Polarization-induced noise in resonator fiber optic gyro - OSA Publishing

Polarization-induced noise in resonator fiber optic gyro Huilian Ma,1,* Zhen Chen,1 Zhihuai Yang,2 Xuhui Yu,1 and Zhonghe Jin1 1

Micro-satellite Research Center, Zhejiang University, Hangzhou 310027, China 2

Institute of Tianjin Navigation Instrument Research, Tianjin 300131, China *Corresponding author: [email protected]

Received 9 May 2012; revised 29 August 2012; accepted 29 August 2012; posted 30 August 2012 (Doc. ID 168291); published 24 September 2012

An optical fiber ring resonator (OFRR) is the key rotation-sensing element in the resonator fiber optic gyro (R-FOG). In comparing between different OFRR types, a simulation model that can apply to all cases is set up. Both the polarization crosstalk and polarization-dependent loss in the coupler are fully investigated for the first time to our knowledge. Three different splicing schemes, including a single 0°, a single 90°, and twin 90° polarization axis rotated spices, are compared. Two general configurations of the OFRR are considered. One is a reflector OFRR, the other is a transmitter OFRR. This leads to six different OFRR types. The output stability of the R-FOG with six OFRR types is fully investigated theoretically and experimentally. Additional Kerr noise due to the polarization fluctuation is discovered. The OFRR with twin 90° polarization axis rotated splices is of lower additional Kerr noise and hence has better temperature stability. As the coupler is polarization dependent, we notice that in a reflector OFRR, the straight-through component of the output lightwave, which can be isolated by a transmitter configuration, would produce large polarization fluctuation–induced noise. The experimental results show that the bias stability of the transmitter OFRR is 8 times improved over that of the reflector OFRR, which is in accord with the theoretical analysis. By the analysis and experiments above, it is reasonable to make a conclusion that an R-FOG based on a transmitter OFRR with twin 90° polarization axis rotated splices is of better temperature stability and smaller additional Kerr effect noise. © 2012 Optical Society of America OCIS codes: 060.2370, 060.2800.

1. Introduction

A resonator fiber optic gyro (R-FOG) is a highaccuracy inertial rotation sensor based on the Sagnac effect [1]. An optical fiber ring resonator (OFRR) is the key rotation-sensing element in the R-FOG. Lightwaves circulate many turns in the OFRR to enhance the Sagnac effect. Compared with the interferometric fiber optic gyro (I-FOG) [2,3], the R-FOG has potential in realizing I-FOG–like performance with a coil length of up to 100× shorter than those of I-FOG in a given performance class [4]. In practice, however, its performance achieved to date is still below expectation due to noises from various effects, among which the noise induced by the backscattering and the polarization fluctuation are the most 1559-128X/12/286708-10$15.00/0 © 2012 Optical Society of America 6708

APPLIED OPTICS / Vol. 51, No. 28 / 1 October 2012

important [5]. These two types of noise limit the gyro sensitivity far greater than the shot noise associated with the photodetectors. The backscattering-induced noise is caused by the nonuniformity of the fiber which constitutes the OFRR. It can be reduced below the shot noise limited level by the double phase modulation technique [6]. The polarization fluctuation– induced noise is dominantly caused by the existence of dual eigenstates of polarization (ESOP) in the OFRR and by the temperature-sensitive birefringence of the fiber [7]. Researchers have proposed several structures of the OFRR with polarizationmaintaining (PM) fiber and coupler, such as an OFRR with a single 90° polarization axis rotated splice [7–9], an OFRR with twin 90° polarization axis rotated splices [10], etc. It has been shown that the OFRR with twin 90° polarization axis rotated splices has advantages in suppressing the polarization fluctuation–induced noise in the R-FOG [10].

