1. Introduction

Environmental sensing based on optical fibers has several advantages over methods based on electrical signals. These advantages include durability, small size, and immunity to electromagnetic interference, which make optical fibers ideal for collecting data in harsh environments. Fiber optical sensing systems detect changes in the surrounding environment by measuring variations in the amplitude, phase, frequency or polarization of light propagating in fibers. Particularly, variations in the state of polarization (SOP) of light caused by altered birefringence in the glass has been used to measure changes in applied strain, magnetic fields and ambient temperature [1–3]. To determine if two laser signals are co-polarized, it is typically necessary to launch them into a polarimeter, where their SOPs are determined with reference to physical polarizers whose orientations in space remain fixed. Directly measuring the degree of co-polarization of two lasers with respect to each other without the need for an external reference could enable novel sensing schemes and new approaches to polarimeter instrumentation.

All-optical signal processing refers to a collection of techniques for manipulating light with light, which is used in a variety of applications for overcoming limitations of conventional sensing methods [4–6]. For example, sidebands generated via nonlinear self-induced phase modulation have been used for extending the sensing range in chirped pulse optical time domain reflectometry (CP-OTDR) [7], refining the resolution of optical frequency domain reflectometry (OFDR) [8] and fiber optical sensors [9], increasing the sensitivity of fiber Bragg grating (FBG) based thermometers [10], and increasing the extinction ratio of pulses used for long range OTDR [11]. When attenuation and chromatic dispersion are neglected, the power of each sideband as a function of the powers of the input lasers can be calculated analytically [12], allowing for deeper insight into the behavior of sinusoidally modulated optical signals in a nonlinear medium and enabling a variety of novel applications. Furthermore, if the wavelength dependence of randomly varying birefringence in the nonlinear medium is low, the power of the $n^{th}$ order sideband is proportional to $\cos ^{2n}(\alpha /2)$, where $0{\circ }\leq \alpha \leq 180{\circ }$ is the angle between the vectors representing the SOPs of the input lasers on the Poincaré sphere. This proportionality relation has been exploited to enhance the sensitivity of a polarimetric strain sensor [13]. In addition, the dependence on $\alpha$ suggests that the SOP of a signal laser can be recovered by launching it into a nonlinear medium along with a reference laser and measuring the power of the generated sidebands.

This paper presents an all-optical approach for polarimeter implementation, where the SOP of a signal laser is measured relative to the SOP of a reference laser without using polarizers for detection. An analytical model is developed for describing the impact of relative polarization between the two lasers on frequency sidebands generated by the Kerr effect. The results of this model are utilized to develop a new method for extracting the SOP of the signal laser from measurements of sideband power. The new polarimetric method is experimentally demonstrated by measuring the SOP of the signal laser with an error on the Poincaré sphere below 10.22°, corresponding to $0.8\%$ of the sphere’s total area. Limitations of the proposed technique are discussed along with potential applications for distributed polarimetric sensing, innovative polarization based multiplexing schemes for telecommunications and all-optical polarimeter instrumentation.

2. Theory

A detailed derivation of the following expressions is presented in [13]. Launching lasers at two different angular frequencies, $\omega _r$ and $\omega _s$ with $\omega _r>\omega _s$, into an optical fiber with a length, $L$, and a waveguide nonlinearity parameter, $\gamma$, the electric field amplitude at the input is given by

 

Fig. 1. (a) Visualization of the angle, $\alpha$, between the SOP of a reference pulse and the SOP of the signal laser represented on the Poincaré sphere. (b) Optical spectrum measured after the Kerr medium.

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Neglecting dispersion, attenuation and random birefringence along the fiber, the Nonlinear Schrödinger Equation (NLSE) reduces to $\partial _z|A(z)\rangle = i\gamma \langle A|A\rangle |A(z)\rangle$, which is solved analytically to obtain an expression for the power of the $n^{th}$ order sideband, $P_{n}=\langle A_{n}|A_{n}\rangle =P_s J^2_{n+1}(\phi _{\text {NL}}) + P_r J^2_{n}(\phi _{\text {NL}})$, where $\phi _{\text {NL}}=2\gamma L \sqrt {P_rP_s}\cos (\alpha /2)$. Introducing the normalization, $x=\gamma L P_r$, $y=\gamma L P_s$, $z_n=\gamma LP_n$, and applying the asymptotic approximation of the Bessel functions for $n>0$ and $0<\phi _{\text {NL}}<\sqrt {1+n}$ leads to

