{2}\alpha )\cos (2\beta ),\\ S_{\mathrm{2}}=S_{\mathrm{0}}\cos (\mathrm{2}\alpha )\sin (\mathrm{2}\beta ),\\ S_{\mathrm{3}}=S_{\mathrm{0}}\sin (\mathrm{2}\alpha ).\end{array}

In our proposed Stokes parameter coding schemes, S_{\mathrm{2}} or S_{\mathrm{3}} can be encoded by modulating angels of \alpha and \beta, through EOM and magneto-optical modulation (MOM).

Now, the Stokes components of signal light after the EOM can be seen as [11]

(6)\begin{array}{l}{S}_{\mathrm{2}}={\widehat{\mathit{E}}}_{\mathrm{E}\mathrm{O}\mathrm{M}}{\mathit{P}}_{\mathrm{3}}{\mathit{E}}_{\mathrm{E}\mathrm{O}\mathrm{M}},\\ S_{\mathrm{3}}={\widehat{\mathit{E}}}_{\mathrm{E}\mathrm{O}\mathrm{M}}{\mathit{P}}_{\mathrm{4}}{\mathit{E}}_{\mathrm{E}\mathrm{O}\mathrm{M}},\end{array}

where {\mathit{P}}_{\mathrm{3}}, {\mathit{P}}_{\mathrm{4}} are the sandwich matrixes, which are

(7){\mathit{P}}_{\mathrm{3}}=\left[\begin{array}{ll}\mathrm{0}& \mathrm{1}\\ \mathrm{1}& \mathrm{0}\end{array}\right],{\mathit{P}}_{\mathrm{4}}=\left[\begin{array}{ll}\mathrm{0}& \mathrm{i}\\ -\mathrm{i}& \mathrm{0}\end{array}\right].

So S_{\mathrm{2}} and S_{\mathrm{3}} can be described as follows:

(8)\begin{array}{l}{S}_{\mathrm{2}}=-E_x^{\mathrm{2}}{\sin }^{\mathrm{2}}(\tau /\mathrm{2})\sin \mathrm{4}\phi ,\\ S_{\mathrm{3}}=E_x^{\mathrm{2}}\sin (\mathrm{2}\phi )\sin \tau .\end{array}

Here, it is noted that when \phi ={\mathrm{45}}^{\circ }, we can set S_{\mathrm{2}} component to zero and modulate the S_{\mathrm{3}} component independently. Then, we will get S_{\mathrm{3}}=E_x^{\mathrm{2}}\sin \tau. The experimental setup is shown in Fig. 2.

Fig.2 Position of electro-optic crystal

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Taking \phi ={\mathrm{45}}^{\circ } into Eq. (8), the SOP of the signal after EOM can be derived as

(9){\mathit{E}}_{\mathrm{o}\mathrm{u}\mathrm{t}}={\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}{E}_{\mathrm{0}}\left(\begin{array}{c}\cos (\tau /\mathrm{2})\\ -\mathrm{i}\sin (\tau /\mathrm{2})\end{array}\right).

According to Eq. (9), when \tau is small, the intensity of the horizontal component will be much larger than that of the vertical one. In our scheme, we use the stronger horizontal polarized component as the LO light, and use the vertical component as the signal light. The signal and the LO light were then transmitted along on a single spatial transmission mode.

After the EOM, S_{\mathrm{2}} components can be modulated by a MOM to carry the signal.

The simulation shows that when half-wave plate (HWP) rotates on the azimuth, the S_{\mathrm{2}} component of polar light will be changed such as the MOM. When modulating the signal, we consider the delay between the EOM and HWP. Thus the signal and LO are synchronized at the output of HWP. Since the Jones matrix of HWP is [12]

(10){\mathit{E}}_{\mathrm{H}}=-\mathrm{i}\left(\begin{array}{ll}\cos (\mathrm{2}\varphi )& \sin (\mathrm{2}\varphi )\\ \sin (\mathrm{2}\varphi )& -\cos (\mathrm{2}\varphi )\end{array}\right),

where \varphi presents the included angle between the crystal axis and horizontal component of signal light, the optical signal after the MOM can be written as

(11){\mathit{E}}_{\mathrm{o}\mathrm{u}\mathrm{t}\mathrm{2}}={\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}{E}_x\left(\begin{array}{l}-\mathrm{i}\cos (\mathrm{2}\varphi )\cos (\tau /\mathrm{2})-\sin (\mathrm{2}\varphi )\sin (\tau /\mathrm{2})\\ -\mathrm{i}\sin (\mathrm{2}\varphi )\cos (\tau /\mathrm{2})+\cos (\mathrm{2}\varphi )\sin (\tau /\mathrm{2})\end{array}\right).