There are two general configurations of the OFRR used in the R-FOG. One is a reflector OFRR comprising a sensing loop and a coupler; the other is a transmitter OFRR comprising a sensing loop and two couplers. The two configurations and three different polarization axes rotated splicing structures leads to six different OFRR types. In order to further analyze and better design the R-FOG system, a deep analysis of the polarization fluctuation–induced noise is fully developed in this paper. A simulation model based on the transfer matrices is set up. A typical PM coupler, which has polarization-dependent coupling, losses, and crosstalk, is fully investigated in comparative analysis of the polarization-induced noise in OFRRs with three different splicing schemes, including a 0° polarization axis rotated splice, a single 90° polarization axis rotated splice, and twin 90° polarization axis rotated splices. Compared with the OFRR with a 0° polarization axis rotated splice, the latter two OFRRs show significant improvements in temperature stability. Furthermore, an additional Kerr effect due to the polarization fluctuation–induced noise is discovered and investigated for the first time. The OFRR with twin 90° polarization axis rotated splices is of lower additional Kerr effect and hence is of better temperature stability. As the polarizationdependent characteristics of the coupler is considered, we notice that in a reflector OFRR, the straight-through component of the output lightwave, which can be isolated by a transmitter configuration, would introduce large polarization fluctuation. Experimental results show that the bias stabilities of the R-FOG with a transmitter OFRR and a reflector OFRR are 0.018°=s and 0.14°=s over 6000 s, respectively. It is a nearly 8 times improvement in the bias stability, which is in accord with the theoretical expected value. 2. Principles and Analysis

Figure 1 shows two general configurations of the OFRR used in the R-FOG. Figure 1(a) is a reflector OFRR. A transmitter OFRR is shown in Fig. 1(b). The reflector OFRR will be used as a model to analyze the polarization-induced noise in the R-FOG in detail. The coupler is considered to be polarization dependent. The relationships between input and output electric fields at each port of the coupler C1 is given by [11]

E3j E4j

Cr Ct

Ct Cr

Cl 0

0 Cl

E1j : E2j

(1)

The symbol j denotes the CW lightwave and CCW lightwave alternatively. The excess loss matrix Cl , the transmission matrix Ct , and the reflector matrix Cr of the coupler are expressed, respectively, as p 1 − αcx p 0 ; Cl 0 1 − αcy p j kcx p0 ; Ct 0 j kcy p 1 − kcx p 0 ; Cr 0 1 − kcy

(2)

where αcx and αcy are the loss at the coupler C1 for the x and y polarized lightwave, respectively. kcx and kcy are the coupling coefficient of the coupler C1 for the x and y polarized lightwave. Taking into account the mismatch of the polarization axes between the coupler and the fiber loop, Eq. (1) is amended as

Ct Cl T tcw Cr Cl T rccw E1cw ; Cr Cl T rcw Ct Cl T tccw E2cw (3) Ct Cl T tcw Cr Cl T rcw E1ccw E3ccw ; E4ccw Cr Cl T rccw Ct Cl T tccw E2ccw E3cw E4cw

where the polarization crosstalk matrices T rcw, T tcw T rccw , and T tccw are expressed, respectively, as T rcw

cos θcr − sin θcr

sin θcr ; cos θcr

(4a)

cos θct − sin θct

sin θct ; cos θct

(4b)

cos θcr sin θcr

− sin θcr ; cos θcr

(4c)

cos θct sin θct

− sin θct : cos θct

(4d)

T tcw T rccw T tccw

Polarization crosstalk in the OFRR is expected to occur primarily at the coupler. θcr and θct are the equivalent angular deviations to describe the crosstalk in the coupler for the throughport and crossport, respectively. The energy conservation at the coupler is given by [11] jCl E1j j2 jCl E2j j2 jE3j j2 jE4j j2 : Fig. 1. (Color online) Two general configurations of the OFRR. (a) reflector OFRR: L LR LL and ΔL LR − LL ; (b) transmitter OFRR: L L1 L2 L3 L4 and ΔL L1 L4 − L2 − L3 .

(5)

The output intensity at the OFRR is derived, from Eqs. (1)–(5), as 1 October 2012 / Vol. 51, No. 28 / APPLIED OPTICS

6709

C12 −C21 θu =ηr sinηr z;

I 4j jE4j j2 jCl E1j j2 jCl E2j j2 − jE3j j2 H H H H H H H EH 1j fCl Cl − T tj Cl Ct Sj I − F j Cl Cl F j

× Sj Ct Cl T tj gE1j ; Scw

∞ X

S¯ cw n

n0

Sccw

∞ X

(6)

∞ X

Cr Cl T rccw F cw n ;

(7a)

n0

S¯ ccw n

n0

∞ X

Cr Cl T rcw F ccw n :

β

The symbol H denotes complex conjugate transpose and I is a unit matrix. The matrix S¯ j in Eqs. (7a) and (7b) means the one-turn transfer matrix of the OFRR. F j is the Jones matrix of the fiber loop given by F cw Bcw Rcw Acw ;

