Abstract

Single mode fiber (SMF) birefringence effects have been a limiting factor for a variety of Sagnac applications over longer distance SMF links. In this report, we present a new concept of the SMF birefringence compensation in a Sagnac interferometric setup, based on a novel polarization control system. For the destructive interference, our control system guarantees a perfect compensation of both the SMF birefringence and imperfect propagation times matching of the setup’s components. For the stabilization of the constructive interference, we have applied a fiber stretcher and a simple proportional-integral-derivative (PID) controller. The enclosed experimental data of the setup’s visibility confirm validity of our polarization control system. We have also showed that the SMF birefringence model used in a “plug & play” interferometric setup [19], widely cited in the papers on quantum key distribution [11, 12, 13], cannot be applied in SMF Sagnac interferometric setup. However, the SMF birefringence model based on the Kapron equivalence well describes SMF Sagnac.

© 2009 Optical Society of America

1. Introduction

Fiber Sagnac interferometers have mainly been used in rotation sensing and gyroscope applications (described, for instance in [1, 2, 3]). In these applications, the birefringence influence on the measurement accuracy is really not any major issue. However, in remote measurement applications, requiring long fiber Sagnac loops (such as measurements of slowly varying environmental signals [1, 3], distributed Sagnac sensing [1], a fiber Sagnac rotational seismometer [4, 5], a fiber Sagnac gravitational telescope [3], and Sagnac hydrophones [3]) the fiber birefringence is an important design factor. Also in the maturing quantum information area an increased interest could recently be noticed in Sagnac quantum key distribution (QKD) and Sagnac quantum secret sharing over SMF telecom links [6, 7, 8, 9, 10]. In these single photon applications, the SMF birefringence limits the secure transmission distance and also requires use of polarization insensitive phase modulators, which are commercially unavailable [11, 12, 13]. Since SMF birefringence effects have been a limiting factor for the Sagnac applications over longer SMF links, we present here a new concept of the SMF birefringence effects compensation in Sagnac [1, 14, 15, 16], based on a polarization control system. The system has been proven working well in a three-party Sagnac quantum secret sharing application over total Sagnac SMF transmission loop distances of 55–75 km [10 ]. However, the paper [10] has not included a general mathematical model of our polarization control system. Here, we present the model (developed for a classical light source) providing a perfect compensation, in regards to the destructive interference, of both the SMF birefringence and imperfect propagation times’ matching of the setup’s components. For the stabilization of the constructive interference, we have applied a fiber stretcher and a simple PID controller.

2. Birefringence in Sagnac

Figure 1 shows a model of a simplified SMF Sagnac interferometer without attenuation losses and with the coupler coupling ratio k = 0.5, assumed to be the same for both parts (Ex and Ey) of the electric vector E, i.e. kx = ky. The x-part of the electric vector will later be called the horizontal, while the y-part the vertical component. The analysis can easily be extended to Sagnac interferometers with attenuation losses [16]. Our goal for this section is to show how birefringence in Sagnac influences its output signals. A more detailed analysis, without the above assumption of kx = ky, can be found in [16].

The interferometer’s birefringence will be analyzed here by using the Kapron equivalence [16, 17, 18] that states that polarization behavior of any unknown optical system or medium is always equivalent to the effects of the three following basic devices: a rotation of the input light axis (inclination) by an angle Φ to yield the optical axes of the retarder, a linear retardation R, and an additional axis rotation Ω corresponding to a circular retardation, as it is shown in Eq. (1)

Jsys=R(Ω)·eiR/200eiR/2·R(Φ),

where the general rotation matrix is given by

R(Θ)=cosΘsinΘsinΘcosΘ.
 

Fig. 1. SMF Sagnac interferometer without attenuation losses [16]. Ein1, Eout1, Eout4, Eclk2, Ecnt2, Eclk3, and Ecnt3 denote the electric field amplitudes. The CPL coupling ratio k = 0.5 is assumed to be the same for both Ex and Ey, i.e. kx = ky.

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By substituting Eq. (2) into Eq. (1) the latter one can be reexpressed as [16]

Jsys=PQ*QP*,

where

P=cos(R/2)cos(Φ+Ω)isin(R/2)cos(ΦΩ),
Q=cos(R/2)sin(Φ+Ω)+isin(R/2)sin(ΦΩ),

and P *,Q * are complex conjugates of P and Q, respectively.

The Jones matrix for the fiber loop in the counterclockwise direction is the transpose of the clockwise matrix [16]

Jsmfclk=PQ*QP*,
Jsmfcnt=(Jsmfclk)T=PQQ*P*.

The clockwise vector Eclk2 and counterclockwise one Ecnt3 are easily found using the electric field amplitude transmission equation for a symmetrical (kx = ky = 0.5) coupler

E2clk=22E1in,
E3cnt=i22E1in,

where the vector Ein1 is given by

E1in=E1xinE1yin.

The clockwise and counterclockwise vectors Eclk3 and Ecnt2 are given by

E3clk=JsmfclkE2clk,
E2cnt=JsmfcntE3cnt.

Finally, the output vectors Eout1 and Eout4 are found by again applying the electric field amplitude transmission equation for a symmetrical (kx = ky = 0.5) coupler

E1out=i22E3clk+22E2cnt,
E4out=22E3clk+i22E2cnt.

In the ideal case (assuming no birefringence and kx = ky = 0.5), the Eq. (13) corresponds to the constructive interference, while the Eq. (14) corresponds to the destructive one. It can be easily shown (by substituting, first Eq. (4) and Eq. (5) into Eq. (6) and Eq. (7), and finally Eq. (6 ) and Eq. (7) into Eq. (11) and Eq. (12)) that both constructive and destructive interference strongly depends on the asymmetry between the clockwise and counterclockwise birefringence (described by Eqs. (4) – (7)) unless the inclination rotation Φ is assumed to be equal to -Ω (the axis rotation corresponding to a circular retardation). However, such an assumption does not correspond to the general SMF birefringence.

3. Birefringence compensation in the “plug and play” interferometric setup

Figure 2 shows a “plug & play” interferometric setup, widely used in quantum key distribution experiments (QKD) over SMF links [11, 12 , 13]. The setup’s schematics is simplified to show the birefringence compensation part only. In the QKD experiments the laser pulses are always very weak (“faint”) since QKD requires single photons. However, the setup’s intrinsic birefringence compensation feature does not put any special requirement on the power of the laser pulses, assuming a laser for typical telecom applications.

