Influence of laser linewidth and polarization modulator length on polarization shift keying for free space optical communication

Biao Han; Wei Zhao; Xiaoping Xie; Yulong Su; Wei Wang; Hui Hu

doi:10.1364/OE.23.008639

1. Introduction

Polarization shift keying (PolSK) as a candidate for modulation in free space optical communication, has been shown to have 3dB better sensitivity than on−off keying [1–6]. In PolSK format, the digital information is encoded with the state of polarization (SOP) of laser beam. Hence, generating different SOPs as communication signal is one of the key technologies. At present, three schemes are applied to investigate this problem by using two lasers with orthogonal SOPs [5,6], using phase modulator [7], and using polarization modulator [8–10].

The first scheme is shown in Fig. 1. Two orthogonal linearly polarized lasers, which are controlled by the high level and low level of the initial signal respectively, are combined with a polarization beam combiner (PBC) to generate optical communication signal.

Fig. 1 Schematic diagram of PolSK modulation using two orthogonal linearly polarized lasers.

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The second method is shown in Fig. 2. Firstly, the linearly polarized incident light is split into two beams with equal energy and orthogonal polarized direction by polarization beam splitter (PBS). Then, one of the beams is phase modulated with a phase modulator controlled by the initial signal to generate modulated phase difference between the two beams. Lastly, the two beams are recombined with a PBC. As they are coherent, the SOP of the combined light is modulated.

Fig. 2 Schematic diagram of PolSK modulation using phase modulator.

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The last approach is shown in Fig. 3. A polarization modulator (PolM) is employed to generate PolSK signal directly, which is usually made of birefringent crystal and has two orthogonal eigen polarization modes. The incident light into PolM is linearly polarized with 45̊ angle relative to the optical axis. When laser propagating in PolM, because the two eigen polarization modes have equal energy and their phase difference is controlled by the initial signal, its SOP is modulated.

Fig. 3 Schematic diagram of PolSK system using polarization modulator.

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Comparing these three means above, evidently, generating SOP with PolM is the simplest one. However, since laser source is quasi monochromatic with linewidth and PolM is made of anisotropy birefringent crystal, the degree of polarization (DOP) will degraded after modulation because of polarization mode dispersion (PMD) [11,12]. To the authors’ knowledge, the influence of this change on PolSK has never been investigated. In this work we will analyze this issue with coherency matrix, and conclude a laser linewidth requirement for PolSK communication application.

The organization of this paper is as follows: Section 2 builds up a theoretical model to analyze the influence of PMD in PolM on PolSK. Section 3 simulates and discusses the influence on PolSK with different laser linewidth and PolM length. Section 4 gives a conclusion.

2. Theoretical model

In practice, light emitted from laser source is quasi monochromatic with narrow linewidth. Its polarization properties can be described by coherency matrix [13,14].

J = [\begin{matrix} J_{x x} & J_{x y} \\ J_{y x} & J_{y y} \end{matrix}] = [\begin{matrix} 〈 E_{x} E_{x}^{*} 〉 & 〈 E_{x} E_{y}^{*} 〉 \\ 〈 E_{y} E_{x}^{*} 〉 & 〈 E_{y} E_{y}^{*} 〉 \end{matrix}]

Here, J is coherency matrix, J_αβ (α, β = x, y) is matrix elements, E_x and E_y are the two quadrature field components of light, “*” and “<>” denote complex conjugate and average respectively.