The Stokes components of the output light are

(12)S_{\mathrm{2}}={\displaystyle \frac{\mathrm{1}}{\mathrm{4}}}{E}_x^{\mathrm{2}}\cos \tau \sin (\mathrm{4}\varphi ),

(13)S_{\mathrm{3}}={\displaystyle \frac{\mathrm{1}}{\mathrm{4}}}{E}_x^{\mathrm{2}}\sin \tau .

According to Eq. (12), when the value of \tau is fixed, we can modulate the S_{\mathrm{2}} by changing the angle \varphi of MOM. Equation (13) indicates that the MOM will not affect the value of S_{\mathrm{3}}.

Experiments

Optical path design

Based on Section 2, we design a free space continuous variables coherent optical communication setup, which is shown in Fig. 3. At Alice end, a light beam at 808 nm was focused by a convex with a focus length of 8.5 mm and passes through an isolator. Then, the horizontal polarizer and two PBS ware used to convert this light into the horizontal polarization status (E_{\mathrm{1}}). After that, we used an EOM (New Focus 4102M) encodes the S_{\mathrm{3}} component. Here, we rotated a HWP to control the S_{\mathrm{2}} component, which simulates the function of a MOM.

At the Bob end, the measurement base is first selected by a Q-Q-H SU(2) convert box which composes one HWP and two quarter-wave plates (QWP). Then, a PBS splits the light into a horizontal polarized component (E_{\mathrm{2}}) and a vertical polarized component (E_{\mathrm{3}}) for detection. A balance homodyne detection circuit was used to detect the optical signal for further digital signal processing using LabVIEW [13].

Fig.3 Integrated optical path structure

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Stokes parameters detection system

At the detection end, the measurement base is selected by rotating the Q-Q-H wave plate. {\widehat{a}}_x, {\widehat{a}}_y are set as the coherent polarization status of input light, and {\widehat{c}}_x, {\widehat{c}}_y as the coherent polarization status of output light. The detecting optical path is shown in Fig. 4.

Fig.4 SU(2) convert box

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Since Stokes component follows the following:

(14)\begin{array}{l}{S}_{\mathrm{2}}=\langle a_x^{†}{a}_y\rangle +\langle a_y^{†}{a}_x\rangle ,\\ S_{\mathrm{3}}=\mathrm{i}(\langle a_y^{†}{a}_x\rangle -\langle a_x^{†}{a}_y\rangle )\mathrm{.}\end{array}

The homodyne detection light density is

(15)I=\langle c_y^{†}{c}_x\rangle -\langle c_x^{†}{c}_y\rangle .

The SU(2) convert box can be regarded as a Jones matrix. The output {\mathit{E}}_{\mathrm{s}\mathrm{u}} is described as

(16){\mathit{E}}_{\mathrm{s}\mathrm{u}}={\mathit{E}}_{\mathrm{Q}}{\mathit{E}}_{\mathrm{Q}}{\mathit{E}}_{\mathrm{H}},

Where {\mathit{E}}_{\mathrm{Q}} and {\mathit{E}}_{\mathrm{H}} present the Jones matrixes of QWP and HWP, respectively. If the crystal axes of three wave plates set of particular angles different angles with the horizontal direction, respectively, Eq. (16) will satisfy the following equations:

(17)I=S_{\mathrm{2}},{\mathit{E}}_{\mathrm{s}\mathrm{u}}={\displaystyle \frac{\sqrt{\mathrm{2}}}{\mathrm{2}}}\left(\begin{array}{ll}\mathrm{1}& \mathrm{1}\\ -\mathrm{1}& \mathrm{1}\end{array}\right),

(18)I=S_{\mathrm{3}},{\mathit{E}}_{\mathrm{s}\mathrm{u}}={\displaystyle \frac{\sqrt{\mathrm{2}}}{\mathrm{2}}}\left(\begin{array}{ll}\mathrm{1}& \mathrm{i}\\ \mathrm{i}& \mathrm{1}\end{array}\right).