 

Fig. 2. “Plug & play” interferometric setup. 1 denotes the input, T transmitted output, and R reflected output of the polarization beam splitter, while 1 denotes the input, T the transmitted output in direction 1 → T, and R transmitted output in direction T→R of the circulator. The long arm includes the delay line DL, which causes an additional signal delay compared to the short arm, which does not have such a device.

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In detail, a laser pulse is sent through the circulator CIR to the 50/50 coupler CPL that splits it into two pulses propagating through a short arm and a long arm (that includes a delay line), respectively. The pulses arrive to a polarization beam splitter (PBS) and leave the interferometer INT by the same PBS port (1): the long arm pulse vertically and the short one horizontally polarized. Then, the long arm and short arm pulses are transmitted over SMF link and reflected at the Faraday mirror, that rotates their polarizations by 90°, and transmitted back. Both pulses continue their backward propagation over SMF link and are directed by the PBS to the opposite arms (in regards to their forward propagation) of the interferometer INT so they arrive at the same time to the coupler CPL, where they interfere. Depending on the phase difference (that could be controlled by one or more phase modulators, not shown here) between the pulses they are detected either in the detector DET2 if the phase difference φ = 0 ± 2, where n is an integer number; or in DET1 if φ = π ± 2; or in both detectors for other φ values. Thus, the “plug & play” system is auto-compensated for propagation time difference between the long arm and short arm pulses since they travel the same path length. For a high interference visibility, the long arm pulse should arrive to the PBS horizontally and the short arm vertically polarized. Thus, the SMF link’s birefringence should be analyzed in order to find how it influences the polarization change of the pulses. Furthermore, the pulses should arrive to the coupler CPL at the same amplitude.

In [19] a similar analysis to the one mentioned in Section 2 (i.e. assuming Ω = -Φ) of the SMF birefringence has been used to show that a Faraday mirror terminating a SMF link provides a full compensation of the link birefringence for a light signal propagating forward from the light source to the Faraday mirror and returning backward to the source. Since the model assumes Ω = - Φ, the Jones matrix for the forward SMF birefringence simplifies to the same matrix as in [19]

Jf=PQQP*,
P=cos(R/2)isin(R/2)cos(2Φ),
Q=isin(R/2)sin(2Φ),
P*=cos(R/2)+isin(R/2)cos(2Φ).

The Jones matrix for the backward SMF birefringence is given by

Jb=PQQP*.

The matrix has been obtained by transposing the forward one. Its off-diagonal components are negated since the matrix describes the backward propagation through the SMF link. The negation follows the standard convention for backward propagating fields described in a right-handed coordinate system with reversed z and x axes [12].

The entire system Jones matrix (for the forward and backward propagation through the SMF) is thus given by

Jsys=JbJFMJf,

where the Jones matrix for Faraday mirror is given by [19]

JFM=011o.

By substituting Eq. (18) into Eq. (17) we are finally getting

Jsys=PQQP*0110PQQP*=0110

Equation (19) shows that the Jones matrix for the entire system (forward and backward) has reduced to the Jones matrix for Faraday mirror, which obviously shows a desired birefringence compensation. However, the birefringence matrices in the equation have been incorrectly simplified by assuming Ω = -Φ. This, not correct, birefringence model of the SMF has been widely cited in the papers dedicated to quantum key distribution. Finally for the “plug & play” birefringence compensation, it should be mentioned that the horizontal and vertical parts of the electric field vector sent forward are swapped upon its arrival to the interferometer INT (after the forward and backward propagation). Nothing similar can be found in Sagnac interferometer (see Section 2).

Finally for this section, it should be emphasized that the birefringence matrices with the general P (Eq. (4)) and Q (Eq. (5)) components (also without assuming Ω = -Φ) confirm validity of the birefringence compensation in the “plug & play” setup.

4. Analysis of single mode fiber birefringence compensation in Sagnac

Figure 3 shows a SMF Sagnac interferometer with birefringence compensation (see Section 5 for a detailed description of the setup’s components). The setup is analyzed here on a classical (non-quantum) signal level. However, it works equally well on quantum levels (see [10] describing Sagnac quantum secret sharing). A coherent light source (laser) is connected through the circulator (CIR) to the coupler (CPL3) with the coupling ratio k = 0.5, equally dividing the light into clockwise and counterclockwise parts that later enter two interferometers INT1 and INT2, respectively.

 

Fig. 3. SMF birefringence compensation in Sagnac interferometer. All the components inside the boxes, marked with dot dashed lines, are polarization maintaining. Couplers CPL1, CPL2, and CPL3 coupling ratios are k=0.5. The delay line’s (DL) length is matched to the stretcher’s (STR) length. The SMF denotes single mode fiber. For the circulator, 1 denotes the input, T the transmitted output in direction 1 → T, and R transmitted output in direction T → R. For the polarization beam splitters PBS1 and PBS2, with the outputs aligned to the slow (horizontal) axis, 1 denotes the input, T transmitted output, and R reflected output. The signal transmitted from the output R into the input 1 shifts its polarization (from the horizontal to the vertical), while there is no such shift for the signal transmitted from the output T into the input 1. DET1, DET2, and DET3 denote photo detectors.

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The outputs of the interferometers are terminated with a SMF loop. The interferometer INT2 contains a polarizing beam splitter (PBS2) and a 50/50 coupler (CPL2). The INT1 has additionally a fiber stretcher (STR), controlled by a digital acquisition card (DAQ), and a delay line (DL), whose length is matched to stretcher’s fiber length. The interferometers are used for removing the SMF birefringence effects on the clockwise and counterclockwise pulses by converting their polarization (into the horizontal one) after they have propagated over the Sagnac loop. The conversion is necessary since the birefringent SMF, as already mentioned, differently changes polarization of the clockwise and counterclockwise pulses, which would decrease interference’s visibility in the coupler CPL3, being a part of the INT3. The stretcher’s main function is to minimize the amount of energy loss in the couplers’ CPL1 and CPL2 arms not connected into the coupler CPL3. This energy is monitored by an additional detector (DET3) connected to the coupler CPL1. The detector’s output is read, once per reading period TR, by the DAQ card. After the reading process, a simple proportional-integral-derivative (PID) controller adjusts stretcher’s length, thus the phase of the signal in stretcher’s arm. This arrangement maximizes the amount of energy flowing into the coupler CPL3 and minimizes the interference losses in the couplers CPL1 and CPL2.