For an arbitrary SOP of the signal laser represented using the normalized Stokes formalism, $\hat {S}=(a,b,c)^T$, and a polarization for the reference laser along the X-direction, $\hat {X}=(1,0,0)^T$, the relative angle is $\alpha _X=\arccos (\hat {S}\cdot \hat {X})=\arccos (a)$. The power of the first order sideband given a reference laser SOP along $\hat {X}$ is $P_{1,X}$, abbreviated as $P_X$, and is given by $P_X=P_{\text {max}}\cos ^2(\alpha _X/2)$, where $P_{\text {max}}=Z_{\text {max,1}}/(\gamma L)$, implying that $\alpha _X=2\arccos (\sqrt {P_X/P_{\text {max}}})=\arccos (a)$. Using the identity, $2\arccos (u)=\arccos (2u^2-1)$ [14], shows that $a=2P_X/P_{\text {max}}-1$. Similarly, a reference laser SOP along the Y-direction, $\hat {Y}=(-1,0,0)^T$, yields $-a=2P_Y/P_{\text {max}}-1$. Adding the expressions for $a$ and $-a$ shows that $P_{\text {max}}=P_X+P_Y$. Therefore, if the sideband powers, $P_X$ and $P_Y$, generated by reference lasers with SOPs corresponding to $\hat {X}$ and $\hat {Y}$ are measured, the Stokes component, $a$, of the signal SOP can be calculated from $a=2 P_X/(P_X+P_Y)-1$. This analysis can be repeated for $\hat {D}=(0,1,0)$ and $\hat {A}=(0,-1,0)$ corresponding to light polarized along the +45° diagonal and –45° anti-diagonal respectively, as well as $\hat {R}=(0,0,1)$ and $\hat {L}=(0,0,-1)$ corresponding to right hand circular light and left hand circular light. Doing so yields the following expression for the measured SOP of the signal laser,

3. Experimental setup and results

The experimental setup utilized to conduct all-optical measurements of the SOP of a signal laser is presented in Fig. 2(a). Figures 2(b)-(e) show the power as a function of time and the power spectral density (PSD) of the light at different points along the optical path. The reference signal is generated by a laser (Laser-R, PS-NLL-1550, TeraXion) emitting at 193397.436 GHz (1550.217 nm) and passed through two Electro Optical Modulators (EOM1, MXPE-LN-10, Photline) (EOM2,OC-192 Modulator, JDS uniphase) driven by a dual pulse generator (DPG, 8130A, Hewlett Packard). To minimize the pedestal of the generated pulses and maximize the extinction ratio, the DPG output is disabled and the DC bias of each modulator is adjusted to minimize the power transmitted through the EOMs, as monitored using a dual port power meter (DPM, ML910B, Anritsu). The DPG is then set to emit 1 ns electrical pulses from both output ports, and the time delay between the two electrical pulses is tuned to ensure that EOM2 only transmits when it receives an optical pulse from EOM1. Using two EOMs further ensures that leakage of continuous wave (CW) light from Laser-R is minimized, leading to a reduction in measurement distortion. The reference pulse is amplified using a polarization maintaining erbium doped fiber amplifier (PM-EDFA, EDFA100P, Thorlabs) and a high power EDFA (HP-EDFA1, APEDFA-C-10-B-FA, Amonics). Subsequently, the reference light passes through a bandpass filter (BPF1, Ultra Narrow Filter, AOS) and is aligned with a polarizer (POL-R) by adjusting a polarization controller (PC-R). Using a primary 1x6 coupler, the reference pulse is split into six different branches, each containing a polarization controller (PC-(1-6)) and a variable optical attenuator (VOA-(1-6)), whereupon a secondary 1x6 coupler recombines the pulses from all six branches. The lengths of the branches are tailored such that the six 1 ns pulses exit the secondary 1x6 coupler with a delay of 5 ns between neighbouring pulses, as illustrated in Fig. 2(b), to minimize power fluctuations caused by interference due to temporal overlap. The six reference pulses are then amplified by an EDFA (AEDFA-PA-25-B-FA, Amonics) followed by a HP-EDFA (HP-EDFA2, AEDFA-33-B-FA, Amonics) before passing through a 3 GHz bandpass filter (BPF2, TFC, TeraXion). Upon reaching a 1:99 coupler (C1), 1% of the propagating light is sent to a photodiode (PD1, DSC10H, Discovery Semiconductors Inc.) attached to an oscilloscope (infiniium DSO81204B, Agilent) sampling at 40 GSa/s. Using the trace measured by the scope, the VOA’s are adjusted to ensure that all six pulses have identical peak powers. The remaining 99% of the reference light from C1 is passed to a 50:50 coupler (C2).