The DOP of light P is defined by the ratio of polarized component intensity I_pol to total intensity I_tot. With the aid of coherency matrix, it is found to be

P = \frac{I_{pol}}{I_{tot}} = \sqrt{1 - \frac{4 \det J}{tr J^{2}}}

Where “det” and “tr” denote determinant and trace respectively. Therefore, the key factor to calculate DOP is to find out the expression of field components E_x and E_y. For any light, the field component can be represented in terms of a complex analytic signal E(t). It can be expressed as [14]

E (t) = \frac{1}{π} \int_{0}^{\infty} υ (ω) \exp (j ω t) d ω

Here, ω is circular frequency of light, υ(ω) is amplitude spectrum which is represented by means of Fourier transform as

υ (ω) = {\begin{matrix} \int_{- \infty}^{\infty} E (t) \exp (- j ω t) d t; ω \geq 0 \\ 0 ω < 0 \end{matrix}

Before producing the analyze model, as shown in Fig. 3, we suppose the amplitude of the incident light into PolM is E₀(t), and establish coordinate system with x and y axis being parallel to the polarized direction of ordinary and extraordinary light in PolM respectively. As the incident light has a 45̊ polarized angle to x and y axis, by using Jone Matrix, its field E₀ can be written as

E_{0} = \frac{1}{\sqrt{2}} [\begin{matrix} E_{0} (t) \\ E_{0} (t) \end{matrix}]

with

E_{0} (t) = \frac{1}{π} \int_{0}^{\infty} υ_{0} (ω) \exp (j ω t) d ω

Where υ₀(ω) is the spectrum of laser source defined similar to Eq. (4). The incident light intensity I₀ can be calculated by

I_{0} = tr J = tr 〈 E_{0} \cdot E_{0}^{*} 〉 = \frac{1}{π^{2}} \int_{0}^{\infty} {| υ_{0} (ω) |}^{2} d ω

For simplicity, supposing PolM is lossless, PolSK signal E_a after modulating can be expressed as

E_{a} = [\begin{matrix} E_{x} (t) \\ E_{y} (t) \end{matrix}] = [\begin{matrix} \frac{\exp [j φ (t)]}{\sqrt{2} π} \int_{0}^{\infty} υ_{0} (ω) \exp {(j(ω - ω_{0}) [t - (\frac{d β_{x}}{d ω}) L]} d ω \\ \frac{1}{\sqrt{2} π} \int_{0}^{\infty} υ_{0} (ω) \exp {(j(ω - ω_{0}) [t - (\frac{d β_{y}}{d ω}) L]} d ω \end{matrix}]

In the above equation, ω₀ is center circular frequency of laser source and L is PolM length. φ(t) modulated by the initial signal is the phase difference between x and y field component after PolM, whose propagation constant are β_x and β_y respectively.

The determinant D of the coherency matrix J for PolSK signal is

D = \det J = det 〈 E_{0} \cdot E_{0}^{*} 〉 = \frac{1}{4} (I_{0}^{2} - {| S_{1} |}^{2})

with

S_{1} = \frac{1}{π^{2}} \int_{0}^{\infty} {| υ_{0} (ω) |}^{2} \exp [j(ω - ω_{0}) δ τ_{g} L] d ω

Here, δτ_g denotes the group delay difference between x and y component, which is defined by

δ τ_{g} = \frac{d β_{x}}{d ω} - \frac{d β_{y}}{d ω} \approx \frac{d [(n_{x} - n_{y}) | k |]}{d (c | k |)} = \frac{1}{c} [(n_{x} - n_{y}) - λ \frac{d (n_{x} - n_{y})}{d λ}]

In Eq. (11), λ is the central wavelength of laser source, k is the corresponding wave vector with |k| = 2π/λ, c is light velocity in vacuum, n_x and n_y are the refractive indices of the two eigen polarization modes in PolM.

Substituting Eqs. (7), (9), (10) and (11) into Eq. (2), the DOP of PolSK signal is obtained.

P = \frac{| S_{1} |}{I_{0}} = | γ |

Here, γ is mutual correlation function of x and y component being related to the spectrum of laser source which is usually approximated with Gaussian profile as

{| υ_{0} (ω) |}^{2} = \exp [- (\ln 2) \frac{ω - ω_{0}}{δ ω}]

Here, δω is laser linewidth expressed in circular frequency, with

δ ω = \frac{2 πc}{λ^{2}} δ λ

Where δλ is laser linewidth expressed in wavelength. In this situation, γ can be expressed as [13]

γ = \exp [- {(\frac{δ ω δ τ_{g} L}{2 \sqrt{\ln 2}})}^{2}] = \exp [- {(\frac{πcδ λ δ τ_{g} L}{λ^{2} \sqrt{\ln 2}})}^{2}]

Substituting Eq. (15) into Eq. (12), the DOP of PolSK signal is found.