All components of the interferometers INT1, INT2, and INT3 are polarization maintaining and aligned to the horizontal (slow) axis. Also outputs of the polarizing beam splitters PBS1 and PBS2 are aligned to the horizontal axis as well as the interconnecting fiber cords. The signal transmitted from the polarizing beam splitter’s output R into the input 1 shifts its polarization (from the horizontal to the vertical), while there is no such shift for the signal transmitted from the output T into the input 1. The clockwise and counterclockwise pulses leaving the interferometers Int.1 and Int.2 (see Fig. 3) are generally elliptically polarized (due to the propagation timing skews δ 1 and δ 2, discussed below), which is shown in the following Jones vector

J=ExeiδxEyeiδy.

An imperfect matching of the propagation times over the upper and lower arms of the interferometers Int.1 and Int.2 contributes to timing skews between the horizontal and vertical components, which has been emphasized in Fig. 3 by two arrows showing phase differences (δ 1 and δ 2) between the arms in the interferometers Int.1 and Int.2, respectively.

The analysis of the SMF Sagnac interferometer with birefringence compensation (see Fig. 3) will be carried out in a similar way to the analysis of the Sagnac interferometer model in Section 1. The electrical vectors’ indices will be denoted with the small characters (a, b, c, d) in accordance with the Fig. 3.

The clockwise vector Eclka and counterclockwise one Ecntd are found by applying the equation for the transmission of the electric field amplitude in a symmetrical (k=0.5) coupler

Eaclk=22Ecirclk,
Edcnt=i22Ecirclk,

where the vector Eclkcir , the clockwise electric field amplitude at the CIR output

Ecirclk=EcirclkEciryclk=Ecirxclk0,

since both the circulator CIR and the laser are aligned to the horizontal axis. The Jones matrix for the clockwise light propagation over the interferometer INT1 is given by

Jint1clk=2210ieiδ10.

Thus, the clockwise vector Eclkb is given by

Ebclk=Jint1clkEaclk=22Jint1clkEcirclk.

After passing the SMF link the clockwise vector changes to

Ecclk=JsmfclkEbclk=22JsmfclkJint1clkEcirclk.

The Jones matrix for the clockwise light propagation over the interferometer INT2 is given by

Jint2clk=22ieiδ200.

The fact that this matrix is not a transpose of the Jones matrix for Interferometer INT1 (Eq. (24)) depends on the order in which the inputs of the couplers CPL1 and CPL2 have been connected into the coupler CPL3. The operator i shows that the clockwise signal transmitted from the port R of the PBS2 into the port d of the coupler CPL3 is phase shifted by π/2 relative to the signal on the port T. The opposite applies for the interferometer INT1 (Eq. (24)).

The clockwise vector Eclkd is given by

Edclk=Jint2clkEcclk=22Jint2clkJsmfclkJint1clkEcirclk.

Similarly to the analysis of the clockwise vector, the counterclockwise one is, after passing over the Sagnac interferometer, given by

Eacnt=i22Jint2cntJsmfcntJint2cntEcirclk,

where the Jcnt int1 and Jcnt int2 are the transpose matrices of the matrices Jclk int1 (Eq. (24)) and Jclk int2 (Eq. (27)), respectively [16]

Jint1cnt=221ieiδ100,
Jint2cnt=2210eiδ20.

Finally, after both signals (Eclkd and Ecnta) have interfered in the CPL3, we are getting the following vectors at the detectors DET1 and DET2

EDET1=22Edclk+122Eacnt,
EDET2=i22Edclk+22Eacnt,

After substituting all parts (already calculated in this section) one can finally get the following expression for the EDET1 and EDET2

EDET1=00,
EDET2=ED2x0,

where

ED2x=12Ecirxclk{sinR2[(eiδ1eiδ2)sin(ΦΩ)
i(ei(δ1+δ2)1)cos(ΦΩ)]
cosR2[(ei(δ1δ2)+1)cos(ΦΩ)
i(eiδ1+eiδ2)sin(Φ+Ω)]}.

Equations (34) – (36) show an expected result regarding polarization of the vectors EDET1 and EDET2 . Both have horizontal components only, which has been anticipated taking into account the polarization conversion in the interferometers INT1 and INT2, see Fig. 3. What really astonish is the fact that the horizontal component E D1x of the vector EDET1 has vanished, providing a perfect destructive interference, independent of the birefringence and the propagation time skews δ 1 and δ 2! This counterintuitive finding is our main result reported here. It opens the door to many Sagnac applications over longer distance SMF links, including high accuracy measurements, quantum key distribution, and quantum secret sharing [10].

The constructive interference does not show the perfect features of the destructive one (except for the fact that the vertical component E D1y of the vector EDET2 has vanished). Instead, the horizontal component E D2x of the vector EDET2 depends on both the birefringence and the propagation time skews δ 1 and δ 2. However, the visibility of the interference remains perfect (i.e. the visibility V= 1) since the destructive interference, as already mentioned, does neither depend on the SMF birefringence nor on the propagation time skews in the interferometers INT1 and INT2. In order to control the stability of the constructive interference we have, as already mentioned, applied a fiber stretcher, see Fig. 3.

The experimental data of the setup’s visibility are presented in the next section.

5. Experimental data

In Section 4 we have shown the principle of our birefringence compensation in the SMF Sagnac interferometer (see Fig. 3). In our experimental setup, we have used a pulsed 1550 nm laser generating 1 mW, 500 ps wide pulses at 2 MHz repetition rate. The fiber stretcher (from General Photonics Inc.) was controlled by the NI6602, a digital acquisition card (DAQ) from National Instrument. As photo detectors, we have used InGaAs avalanche photodiodes from Princeton Lightwave Inc. at the classical (non-quantum) signal level (see [10] for quantum experimental results describing Sagnac quantum secret sharing), with the gating pulse’ width of 2 ns.

The system timing was controlled by the pulse generator from Quantum Composers Inc. in the burst mode, with the duty cycle of 20 %. Thus the effective laser pulse repetition rate was 400 kHz. The total SMF-28 fiber (from Corning Inc.) loop length was 50 km. The measurement density has been set to the one detectors’ (DET1 and DET2) reading per second due to the fact that the DAQ card’s reading period of the photo detector DET3, being a part of our PID regulator, was set to 1 sec. Finally, it should be emphasized that the measurements were carried out in a noisy environment due to the temporary construction works, close to our lab.