F ccw Accw Rccw Bccw :

(8)

The transfer matrix before and after the splice can be written as:

Acw Fθu jzLR ; Bcw Fθu jzLL

Accw F−θu jzLR ; Bccw F−θu jzLL

(9)

where θu θs =L. L is the total fiber loop length. LR and LL is the fiber length before and after the splice along the CCW direction. θs is the rotation angle at the splice in the fiber loop. For the analysis, we use a model in which the uniform twist of the fiber θu per unit length is caused by the misalignment θs . The splice matrix for the CW and CCW lightwave in the OFRR is expressed as follows [12]: Rcw Rccw

cos θs sin θs ; − sin θs cos θs p cos θs − sin θs ; 1 − αs · sin θs cos θs

p 1 − αs ·

(10)

where αs is the excess loss at the splice. The transfer matrix of the fiber with the length of z and the twist angle of θs is expressed as [12] Fθu exp−jβz 0 exp−αf x z=2 Cθu ; (11) · 0 exp−αf y z=2 where αf x and αf y are the fiber propagation loss for x and y polarized lightwaves, respectively. Cφ

C11 C21

C12 ; C22

C11 C22 cosηr z − j · Δβ=2ηr sinηr z; 6710

(12a)

(12b)


q ηr Δβ=22 θ2u ;

(12d)

θu θs =z;

(12e)

where θu is the angle of twist per unit of length. β is the average propagation constant of the fiber given by

(7b)

n0

(12c)

β x βy π nx ny ; 2 λ0

Δβ βx − βy

2π 2π n − ny Δn; λ0 x λ0

(12f)

(12g)

where βi and ni (i x, y) are the propagation constant and the refractive index of each polarization mode of the fiber, respectively. λ0 is the wavelength. 3. Polarization-Induced Noise in the R-FOG A. Influence of the Polarization Crosstalk at Coupler

To attenuate the polarization-induced noise, the OFRR used in the R-FOG should be composed, at least, of the PM fiber. It has two special states of polarization (SOP) that do not change the direction of polarization after one round propagation, which are called ESOPs. Each ESOP produces the resonance independently. When the excess loss of the coupler is polarization independent, two ESOPs are orthogonal to each other. The overall output of the OFRR can be expressed as a linear combination of the two ESOPs. The relationship between the two ESOPs depends on the characteristics of the OFRR. By analyzing the resonance curves of the two ESOPs separately, the overall OFRR resonance curve can be obtained. This is the basic method to investigate the polarization-induced noise in the R-FOG. The noise induced by the angular misalignment at the in-loop splice, which can be controlled within 1° accuracy, has been investigated by Takiguchi and Hotate [8,12]. However, the polarization crosstalk of the OFRR is induced dominantly by the angular misalignment of the coupler, e.g., a coupler with a polarization extinction ratio (PER) of 20 dB, an equivalent angular misalignment is about 6°, much greater than the misalignment at the in-loop splice. To simplify the analysis, the angular misalignment at the in-loop splice is neglected, as is the in-loop fiber twist caused by it. 1. OFRR without Polarization Axis Rotated Splice Consider that the rotation angle at the in-loop splice is zero. The coupler is assumed to be polarization independent. The one-turn transfer matrix of the OFRR can be simplified as

S¯ cw Cr Cl T rccw F cw S11cw tf exp−jβL S21cw

S12cw S22cw

;

(13a)

S¯ ccw S¯ cw t Cr Cl T rcw F ccw S11ccw S12ccw tf exp−jβL ; S21ccw S22ccw

βL ξ represents the round-trip phase shift for the two ESOPs in the OFRR. Equation (15) shows that the output of the OFRR is a linear combination of the two ESOPs’ resonance curves. Let us suppose that the input polarization Ej xj ; yj T j CW; CCW onto each of the two ESOPs. V 1j and V 2j , are aj and bj , respectively, given by

(13b)

where ΔβL ; exp −j 2

S11cw S11ccw cos θcr

(14a)

S22cw S22ccw S11cw S11ccw cos θcr ΔβL ; × exp j 2

(14b)

S12cw −S21cw −S12ccw S21ccw ΔβL ; − sin θcr exp j 2

(14c)