 

Fig. 2. (a) The experimental setup used for measuring the state of polarization, $\hat {S}$, of the signal laser using sidebands generated via the Kerr effect. (b-e): The light in the time- and angular frequency domains at different points along the optical path. Six reference pulses with polarizations aligned along the cardinal directions on the Poincaré sphere will experience different amounts of nonlinear phase modulation when interfering with the signal as described by Eq. (2). From the relative powers of the pulses measured when filtering out the 1st order sideband, the state of polarization of the signal can be recovered using Eq. (3).

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The signal is generated by a distributed feedback laser (DFB) (Laser-S, NLK1556STG, NEL) emitting CW light at 193388.123 GHz (1550.211 nm). The signal polarization is aligned with the transmission axis of POL-S by adjusting PC-S, and then is passed through a programmable polarization controller (PPC, 11896A, Hewlett Packard). A coupler directs 10% of the signal power to a conventional polarimeter (IPM5300, Thorlabs) and the remaining 90% is combined with the six reference pulses using C2. The power of the CW signal is low enough to avoid stimulated Brillouin scattering (SBS) that can distort the measurement. The combined signal and reference light is then passed through an optical isolator to a 5.7 km highly nonlinear fiber (418SG 04611A, Draka Comteq) that leads to the generation of frequency sidebands with a spacing of 9.312 GHz equal to the frequency separation between the signal- and the reference laser. Using a bandpass filter (BPF3, XTM-50, EXFO) light in the 1st order sideband at 193406.749 GHz (1550.062 nm) is extracted and detected using a photodiode (PD2, Model 1592 3.5GHz, New Focus) connected to the oscilloscope, which averages 512 traces to reduce the impact of electrical noise and optical power fluctuations.

To calibrate the setup, the PPC is adjusted such that the SOP of the signal as measured by the conventional polarimeter is along $\hat {X}$. Then, PC-1 and PC-2 are adjusted until the detected power of the first sideband pulse is maximized and the detected power of the second sideband pulse is minimized. For additional accuracy, the signal SOP is changed to $\hat {Y}$ using the PPC, whereupon the power of the first sideband pulse is minimized and the power of the second sideband pulse is maximized. Setting the signal SOP to $\hat {D}$ and $\hat {A}$, the procedure is repeated for sideband pulses 3 and 4. Finally, the same procedure is carried out for SOPs corresponding to $\hat {R}$ and $\hat {L}$ to correctly align sideband pulses 5 and 6. (Dataset 1, Ref. [15]).

Having calibrated the polarizations of the reference pulses, the setup is ready to measure any SOP of the signal. A set of 20 consecutive signal SOPs is generated using the PPC and for each SOP, the peak powers of the six sideband pulses are extracted, whereupon the Stokes parameters are determined using Eq. (3). The Stokes parameters of each signal SOP are also measured by the conventional polarimeter and presented in Figs. 3(a)-(b), showing close agreement with the measurements obtained using the proposed setup. To characterize the measurement error, the angle error, $\alpha _{\text {err}}$, between the Stokes vectors obtained using the conventional polarimeter and the proposed setup is obtained for 190 random signal SOPs, showing that $\alpha _{\text {err,max}}=10.22^{\circ }$. The surface area of a spherical cap for a cone with an apex angle of $2\alpha _{\text {err,max}}$ comprises an error area corresponding to $\Omega _{\text {err,max}}/4\pi =0.8\%$ of the surface of the Poincaré sphere. The error area implies that for a Stokes vector measured using the current setup, the true Stokes vector will at worst be located within a patch centered at the measurement corresponding to 0.8% of the total area of the sphere. In Fig. 3(c), a histogram of the 190 measured values of $\Omega _{\text {err}}/4\pi$ in percentage is presented, which indicates that approximately $70\%$ of measurements have $\Omega _{\text {err}}/4\pi$ below $0.4\%$.