The angles between the crystal axis and horizontal direction of three wave plates Q-Q-H set as α, β, ϕ, respectively.

\alpha =\mathrm{0},\beta =\mathrm{0},\phi =-{\displaystyle \frac{\mathrm{3}}{\mathrm{8}}}\mathrm{\pi },\mathrm{m}\mathrm{e}\mathrm{a}\mathrm{s}\mathrm{u}\mathrm{r}\mathrm{e}{S}_2\mathrm{c}\mathrm{o}\mathrm{m}\mathrm{p}\mathrm{o}\mathrm{n}\mathrm{e}\mathrm{n}\mathrm{t};

\alpha =\mathrm{0},\beta =-{\displaystyle \frac{\mathrm{1}}{\mathrm{4}}}\mathrm{\pi },\varphi =-{\displaystyle \frac{\mathrm{1}}{\mathrm{8}}}\mathrm{\pi },\mathrm{m}\mathrm{e}\mathrm{a}\mathrm{s}\mathrm{u}\mathrm{r}\mathrm{e}{S}_{\mathrm{3}}\mathrm{c}\mathrm{o}\mathrm{m}\mathrm{p}\mathrm{o}\mathrm{n}\mathrm{e}\mathrm{n}\mathrm{t}.

From selecting the measurement base, the balance homodyne detection is implemented [14,15].

In Eq. (13), S_{\mathrm{3}} parameter is a sine function of \tau, here \tau ={\displaystyle \frac{\mathrm{\pi }}{V_{\mathrm{\pi }}}}V and V is the modulation voltage of EOM. The result of sine relationship between S_{\mathrm{3}} and V is shown in Fig. 5, where x-axis and y-axis present the drive voltage V and S_{\mathrm{3}} parameter, respectively. The modulation voltage V of the electro-optical crystal is a 1.6 MHz sinusoidal oscillatory wave with a V_{\mathrm{0}} direct current (DC) bias voltage. It can be written as

(19)V=V_{\mathrm{0}}+V_x\sin (\omega t),

where \omega is the frequency of sine wave. V_x is the input drive voltage of EOM.

According to Eqs. (13) and (19), the relationship between S_{\mathrm{3}} and V is described as follows:

(20)S_{\mathrm{3}}={\displaystyle \frac{\mathrm{1}}{\mathrm{4}}}{E}_x\sin {\displaystyle \frac{\mathrm{\pi }}{V_{\mathrm{\pi }}}}[V_{\mathrm{0}}+V_x\sin (\omega t)].

When the change of V_x is fairly small:

(21)S_{\mathrm{3}}\approx {\displaystyle \frac{k}{\mathrm{4}}}{E}_x{\displaystyle \frac{\mathrm{\pi }}{V_{\mathrm{\pi }}}}[V_{\mathrm{0}}+V_x\sin (\omega t)].

Since {\displaystyle \frac{k}{\mathrm{4}}}{E}_x{\displaystyle \frac{\mathrm{\pi }}{V_{\mathrm{\pi }}}} is a constant, let M={\displaystyle \frac{k}{\mathrm{4}}}{E}_x{\displaystyle \frac{\mathrm{\pi }}{V_{\mathrm{\pi }}}}, Eq. (21) is simplified as follows:

(22)S_{\mathrm{3}}\approx M[V_{\mathrm{0}}+V_x\sin (\omega t)]=M{V}_{\mathrm{0}}+M{V}_x\sin (\omega t).

When the detection signal passes through the bandpass filter at the end of the detection circuit, the DC signal is filtered, and then

(23)S_{\mathrm{3}}-M{V}_{\mathrm{0}}=M{V}_x\sin (\omega t).