6. Conclusions

Our new concept of the SMF birefringence effects compensation (based on a novel polarization control system) in a Sagnac interferometric setup opens the door to a variety of Sagnac-based applications over longer SMF links such as precise optical sensing, dispersion characteristics of optical fibers, acoustic and strain sensing, and, generally, sensing of any time varying phenomenon [1] as well as to quantum information application such as quantum key distribution and secret sharing [10]. We have also showed that the SMF birefringence model used in a “plug & play” interferometric setup [16], widely cited in the papers on quantum key distribution, cannot be applied in SMF Sagnac interferometric setup. However, the SMF birefringence model, based on the Kapron equivalence [17, 16, 18], well describes SMF Sagnac.

 

Fig. 4. Sagnac visibility measurement result for the total SMF fiber loop length of 50 km. The measurements were carried out during one hour at the 1550 nm laser wavelength with pulse repetition rate of 2 MHz in the burst mode with the burst duty cycle of 20 %. The laser pulse level was set to a classical (non-quantum) signal level. The measurement density is one result/sec since the DAQ card’s reading period of the photo detector DET3 was set to 1 sec.

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Acknowledgments

This work was supported by Swedish Defence Material Administration (FMV) and Swedish Research Council (VR).

References and links

1. E. Udd, “Sensing and instrumentation applications of the Sagnac fiber optic interferometer,” in Interferometric Fiber SensingProc. SPIE 2341, 52–59 (1994). [CrossRef]  

2. J. Zheng, “Differential singlemode fibre frequency-modulated continuous-wave Sagnac gyroscope,” Electron. Lett. 41, 727–728 (2005). [CrossRef]  

3. B. Culshaw, “The optical fibre Sagnac interferometer: an overview of its principles and applications,” Meas. Sci. Technol. 17, 1–16 (2006). [CrossRef]  

4. L. R. Jaroszewicz, “Fiber-optic Sagnac Interferometer as real sensor of the physical quantities,” in Proceedings of the Symposium on Photonics Technologies (Wroclaw, Poland, 2006), pp. 99–101.

5. L. R. Jaroszewicz and Z. Krajewski, “Application of fiber-optic Sagnac interferometer for detection of rotational seismic events,” Molecular and Quantum Acoustics 22, 133–134 (2001).

6. T. Nishioka, H. Ishizuka, T. Hasegawa, and J. Abe, “Circular type quantum key distribution,” IEEE Photon. Technol. Lett. 14, 576–578 (2002). [CrossRef]  

7. C. Zhou, G. Wu, L. Ding, and H. Zeng, “Single-photon routing by time-division phase modulation in a Sagnac interferometer,” Appl. Phys. Lett. 83, 15–17 (2003). [CrossRef]  

8. B. Qi, L. L. Huang, H. K. Lo, and L. Qian, “Quantum key distribution based on a Sagnac loop interferometer and polarization-insensitive phase modulators,” in IEEE International Symposium on Information Theory (Institute of Electrical and Electronics Enginners, 2006), pp. 2090–2093. [CrossRef]  

9. W. A. de Brito and R. V. Ramos, “Quantum information technology with Sagnac interferometer: interaction-free measurement, quantum key distribution and quantum secret sharing,” J. Mod. Opt. 55, 1231–1241 (2008). [CrossRef]  

10. J. Bogdanski, J. Ahrens, and M. Bourennane, “Sagnac secret sharing over telecom fiber networks,” Opt. Express 17, 1055–1063 (2009). [CrossRef]   [PubMed]  

11. N. Gisin, G. Ribordy, W. Tittel, and H Zbinden, “Quantum cryptography,” Rev. Mod. Phys. 74, 145–195 (2002). [CrossRef]  

12. D. S. Bethune and W. P. Risk, “Autocompensating quantum cryptography,” New J. Phys. 4, 42.1–42.15 (2002). [CrossRef]  

13. A. Muller, T. Herzog, B. Huttner, W. Tittel, H. Zbinden, and N. Gisin, “Plug and play systems for quantum cryptography,” Appl. Phys. Lett. 70, 793–795 (1997). [CrossRef]  

14. A. Kuzin, H. Cerecedo Nún̈ez, and N. Korneev, “Alignment of a birefringent fiber Sagnac interferometer by fiber twist,” Opt. Commun. 160, 37–41 (1999). [CrossRef]  

15. B. Ibarra-Escamilla, E. A. Kuzin, O. Pottiez, J. W. Haus, F. Gutierrez-Zainos, R. Grajales-Coutin, and P. Zaca-Moran, “Fiber optical loop mirror with a symmetrical coupler and a quarter-wave retarder plate in the loop,” Opt. Commun. 242, 191–197 (2004). [CrossRef]  

16. D. B. Mortimore, ”Fiber loop reflectors,” J. Lightwave Technol. 6, 1217–1224 (1988). [CrossRef]  

17. F. P. Kapron, N. F. Borrelli, and D. B. Keck, “Birefringence in. dielectric optical waveguides,” IEEE J. uantum Electron. QE-8, 222–230 (1972) [CrossRef]  

18. C. Tsao, Optical fibre waveguide analysis (Oxford Science Publ. 1992).

19. M. Martinelli, “A universal compensator for polarization changes induced by birefringence on a retracing beam,” Opt. Commun. 72, 341–344 (1989). [CrossRef]  