 

Fig. 3. (a) Stokes parameters for a 20-step sweep of polarizations with lines indicating the SOP determined by a conventional polarimeter and the data points representing the SOP determined from measurements according to Eq. (3). (b) The SOP according to the polarimeter and the measured SOP represented on the Poincaré sphere. Figure generated using [16]. (c) Histogram of $\text {N}=190$ values of the area created by a spherical cap with an apex angle of $2\alpha _{\text {err}}$ compared to the total area of the unit sphere, where $\alpha _{\text {err}}$ is the angle on the Poincaré sphere between the measured SOP and the SOP according to the polarimeter. The maximum value of $\alpha _{\text {err}}$ is $10.22^{\circ }$, corresponding to $0.8\%$ of the area of the Poincaré sphere. The discrepancy in (a) between the value of $S_3$ measured by the conventional polarimeter and the value obtained using the presented method as well as the maximum value of $\alpha _{\text {err}}$ are attributed to an imperfect calibration of the SOPs of the referenced pulses caused by randomly varying birefringence in the Kerr medium

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4. Discussion

The model for polarization dependent sideband generation relies on simplifying assumptions. Attenuation in the Kerr medium is ignored, but can be accounted for by replacing the length of the fiber, $L$, with the effective length, $L_{\text {eff}}=(1-\exp (-\alpha _LL))/\alpha _L$, where $\alpha _L$ is the attenuation coefficient. Dispersion is assumed to be negligible, which is a valid assumption when the dispersion length, $L_D$, is much greater than the nonlinear length, $L_{\text {NL}}$, where $L_D=2\pi cT_0^2/\lambda ^2D$, $c$ is the speed of light, $T_0$ is the pulse duration, $\lambda$ is the wavelength of the light, $D$ is the chromatic dispersion, and $L_{\text {NL}}=1/\gamma P$. Using $T_0=2\pi /4\omega _d=27$ ps corresponding to the full width at half maximum of one beat cycle, $\lambda =1550$ nm and $D=3$ ps/(nm$\cdot$km) as reported in [17] leads to $L_D\approx 190$ km. In comparison, the presented model for sideband generation is valid when $0<\phi _{\text {NL}}<\sqrt {1+n}$, implying that using $2\gamma LP=\phi _{\text {NL}}<1$ with $P=\sqrt {P_rP_s}$ leads to $L_{\text {NL}}=1/\gamma P\approx 2L\approx 11.4~$km, indicating that chromatic dispersion can be neglected. The model also neglects the impact of wavelength dependent random birefringence in the Kerr medium. Previous results indicate that for $\omega _d/2\pi \approx 10$ GHz, the total drift in the SOP at the end of the Kerr medium utilized in this experiment amounts to at most 8.58° on the Poincaré sphere [13]. Therefore, neglecting the drift in the relative SOP when the signal and reference propagate through the Kerr medium is valid for the purpose of deriving Eq. (2). However, when calibrating the SOP of the reference pulses by minimizing their power, the SOP drift introduced by wavelength dependent random birefringence means that even if the SOPs are orthogonal at the input of the fiber, some sideband power will always be produced at its output. Therefore, it is not possible to consistently achieve a perfect calibration of the reference pulses. Ideal calibration of the reference pulses can be achieved using a Kerr medium with less wavelength dependent randomly varying birefringence or simply by using a shorter fiber length.

The presented experiment utilizes six reference pulses polarized along the cardinal directions on the Poincaré sphere to recover the SOP of the signal. Utilizing six pulses ensures that the signal SOP is always close to at least one of the references, guaranteeing the measurement of a strong signal inside the dynamic range of the detector. Furthermore, the use of six reference pulses makes the derivation of Eq. (3) and the calibration of the setup simple, but overdetermines the system of equations to be solved. An alternate configuration using only four reference pulses whose SOPs on the Poincaré sphere form a tetrahedron also allows for the determination of the SOP of the signal laser. However, using only four reference pulses increases the complexity of the analysis and increases the measurement error as the signal SOP is less likely to be close to one of the reference SOPs.