P = \exp [- {(\frac{πcδ λ δ τ_{g} L}{λ^{2} \sqrt{\ln 2}})}^{2}]

As shown in Fig. 3, at the communication receiver, firstly the SOP of PolSK signal is transformed into linearly polarized whose direction is parallel to one of the eigen axes of PBS by a polarization controller (PC). Then the signal is split by PBS and detected by a balanced detector which integrates two photodetectors (PDs) in it. If PolSK signal is ideal polarized, it can only be detected by one of the PDs. Ignoring the change of light DOP and SOP during free space transmission, the output signal of receiver is

U = R_{u} (t_{FR} {| E_{x} (t) |}^{2} - t_{FR} {| E_{y} (t) |}^{2}) = {\begin{matrix} R_{u} (t_{FR} I_{0} - 0) = + R_{u} t_{FR} I_{0} = + R_{u} I_{R} | E_{x} (t) | \geq | E_{y} (t) | \\ R_{u} (0 - t_{FR} I_{0}) = - R_{u} t_{FR} I_{0} = - R_{u} I_{R} | E_{x} (t) | < | E_{y} (t) | \end{matrix}

Where t_FR is the transmittance of PolSK signal propagating through free space, I_R is the light intensity arriving at the receiver, R_u is voltage responsivity of balanced detector, U is the converted voltage. The bit error rate (BER) B_ER of PolSK communication can be estimated by the following equation [21]

B_{ER} = \frac{1}{2} erfc (\frac{| U |}{\sqrt{2} σ_{n}}) = \frac{1}{2} erfc (\frac{S_{N}}{\sqrt{2}})

Here, |U| is the amplitude of U, σ_n is the standard deviation of noise in receiver, S_N is the signal to noise ratio (SNR) in the ideal situation with no depolarization by PolM, erfc(*) is complementary error function which is defined as

erfc(z) = \frac{2}{\sqrt{π}} \int_{z}^{\infty} \exp (- t^{2}) d t

If PolSK signal is partial polarized with a DOP P, both of the two PDs have voltage outputs. The detected signal U_dep is

U_{dep} = {\begin{matrix} R_{u} {[P I_{R} + \frac{(1 - P) I_{R}}{2}] - \frac{(1 - P)}{2} I_{R}} = + P | U |, | E_{x} (t) | \geq | E_{y} (t) | \\ R_{u} {\frac{(1 - P)}{2} I_{R} - [P I_{R} + \frac{(1 - P) I_{R}}{2}]} = - P | U |, | E_{x} (t) | < | E_{y} (t) | \end{matrix}

Comparing Eqs. (18) and (20), the depolarization by PolM results in extra loss for PolSK communication. We define this depolarization loss L_dep as

L_{dep} = \frac{U - U_{dep}}{U} = 1 - P

The corresponding BER B_ERd is

B_{ERd} = \frac{1}{2} erfc (\frac{| U_{dep} |}{\sqrt{2} σ_{n}}) = \frac{1}{2} erfc (\frac{P S_{N}}{\sqrt{2}})

As erfc(*) is a monotone decreasing function, the depolarization loss will increase the BER.