References

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  1. E. Udd, "Sensing and instrumentation applications of the Sagnac fiber optic interferometer," in Interferometric Fiber Sensing, Proc. SPIE 2341, 52-59 (1994).
    [CrossRef]
  2. J. Zheng, "Differential singlemode fibre frequency-modulated continuous-wave Sagnac gyroscope," Electron. Lett. 41, 727-728 (2005).
    [CrossRef]
  3. B. Culshaw, "The optical fibre Sagnac interferometer: an overview of its principles and applications," Meas. Sci. Technol. 17, 1-16 (2006).
    [CrossRef]
  4. L. R. Jaroszewicz, "Fiber-optic Sagnac Interferometer as real sensor of the physical quantities," in Proceedings of the Symposium on Photonics Technologies (Wroclaw, Poland, 2006), pp. 99-101.
  5. L. R. Jaroszewicz and Z. Krajewski, "Application of fiber-optic Sagnac interferometer for detection of rotational seismic events," Molecular Quantum Acoustics 22, 133-134 (2001).
  6. T. Nishioka, H. Ishizuka, T. Hasegawa, and J. Abe, "Circular type quantum key distribution," IEEE Photon. Technol. Lett. 14, 576-578 (2002).
    [CrossRef]
  7. C. Zhou, G. Wu, L. Ding, and H. Zeng, "Single-photon routing by time-division phase modulation in a Sagnac interferometer," Appl. Phys. Lett. 83, 15-17 (2003).
    [CrossRef]
  8. B. Qi, L. L. Huang, H. K. Lo, and L. Qian, "Quantum key distribution based on a Sagnac loop interferometer and polarization-insensitive phase modulators," in IEEE International Symposium on Information Theory (Institute of Electrical and Electronics Engineers, 2006), pp. 2090-2093.
    [CrossRef]
  9. W. A. de Brito and R. V. Ramos, "Quantum information technology with Sagnac interferometer: interaction-free measurement, quantum key distribution and quantum secret sharing," J. Mod. Opt. 55, 1231-1241 (2008).
    [CrossRef]
  10. J. Bogdanski, J. Ahrens, and M. Bourennane, "Sagnac secret sharing over telecom fiber networks," Opt. Express 17, 1055-1063 (2009).
    [CrossRef] [PubMed]
  11. N. Gisin, G. Ribordy, W. Tittel, and H.  Zbinden, "Quantum cryptography," Rev. Mod. Phys. 74, 145-195 (2002).
    [CrossRef]
  12. D. S. Bethune and W. P. Risk, "Autocompensating quantum cryptography," New J. Phys. 4, 42.1-42.15 (2002).
    [CrossRef]
  13. A. Muller, T. Herzog, B. Huttner, W. Tittel, H. Zbinden, and N. Gisin, "Plug and play systems for quantum cryptography," Appl. Phys. Lett. 70, 793-795 (1997).
    [CrossRef]
  14. A. Kuzin, H. Cerecedo Nunez, and N. Korneev, "Alignment of a birefringent fiber Sagnac interferometer by fiber twist," Opt. Commun. 160, 37-41 (1999).
    [CrossRef]
  15. B. Ibarra-Escamilla, E. A. Kuzin, O. Pottiez, J. W. Haus, F. Gutierrez-Zainos, R. Grajales-Coutin, and P. Zaca-Moran, "Fiber optical loop mirror with a symmetrical coupler and a quarter-wave retarder plate in the loop," Opt. Commun. 242, 191-197 (2004).
    [CrossRef]
  16. D. B. Mortimore, "Fiber loop reflectors," J. Lightwave Technol. 6, 1217-1224 (1988).
    [CrossRef]
  17. F. P. Kapron, N. F. Borrelli, and D. B. Keck, "Birefringence in. dielectric optical waveguides," IEEE J. Quantum Electron. QE-8, 222-230 (1972)
    [CrossRef]
  18. C. Tsao, Optical fibre waveguide analysis (Oxford Science Publ. 1992).
  19. M. Martinelli, "A universal compensator for polarization changes induced by birefringence on a retracing beam," Opt. Commun. 72, 341-344 (1989).
    [CrossRef]

2009 (1)

2008 (1)

W. A. de Brito and R. V. Ramos, "Quantum information technology with Sagnac interferometer: interaction-free measurement, quantum key distribution and quantum secret sharing," J. Mod. Opt. 55, 1231-1241 (2008).
[CrossRef]

2006 (1)

B. Culshaw, "The optical fibre Sagnac interferometer: an overview of its principles and applications," Meas. Sci. Technol. 17, 1-16 (2006).
[CrossRef]

2005 (1)

J. Zheng, "Differential singlemode fibre frequency-modulated continuous-wave Sagnac gyroscope," Electron. Lett. 41, 727-728 (2005).
[CrossRef]

2004 (1)

B. Ibarra-Escamilla, E. A. Kuzin, O. Pottiez, J. W. Haus, F. Gutierrez-Zainos, R. Grajales-Coutin, and P. Zaca-Moran, "Fiber optical loop mirror with a symmetrical coupler and a quarter-wave retarder plate in the loop," Opt. Commun. 242, 191-197 (2004).
[CrossRef]

2003 (1)

C. Zhou, G. Wu, L. Ding, and H. Zeng, "Single-photon routing by time-division phase modulation in a Sagnac interferometer," Appl. Phys. Lett. 83, 15-17 (2003).
[CrossRef]

2002 (2)

T. Nishioka, H. Ishizuka, T. Hasegawa, and J. Abe, "Circular type quantum key distribution," IEEE Photon. Technol. Lett. 14, 576-578 (2002).
[CrossRef]

N. Gisin, G. Ribordy, W. Tittel, and H.  Zbinden, "Quantum cryptography," Rev. Mod. Phys. 74, 145-195 (2002).
[CrossRef]

2001 (1)

L. R. Jaroszewicz and Z. Krajewski, "Application of fiber-optic Sagnac interferometer for detection of rotational seismic events," Molecular Quantum Acoustics 22, 133-134 (2001).

1999 (1)

A. Kuzin, H. Cerecedo Nunez, and N. Korneev, "Alignment of a birefringent fiber Sagnac interferometer by fiber twist," Opt. Commun. 160, 37-41 (1999).
[CrossRef]

1997 (1)

A. Muller, T. Herzog, B. Huttner, W. Tittel, H. Zbinden, and N. Gisin, "Plug and play systems for quantum cryptography," Appl. Phys. Lett. 70, 793-795 (1997).
[CrossRef]

1994 (1)

E. Udd, "Sensing and instrumentation applications of the Sagnac fiber optic interferometer," in Interferometric Fiber Sensing, Proc. SPIE 2341, 52-59 (1994).
[CrossRef]

1989 (1)

M. Martinelli, "A universal compensator for polarization changes induced by birefringence on a retracing beam," Opt. Commun. 72, 341-344 (1989).
[CrossRef]

1988 (1)

D. B. Mortimore, "Fiber loop reflectors," J. Lightwave Technol. 6, 1217-1224 (1988).
[CrossRef]

1972 (1)

F. P. Kapron, N. F. Borrelli, and D. B. Keck, "Birefringence in. dielectric optical waveguides," IEEE J. Quantum Electron. QE-8, 222-230 (1972)
[CrossRef]

Abe, J.

T. Nishioka, H. Ishizuka, T. Hasegawa, and J. Abe, "Circular type quantum key distribution," IEEE Photon. Technol. Lett. 14, 576-578 (2002).
[CrossRef]

Ahrens, J.

Bogdanski, J.

Borrelli, N. F.