Results and discussion

When the random selection measurement base is S_{\mathrm{2}}, the relative between the horizontal azimuth \varphi of HWP and the value of S_{\mathrm{2}} component can be described as S_{\mathrm{2}}={\displaystyle \frac{\mathrm{1}}{\mathrm{4}}}{E}_x^{\mathrm{2}}\cos \tau \sin \mathrm{4}\varphi. The corresponding measurement results are shown in Fig. 6.

Fig.5 Relationship between EOM voltage and S_{\mathrm{3}} parameter

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As discussed in Section 3.2, in Eq. (23), we note that the V_x is not linearly relative to the S_{\mathrm{3}}. Thus, we carefully choose a linear operation point of EOM according to the relationship as shown in Fig. 5. To find the linear operation point, we measured the relationship between the detective voltage of S_{\mathrm{3}} and the driving voltage of EOM. According to the measurement, we set the operation point at 2 V.

Fig.6 Relationship between HWP \varphi and S_{\mathrm{2}}

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Then, we modulate the EOM by using a random code at a bit rate of 1.6 MHz which has 16 equal parts and follows the Gaussian distribution. The length of the random code is {\mathrm{10}}^{\mathrm{4}} bits. At the receiver, the signal is linearly demodulated. Figure 7 is the probability distributions of the demodulation signal at different part, which follow the Gaussian distribution.

Fig.7 Gaussian distribution of S_{\mathrm{3}} demodulation signal

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Fig.8 Error code analysis of S_{\mathrm{3}} acquisition signal.

(a) Coding value; (b) decoding value

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Figure 8 shows the input and the output results. We also measure the bit error rate using the array indexing tool in LabVIEW. For {\mathrm{10}}^{\mathrm{4}} samples, the sending-to-receiving bit error rate is measured and lower than {\mathrm{10}}^{-\mathrm{4}}, which indicates that if a linearly modulated operation point can be properly choose in the EOM, the S_{\mathrm{3}} component can be demodulated in our proposed receiver setup.

At the detection end, if we randomly rotate the SU(2) convert box to modulate the signal and singal has quantum properties, then the commutators of S_{\mathrm{2}} and S_{\mathrm{3}} components follow the uncertainty relations [16]:

(24)\mathrm{\Delta }{\widehat{S}}_{\mathrm{2}}*\mathrm{\Delta }{\widehat{S}}_{\mathrm{3}}\ge \left|\langle {\widehat{S}}_{\mathrm{1}}\rangle \right|.

Equation (24) indicates if one Stokes operator is nonzero, the other two Stokes operators cannot be simultaneously measured with certainty because Heisenberg uncertainty principle and quantum no-cloning theorem. According to the Heisenberg uncertainty principle, the quantum broadening effect makes a potential eavesdropper cannot accurately measure the values of S_{\mathrm{2}} and S_{\mathrm{3}} components. Thus the security of the communication is guaranteed. However, we can still select two groups of components ({\widehat{S}}_{\mathrm{2}} and {\widehat{S}}_{\mathrm{3}}) to carry the signal by using the EOM and MOM. Thus in the receiver, we can decode the information by using the relationship of S_{\mathrm{2}}=\langle {\widehat{S}}_{\mathrm{2}}\rangle and S_{\mathrm{3}}=\langle {\widehat{S}}_{\mathrm{3}}\rangle.

Conclusions

In this paper, we proposed a scheme using Stokes parameter coding for a spatial continuous variables coherent optical communication scheme to stabilize optical coupling and phase synchronization. In the receiver, we also demonstrated that an SU(2) convert box can be used to achieve the base selection of the S_{\mathrm{2}} or S_{\mathrm{3}} components. Our experimental results also indicate if the S_{\mathrm{3}} component can be linearly modulated by properly choosing the operation point of EOM, the Stokes parameters can be demodulated in our proposed receiver setup. This scheme resolves the problem that the coherence is hard to be guaranteed in the M-Z interferometer optical path, and it has great advantages in continuous variable coherent optical communication.

Acknowledgements

This work was supported by the National Natural Science Foundation of China (Grant No. 61177072).

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