3. Simulation and discussion

In order to calculate DOP after PolM and its influence on PolSK, the refractive indices of PolM n_x and n_y in Eq. (11) should be confirmed. As PolM is usually made of LiNbO₃, whose two refractive indices are depended on temperature, wavelength and composition which can be expressed by the form of Sellmeier equation [15–20].

n_{i} = \sqrt{A_{i 1} + \frac{A_{i 2} + B_{i 1} (T - T_{0}) (T + T_{0} + 546)}{λ^{2} - {[A_{i 3} + B_{i 2} (T - T_{0}) (T + T_{0} + 546)]}^{2}} + B_{i 3} (T - T_{0}) (T + T_{0} + 546) - A_{i 4} λ^{2}}

Where, A_i₁, A_i₂, A_i₃, A_i₄, B_i₁, B_i₂ and B_i₃ are constants (i = x or y), T₀ and T are reference and operating temperature in degree centigrade respectively. We use the parameters in Ref. 15, where T₀ = 24.5 ̊C, and suppose T = T₀ (room temperature) for simplicity. Then n_x and n_y can be expressed as

n_{x} = \sqrt{4.9048 + \frac{1.1775 \times 10^{- 13}}{λ^{2} - 4.7533 \times 10^{- 14}} - 2.7153 \times 10^{10} λ^{2}}

n_{x} = \sqrt{4.5820 + \frac{0.9921 \times 10^{- 13}}{λ^{2} - 4.4479 \times 10^{- 14}} - 2.1940 \times 10^{10} λ^{2}}

Here λ is central wavelength in meter which is different with Ref. 15. Making use of Eqs. (11), (16), (21), (22), (24) and (25), DOP of PolSK signal, depolarization loss and BER in PolSK communication can be found. It is clear that they depend on PolM length, the central wavelength and linewidth of laser source. As the central wavelength is usually nearby 1550nm for free space optical communication, we just discuss the influence of laser linewidth and PolM length here.

Before simulation, we set SNR of the received signal in the ideal situation S_N = 6, as in this situation BER is 10⁻⁹ which is a common requirement in application. Meanwhile, we set laser central wavelength λ = 1550nm.

3.1 The influence of laser linewidth

We simulated with PolM length L = 50mm, 75mm, 100mm respectively. Figure 4 shows the DOP of PolSK signal and depolarization loss versus laser linewidth. Figure 5 shows the BER of PolSK communication versus laser linewidth. “Ideal” in the two figures means the ideal situation with no depolarization by PolM.

Fig. 4 DOP of PolSK signal and depolarization loss versus laser linewidth.

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Fig. 5 BER versus laser linewidth.

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As shown in Figs. 4 and 5, with the increase of laser linewidth, the general varying trend of DOP is decreasing while the depolarization loss and BER for PolSK communication are increasing. Furthermore, the change curves can be divided into three areas.

1) Area A. Compare with the ideal situation, there is nearly no change of DOP, depolarization loss and BER in this area. It is suitable for PolSK communication to work in this area.
2) Area B. In this area, with the increase of laser linewidth, DOP decreases while depolarization loss and BER increase dramatically. The performance of PolSK communication reduces in this area.
3) Area C. DOP nearly goes to zero, and depolarization loss and BER come to the maximum in this area. PolSK communication is unable to work in this area.

In addition, with the increase of laser linewidth, the longer the PolM length is, the earlier Area B and C emerge.

3.2 The influence of PolM length

We simulated with laser linewidth δλ = 0.005nm, 0.05nm and 0.5nm which are within Area A, B and C respectively as shown in Figs. 4 and 5. The DOP of PolSK signal, depolarization loss and BER of PolSK communication versus PolM length are shown in Figs. 6, 7 and 8 respectively. “Ideal” in the figures means the ideal situation with no depolarization by PolM.

Fig. 6 DOP of PolSK signal versus PolM length.

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Fig. 7 Depolarization loss versus PolM length.

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Fig. 8 BERversus PolM length.

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With the increase of PolM length, the general varying trend of DOP is decreasing while the depolarization loss and BER for PolSK communication are increasing. Meanwhile, it is clear that the influence is disparate with different laser linewidth δλ.

If δλ = 0.005nm, with the variation of PolM length, the change of DOP, depolarization loss and BER is hard to discern. The curves in this situation nearly cannot distinguish from the ideal ones.

When δλ = 0.05nm, with the increase of PolM length, DOP decreases while depolarization loss and BER increase. The performance of PolSK communication reduces in this area.