F. P. Kapron, N. F. Borrelli, and D. B. Keck, "Birefringence in. dielectric optical waveguides," IEEE J. Quantum Electron. QE-8, 222-230 (1972)
[CrossRef]

Bourennane, M.

Cerecedo Nunez, H.

A. Kuzin, H. Cerecedo Nunez, and N. Korneev, "Alignment of a birefringent fiber Sagnac interferometer by fiber twist," Opt. Commun. 160, 37-41 (1999).
[CrossRef]

Culshaw, B.

B. Culshaw, "The optical fibre Sagnac interferometer: an overview of its principles and applications," Meas. Sci. Technol. 17, 1-16 (2006).
[CrossRef]

de Brito, W. A.

W. A. de Brito and R. V. Ramos, "Quantum information technology with Sagnac interferometer: interaction-free measurement, quantum key distribution and quantum secret sharing," J. Mod. Opt. 55, 1231-1241 (2008).
[CrossRef]

Ding, L.

C. Zhou, G. Wu, L. Ding, and H. Zeng, "Single-photon routing by time-division phase modulation in a Sagnac interferometer," Appl. Phys. Lett. 83, 15-17 (2003).
[CrossRef]

Gisin, N.

N. Gisin, G. Ribordy, W. Tittel, and H.  Zbinden, "Quantum cryptography," Rev. Mod. Phys. 74, 145-195 (2002).
[CrossRef]

A. Muller, T. Herzog, B. Huttner, W. Tittel, H. Zbinden, and N. Gisin, "Plug and play systems for quantum cryptography," Appl. Phys. Lett. 70, 793-795 (1997).
[CrossRef]

Grajales-Coutin, R.

B. Ibarra-Escamilla, E. A. Kuzin, O. Pottiez, J. W. Haus, F. Gutierrez-Zainos, R. Grajales-Coutin, and P. Zaca-Moran, "Fiber optical loop mirror with a symmetrical coupler and a quarter-wave retarder plate in the loop," Opt. Commun. 242, 191-197 (2004).
[CrossRef]

Gutierrez-Zainos, F.

B. Ibarra-Escamilla, E. A. Kuzin, O. Pottiez, J. W. Haus, F. Gutierrez-Zainos, R. Grajales-Coutin, and P. Zaca-Moran, "Fiber optical loop mirror with a symmetrical coupler and a quarter-wave retarder plate in the loop," Opt. Commun. 242, 191-197 (2004).
[CrossRef]

Hasegawa, T.

T. Nishioka, H. Ishizuka, T. Hasegawa, and J. Abe, "Circular type quantum key distribution," IEEE Photon. Technol. Lett. 14, 576-578 (2002).
[CrossRef]

Haus, J. W.

B. Ibarra-Escamilla, E. A. Kuzin, O. Pottiez, J. W. Haus, F. Gutierrez-Zainos, R. Grajales-Coutin, and P. Zaca-Moran, "Fiber optical loop mirror with a symmetrical coupler and a quarter-wave retarder plate in the loop," Opt. Commun. 242, 191-197 (2004).
[CrossRef]

Herzog, T.

A. Muller, T. Herzog, B. Huttner, W. Tittel, H. Zbinden, and N. Gisin, "Plug and play systems for quantum cryptography," Appl. Phys. Lett. 70, 793-795 (1997).
[CrossRef]

Huttner, B.

A. Muller, T. Herzog, B. Huttner, W. Tittel, H. Zbinden, and N. Gisin, "Plug and play systems for quantum cryptography," Appl. Phys. Lett. 70, 793-795 (1997).
[CrossRef]

Ibarra-Escamilla, B.

B. Ibarra-Escamilla, E. A. Kuzin, O. Pottiez, J. W. Haus, F. Gutierrez-Zainos, R. Grajales-Coutin, and P. Zaca-Moran, "Fiber optical loop mirror with a symmetrical coupler and a quarter-wave retarder plate in the loop," Opt. Commun. 242, 191-197 (2004).
[CrossRef]

Ishizuka, H.

T. Nishioka, H. Ishizuka, T. Hasegawa, and J. Abe, "Circular type quantum key distribution," IEEE Photon. Technol. Lett. 14, 576-578 (2002).
[CrossRef]

Jaroszewicz, L. R.

L. R. Jaroszewicz and Z. Krajewski, "Application of fiber-optic Sagnac interferometer for detection of rotational seismic events," Molecular Quantum Acoustics 22, 133-134 (2001).

Kapron, F. P.

F. P. Kapron, N. F. Borrelli, and D. B. Keck, "Birefringence in. dielectric optical waveguides," IEEE J. Quantum Electron. QE-8, 222-230 (1972)
[CrossRef]

Keck, D. B.

F. P. Kapron, N. F. Borrelli, and D. B. Keck, "Birefringence in. dielectric optical waveguides," IEEE J. Quantum Electron. QE-8, 222-230 (1972)
[CrossRef]

Krajewski, Z.

L. R. Jaroszewicz and Z. Krajewski, "Application of fiber-optic Sagnac interferometer for detection of rotational seismic events," Molecular Quantum Acoustics 22, 133-134 (2001).

Kuzin, A.

A. Kuzin, H. Cerecedo Nunez, and N. Korneev, "Alignment of a birefringent fiber Sagnac interferometer by fiber twist," Opt. Commun. 160, 37-41 (1999).
[CrossRef]

Kuzin, E. A.

B. Ibarra-Escamilla, E. A. Kuzin, O. Pottiez, J. W. Haus, F. Gutierrez-Zainos, R. Grajales-Coutin, and P. Zaca-Moran, "Fiber optical loop mirror with a symmetrical coupler and a quarter-wave retarder plate in the loop," Opt. Commun. 242, 191-197 (2004).
[CrossRef]

Martinelli, M.

M. Martinelli, "A universal compensator for polarization changes induced by birefringence on a retracing beam," Opt. Commun. 72, 341-344 (1989).
[CrossRef]

Mortimore, D. B.

D. B. Mortimore, "Fiber loop reflectors," J. Lightwave Technol. 6, 1217-1224 (1988).
[CrossRef]

Muller, A.

A. Muller, T. Herzog, B. Huttner, W. Tittel, H. Zbinden, and N. Gisin, "Plug and play systems for quantum cryptography," Appl. Phys. Lett. 70, 793-795 (1997).
[CrossRef]

Nishioka, T.