In this experiment, imperfections in the electronics of the DPG lead to the generation of pulses that have a high power peak with a duration of 1 ns followed by a low power tail. Along with the effect of wavelength dependent random birefringence, the overlap of the tail of one pulse with the peak of next pulse impedes efforts to consistently calibrate the system. The issue of overlapping pulses can be solved by enhancing the extinction ratio of the pulse generated by the DPG [11]. Enhancing the pulse extinction ratio would allow a closer spacing between the reference pulses, which in turn would allow for a higher sampling rate. Using six pulses spaced 5 ns apart implies a theoretical maximum sampling rate of $1/(6\cdot 5~\text {ns})=33.3~\text {MHz}$. If the tails of the 1 ns pulses are eliminated and a 4-pulse scheme is utilized, the maximum theoretical sampling rate would be $1/(4\cdot 1~\text {ns})=250~\text {MHz}$. Alternatively, in a more complex system where the signal is split so that measurements can be conducted in parallel, the measurement rate only depends on the duration of the reference pulse, which for a 1 ns pulse is 1 GHz. Furthermore, the duration of the reference pulse must be longer than $T_d=2\pi /\omega _d$, which cannot be made arbitrarily small because an increased value of $\omega _d$ increases the error that arises from wavelength dependent random birefringence.

The all-optical signal processing method described in this paper allows the SOPs of two lasers to be compared to each other directly without referencing them to polarizers. A conventional measurement, which also avoids polarizers, could be conducted by shining the two lasers onto the same photodiode and adjusting their polarizations to maximize or minimize the recorded power of the generated beat tone, but the largest $\omega _d$ that can be used in such an experiment is limited by smallest cutoff frequency of the employed electronics. Apart from instrumentation of an all-optical polarimeter, another potential application of direct comparison of the SOPs of two lasers is telecommunications. Just as quadrature amplitude modulation (QAM) relies on measuring the phase difference between the signal laser pulses and a reference laser acting as a local oscillator, the presented method enables measuring the SOP of a signal laser with respect to a reference laser allowing for advanced polarization division multiplexing (PDM) where each point on the Poincaré sphere serves as a channel as visualized in Fig. 4 and experimentally investigated in [18]. Even though the experimental setup presented in this work is costly and implementing advanced PDM requires further expensive additions to allow for parallel detection of the Stokes components, this PDM approach will increase the bandwidth of existing systems by at least 1 order of magnitude making the high cost of the system negligible in comparison to the price of installing additional undersea fiber cables. Furthermore, future advances in photonic integrated circuits (PIC) would allow the system to be implemented on a single chip.

 

Fig. 4. Visualization of advanced polarization division multiplexing scheme with channels represented by red dots located at the cardinal directions of the Poincaré sphere and at the center of each octant.

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Enhanced sensitivity to changes in polarization for the power of higher order sidebands was demonstrated in [13]. If a sideband order higher than $n=1$ is used in the present experiment, the Stokes component, $a$, in $\hat {S}=(a,b,c)^T$ is given by $a=2P_X^{1/n}/(P_X^{1/n}+P_Y^{1/n})-1$, which makes the measured SOP more sensitive for values of $\alpha _X$ close to 180° and less sensitive for values closer to 0°. Extracting multiple sidebands separately could therefore provide a way to measure the SOP with enhanced sensitivity for a range of different angles. Such a flexible approach to enhanced sensitivity could be useful in sensing schemes based on polarization-OTDR when detecting small variations in the SOP caused by environmental disturbances [19].

The presented analysis only utilizes the power of the 1st order sideband but the model derived in [13] allows the SOPs of the sidebands to be predicted. The model relates the SOPs of the sidebands to the properties of the incident light including optical powers, initial SOPs and relative phase of the two lasers. The dependence of the SOP of the sidebands on the properties of the signal laser and the reference laser provides versatile approaches for environmental sensing. Additionally, optical pulses with time dependent SOPs have been utilized for the characterization of material surfaces [20]. Further practical advantages of determining the SOPs of the sidebands will be a subject of future investigation.

5. Conclusion

A method for measuring the SOP of a laser based on an analytical model for polarization dependent frequency sideband generation and all-optical signal processing is demonstrated. Results show that the SOP can be determined with an error of 10.22°, which corresponds to $0.8\%$ of the area of the Poincaré sphere. The presented technique allows for the measurement of the SOP of a signal laser using a reference laser with the potential to increase telecommunication data rates, improve the sensitivity of distributed environmental sensors and enable novel approaches for polarimetric instrumentation.