On condition that δλ = 0.5nm, there is nearly no change of DOP, depolarization loss and BER versus PolM length. However, in this case DOP nearly goes to zero while depolarization loss and BER come to the maximum. PolSK communication is unable to work under such condition.

3.3 Analyze and Discussion

From the simulated results above, we conclude that modulating signal using PolM will bring depolarization loss and extra BER in PolSK communication for the reduction of DOP on PolSK signal. Usually, with the increase of laser linewidth and PolM length, DOP decreases while the depolarization loss and BER increase. Meanwhile, with different laser linewidth and PolM length, the impact to PolSK communication is different.

In theory, all of these are caused by PMD in PolM. According to coherency matrix theory [14], the polarization property can be regard as the coherence character of the two perpendicular polarized components of light. There is a positive correlation between DOP and degree of coherence of the two polarized components (DOCPC).

While generating PolSK signal with PolM, at beginning light is polarized with DOP = 100% and DOCPC = 100%. When light propagating through PolM, because of PMD, the propagation velocity of the two eigen polarization modes are different which resulting in an optical path difference (OPD) between them. In general, OPD L_OPD can be estimated with

L_{OPD} = (n_{x} - n_{y}) L

As the existence of OPD, DOCPC reduces and DOP decreases. Usually, the reduction is evaluated by comparing OPD with light coherent length L_CO which is expressed as

L_{CO} = \frac{λ^{2}}{δ λ}

If L_OPD<<L_CO, because the reduction of DOCPC can be neglected, the decrease of DOP hardly can be aware. If L_OPD>>L_CO, as DOCPC reduces to zero, DOP goes to zero too. Otherwise, DOP is between 0 and 100%.

Obviously, shortening L_CO or lengthening L_OPD will exacerbate the depolarization of PolSK signal. Hence, as laser linewidth δλ is inversely proportional to L_CO and PolM length is proportional to L_OPD, increasing laser linewidth or PolM length will aggravate DOP reduction.

Using the parameters in above work, in case of λ = 1550nm, when L = 50mm, 75mm and 100mm, L_OPD are 3.66mm, 5.49mm and 7.32mm respectively. Comparing these parameters with the coherent length at different laser linewidth, Figs. 4 and 5 can be explained.

Corresponding to δλ = 0.005nm, 0.05nm and 0.5nm, the coherent length L_co are 480.5mm, 48.05mm and 4.805mm respectively. Comparing them with OPD at different PolM length, the analyze result is in accordance with Figs. 6, 7 and 8.

Apparently, Modulating PolSK signal with narrower linewidth laser or shorter PolM is useful to PolSK communication. In practice, PolM length is about 100mm [22], and the BER requirement for communication is 10⁻⁹ in common. In order to make the BER increment caused by depolarization in PolM not exceeding 50%, the laser linewidth should be less than 0.008nm.

4. Conclusion

Taking PMD of PolM into account, we analyzed the change of DOP and its influence on PolSK communication with coherency matrix. The simulated result shows that, DOP decreases after PolM, which will result in depolarization loss and extra BER for PolSK communication. With the increase of laser linewidth and PolM length, the influence is aggravated. In practice, laser linewidth should be less than 0.008nm in order to make the BER increment caused by depolarization not exceeding 50%. The theoretical model in this work can be used to analyze the influence of any PolM on PolSK communication. The laser linewidth requirement presented in this work is useful to select a suitable laser source for PolSK free space optical communication.

Acknowledgments

This work is supported by the Foundation of the Chinese Academy of Sciences (Grant No. CXJJ-14-M09) and the National Natural Science Foundation of China (Grant No. 61231012).

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Influence of laser linewidth and polarization modulator length on polarization shift keying for free space optical communication

Abstract

1. Introduction

2. Theoretical model

3. Simulation and discussion

3.1 The influence of laser linewidth

3.2 The influence of PolM length

3.3 Analyze and Discussion

4. Conclusion

Acknowledgments

References and links

Cited By

Figures (8)

Equations (27)

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