T. Nishioka, H. Ishizuka, T. Hasegawa, and J. Abe, "Circular type quantum key distribution," IEEE Photon. Technol. Lett. 14, 576-578 (2002).
[CrossRef]

Pottiez, O.

B. Ibarra-Escamilla, E. A. Kuzin, O. Pottiez, J. W. Haus, F. Gutierrez-Zainos, R. Grajales-Coutin, and P. Zaca-Moran, "Fiber optical loop mirror with a symmetrical coupler and a quarter-wave retarder plate in the loop," Opt. Commun. 242, 191-197 (2004).
[CrossRef]

Ramos, R. V.

W. A. de Brito and R. V. Ramos, "Quantum information technology with Sagnac interferometer: interaction-free measurement, quantum key distribution and quantum secret sharing," J. Mod. Opt. 55, 1231-1241 (2008).
[CrossRef]

Ribordy, G.

N. Gisin, G. Ribordy, W. Tittel, and H.  Zbinden, "Quantum cryptography," Rev. Mod. Phys. 74, 145-195 (2002).
[CrossRef]

Tittel, W.

N. Gisin, G. Ribordy, W. Tittel, and H.  Zbinden, "Quantum cryptography," Rev. Mod. Phys. 74, 145-195 (2002).
[CrossRef]

A. Muller, T. Herzog, B. Huttner, W. Tittel, H. Zbinden, and N. Gisin, "Plug and play systems for quantum cryptography," Appl. Phys. Lett. 70, 793-795 (1997).
[CrossRef]

Udd, E.

E. Udd, "Sensing and instrumentation applications of the Sagnac fiber optic interferometer," in Interferometric Fiber Sensing, Proc. SPIE 2341, 52-59 (1994).
[CrossRef]

Wu, G.

C. Zhou, G. Wu, L. Ding, and H. Zeng, "Single-photon routing by time-division phase modulation in a Sagnac interferometer," Appl. Phys. Lett. 83, 15-17 (2003).
[CrossRef]

Zaca-Moran, P.

B. Ibarra-Escamilla, E. A. Kuzin, O. Pottiez, J. W. Haus, F. Gutierrez-Zainos, R. Grajales-Coutin, and P. Zaca-Moran, "Fiber optical loop mirror with a symmetrical coupler and a quarter-wave retarder plate in the loop," Opt. Commun. 242, 191-197 (2004).
[CrossRef]

Zbinden, H

N. Gisin, G. Ribordy, W. Tittel, and H.  Zbinden, "Quantum cryptography," Rev. Mod. Phys. 74, 145-195 (2002).
[CrossRef]

Zbinden, H.

A. Muller, T. Herzog, B. Huttner, W. Tittel, H. Zbinden, and N. Gisin, "Plug and play systems for quantum cryptography," Appl. Phys. Lett. 70, 793-795 (1997).
[CrossRef]

Zeng, H.

C. Zhou, G. Wu, L. Ding, and H. Zeng, "Single-photon routing by time-division phase modulation in a Sagnac interferometer," Appl. Phys. Lett. 83, 15-17 (2003).
[CrossRef]

Zheng, J.

J. Zheng, "Differential singlemode fibre frequency-modulated continuous-wave Sagnac gyroscope," Electron. Lett. 41, 727-728 (2005).
[CrossRef]

Zhou, C.

C. Zhou, G. Wu, L. Ding, and H. Zeng, "Single-photon routing by time-division phase modulation in a Sagnac interferometer," Appl. Phys. Lett. 83, 15-17 (2003).
[CrossRef]

Appl. Phys. Lett. (2)

C. Zhou, G. Wu, L. Ding, and H. Zeng, "Single-photon routing by time-division phase modulation in a Sagnac interferometer," Appl. Phys. Lett. 83, 15-17 (2003).
[CrossRef]

A. Muller, T. Herzog, B. Huttner, W. Tittel, H. Zbinden, and N. Gisin, "Plug and play systems for quantum cryptography," Appl. Phys. Lett. 70, 793-795 (1997).
[CrossRef]

Electron. Lett. (1)

J. Zheng, "Differential singlemode fibre frequency-modulated continuous-wave Sagnac gyroscope," Electron. Lett. 41, 727-728 (2005).
[CrossRef]

IEEE J. Quantum Electron. (1)

F. P. Kapron, N. F. Borrelli, and D. B. Keck, "Birefringence in. dielectric optical waveguides," IEEE J. Quantum Electron. QE-8, 222-230 (1972)
[CrossRef]

IEEE Photon. Technol. Lett. (1)

T. Nishioka, H. Ishizuka, T. Hasegawa, and J. Abe, "Circular type quantum key distribution," IEEE Photon. Technol. Lett. 14, 576-578 (2002).
[CrossRef]

J. Lightwave Technol. (1)

D. B. Mortimore, "Fiber loop reflectors," J. Lightwave Technol. 6, 1217-1224 (1988).
[CrossRef]

J. Mod. Opt. (1)

W. A. de Brito and R. V. Ramos, "Quantum information technology with Sagnac interferometer: interaction-free measurement, quantum key distribution and quantum secret sharing," J. Mod. Opt. 55, 1231-1241 (2008).
[CrossRef]

Meas. Sci. Technol. (1)

B. Culshaw, "The optical fibre Sagnac interferometer: an overview of its principles and applications," Meas. Sci. Technol. 17, 1-16 (2006).
[CrossRef]

Molecular Quantum Acoustics (1)

L. R. Jaroszewicz and Z. Krajewski, "Application of fiber-optic Sagnac interferometer for detection of rotational seismic events," Molecular Quantum Acoustics 22, 133-134 (2001).

Opt. Commun. (3)

A. Kuzin, H. Cerecedo Nunez, and N. Korneev, "Alignment of a birefringent fiber Sagnac interferometer by fiber twist," Opt. Commun. 160, 37-41 (1999).
[CrossRef]

B. Ibarra-Escamilla, E. A. Kuzin, O. Pottiez, J. W. Haus, F. Gutierrez-Zainos, R. Grajales-Coutin, and P. Zaca-Moran, "Fiber optical loop mirror with a symmetrical coupler and a quarter-wave retarder plate in the loop," Opt. Commun. 242, 191-197 (2004).
[CrossRef]

M. Martinelli, "A universal compensator for polarization changes induced by birefringence on a retracing beam," Opt. Commun. 72, 341-344 (1989).
[CrossRef]

Opt. Express (1)

Proc. SPIE (1)

E. Udd, "Sensing and instrumentation applications of the Sagnac fiber optic interferometer," in Interferometric Fiber Sensing, Proc. SPIE 2341, 52-59 (1994).
[CrossRef]

Rev. Mod. Phys. (1)

N. Gisin, G. Ribordy, W. Tittel, and H.  Zbinden, "Quantum cryptography," Rev. Mod. Phys. 74, 145-195 (2002).
[CrossRef]

Other (4)

D. S. Bethune and W. P. Risk, "Autocompensating quantum cryptography," New J. Phys. 4, 42.1-42.15 (2002).
[CrossRef]

B. Qi, L. L. Huang, H. K. Lo, and L. Qian, "Quantum key distribution based on a Sagnac loop interferometer and polarization-insensitive phase modulators," in IEEE International Symposium on Information Theory (Institute of Electrical and Electronics Engineers, 2006), pp. 2090-2093.
[CrossRef]

L. R. Jaroszewicz, "Fiber-optic Sagnac Interferometer as real sensor of the physical quantities," in Proceedings of the Symposium on Photonics Technologies (Wroclaw, Poland, 2006), pp. 99-101.

C. Tsao, Optical fibre waveguide analysis (Oxford Science Publ. 1992).

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Figures (4)

Fig. 1.
Fig. 1.

SMF Sagnac interferometer without attenuation losses [16]. Ein 1 , Eout 1 , Eout 4 , Eclk 2 , Ecnt 2 , Eclk 3 , and Ecnt 3 denote the electric field amplitudes. The CPL coupling ratio k = 0.5 is assumed to be the same for both Ex and Ey , i.e. kx = ky .

Fig. 2.
Fig. 2.

“Plug & play” interferometric setup. 1 denotes the input, T transmitted output, and R reflected output of the polarization beam splitter, while 1 denotes the input, T the transmitted output in direction 1 → T, and R transmitted output in direction T→R of the circulator. The long arm includes the delay line DL, which causes an additional signal delay compared to the short arm, which does not have such a device.

Fig. 3.
Fig. 3.

SMF birefringence compensation in Sagnac interferometer. All the components inside the boxes, marked with dot dashed lines, are polarization maintaining. Couplers CPL1, CPL2, and CPL3 coupling ratios are k=0.5. The delay line’s (DL) length is matched to the stretcher’s (STR) length. The SMF denotes single mode fiber. For the circulator, 1 denotes the input, T the transmitted output in direction 1 → T, and R transmitted output in direction T → R. For the polarization beam splitters PBS1 and PBS2, with the outputs aligned to the slow (horizontal) axis, 1 denotes the input, T transmitted output, and R reflected output. The signal transmitted from the output R into the input 1 shifts its polarization (from the horizontal to the vertical), while there is no such shift for the signal transmitted from the output T into the input 1. DET1, DET2, and DET3 denote photo detectors.

Fig. 4.
Fig. 4.

Sagnac visibility measurement result for the total SMF fiber loop length of 50 km. The measurements were carried out during one hour at the 1550 nm laser wavelength with pulse repetition rate of 2 MHz in the burst mode with the burst duty cycle of 20 %. The laser pulse level was set to a classical (non-quantum) signal level. The measurement density is one result/sec since the DAQ card’s reading period of the photo detector DET3 was set to 1 sec.

Equations (42)

Equations on this page are rendered with MathJax. Learn more.

J sys = R ( Ω ) · e iR / 2 0 0 e iR / 2 · R ( Φ ) ,
R ( Θ ) = cos Θ sin Θ sin Θ cos Θ .
J sys = P Q * Q P * ,
P = cos ( R / 2 ) cos ( Φ + Ω ) i sin ( R / 2 ) cos ( Φ Ω ) ,
Q = cos ( R / 2 ) sin ( Φ + Ω ) + i sin ( R / 2 ) sin ( Φ Ω ) ,
J smf clk = P Q * Q P * ,
J smf cnt = ( J smf clk ) T = P Q Q * P * .
E 2 clk = 2 2 E 1 in ,
E 3 cnt = i 2 2 E 1 in ,
E 1 in = E 1 x in E 1 y in .
E 3 clk = J smf clk E 2 clk ,
E 2 cnt = J smf cnt E 3 cnt .
E 1 out = i 2 2 E 3 clk + 2 2 E 2 cnt ,
E 4 out = 2 2 E 3 clk + i 2 2 E 2 cnt .
J f = P Q Q P * ,
P = cos ( R / 2 ) i sin ( R / 2 ) cos ( 2 Φ ) ,
Q = i sin ( R / 2 ) sin ( 2 Φ ) ,
P * = cos ( R / 2 ) + i sin ( R / 2 ) cos ( 2 Φ ) .
J b = P Q Q P * .
J sys = J b J FM J f ,
J FM = 0 1 1 o .
J sys = P Q Q P * 0 1 1 0 P Q Q P * = 0 1 1 0
J = E x e i δ x E y e i δ y .
E a clk = 2 2 E cir clk ,
E d cnt = i 2 2 E cir clk ,
E cir clk = E cir clk E cir y clk = E cir x clk 0 ,
J int 1 clk = 2 2 1 0 i e i δ 1 0 .
E b clk = J int 1 clk E a clk = 2 2 J int 1 clk E cir clk .
E c clk = J smf clk E b clk = 2 2 J smf clk J int 1 clk E cir clk .
J int 2 clk = 2 2 i e i δ 2 0 0 .
E d clk = J int 2 clk E c clk = 2 2 J int 2 clk J smf clk J int 1 clk E cir clk .
E a cnt = i 2 2 J int 2 cnt J smf cnt J int 2 cnt E cir clk ,
J int 1 cnt = 2 2 1 i e i δ 1 0 0 ,
J int 2 cnt = 2 2 1 0 e i δ 2 0 .
E DET 1 = 2 2 E d clk + 1 2 2 E a cnt ,
E DET 2 = i 2 2 E d clk + 2 2 E a cnt ,
E DET 1 = 0 0 ,
E DET 2 = E D 2 x 0 ,
E D 2 x = 1 2 E cir x clk { sin R 2 [ ( e i δ 1 e i δ 2 ) sin ( Φ Ω )
i ( e i ( δ 1 + δ 2 ) 1 ) cos ( Φ Ω ) ]
cos R 2 [ ( e i ( δ 1 δ 2 ) + 1 ) cos ( Φ Ω )
i ( e i δ 1 + e i δ 2 ) sin ( Φ + Ω ) ] } .

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