S21cw −S12cw S12ccw −S21ccw ΔβL : sin θcr exp −j 2

(14d)

In this case, the rotation angle at the splice equals zero. The output intensity of the OFRR can be simplified as I 4j jaj

j2 jU

2

−αc

1j

j2

jbj

j2 jU

2j

j2 ;

(15)

where jU 1j j e

f1 − ρj Γj βL − ξg;

(15a)

jU 2j j2 e−αc f1 − ρj Γj βL ξg;

(15b)

gf C2cross ρj q2 ; 1 − tf jS11j j2 jS12j j2

(15c)

gf 1 − exp−αf L1 − αsm 1 − αc ;

(15d)

aj bj

xj V1j yj V2j ;

(16)

where jaj j2 =jbj j2 and jxj j2 =jyj j2 are defined as the PERs of the input lightwave and the lightwave transferred in the OFRR. The initial phase difference between xi and yi depends on the input polarization and the state of the fiber before the OFRR. Since the photodetectors (PDs) are polarization insensitive, the overall resonance curve is distorted by the combination, and thus the detected resonance point is shifted. By calculating the extreme points of Eq. (15), the resonance phase detection error caused by polarization fluctuation can be obtained: 2 4tf 2jbcw j2 sin2ξ 2 1 sin ξ ϕerrorp arcsin 1−tf 2 jacw j2 2jbccw j2 sin2ξ −arcsin jaccw j2 2 4tf 2 sin ξ 1 : (17) 1−tf 2

Furthermore, the gyro bias error induced by ϕerrorp can be expressed as Ωerrorp

cλ0 ϕ : 2πLD errorp

(18)

A straightforward method to evaluate the influence of the coupler misalignment on polarization fluctuation is to compare the error induced by temperature change with different coupler misalignment. Figure 2(a) shows the influence of the round trip phase shift difference ΔβL between the two ESOPs, as a function of temperature change, on the distance and the relative PER of the two ESOPs. The OFRR is set to be with zero rotation at the inloop splice and 8° misalignment at the coupler. In Fig. 2(a), the distance between the two ESOPs’ resonance dips increases simultaneously with ΔβL, as does the relative PER, which is nearly 18 dB when

q2 1 − tf jS11j j2 jS12j j2 : Γj x q2 q 2 2 2 2 2 1 − tf jS11j j jS12j j 4tf jS11j j jS12j j sin −x=2

1 October 2012 / Vol. 51, No. 28 / APPLIED OPTICS

(15e)

6711

20

2.0

12

1.5 8 1.0 single 0 deg splice

4

0.5 0.0 0.0

0 0.5

1.0

1.5

2.0

2.5

3.0

temperature change ( ∆ β L )

(a)

10

polarization error / deg/s

16

2.5

100

relative PER of ESOPs / dB

distance of ESOPs / rad

3.0

1

8 degree 1.8 degree

0.1 0.01 1E-3 1E-4 1E-5

0 degree splice

1E-6 1E-7 0.0

0.5

1.0

1.5

2.0

2.5

3.0

temperature change ∆β L / rad

(b)

Fig. 2. (Color online) Gyro bias error caused by the polarization fluctuations. (a) resonance point separation and PER between the two ESOPs. (b) output error of the R-FOG.

the two ESOPs’ resonance dips are far from each other and drops to 0 dB when the two ESOPs’ resonance frequencies coincide, which indicates that the two ESOPs divide the output power equally. As the two ESOPs of the input lightwave are of the same frequency, if the primary ESOP (P-ESOP) is at resonance, the secondary ESOP (S-ESOP) is in a nonresonant state. Since the PD is polarization independent, the power of resonant and nonresonant lightwaves will be both converted to a voltage output, and the detected signal is hence proportional to the sum of output magnitudes of the two ESOPs. Thus, it is natural that the extreme point of the overall resonance curve will be shifted. The shift of the extreme point ϕerrorp would be transferred to a gyro bias error Ωerrorp. Figure 2(b) shows the bias error of the R-FOG as a function of ΔβL. The total ring length of the OFRR is 10 m. The finesse of the OFRR is 50. Two different PER values of the coupler C1 are calculated. As can be seen in Fig. 2(b), it can be found that the polarization-induced noise decreases as the resonance dips apart from each other in the range of 0–π rad. When the misalignment of the coupler is 8°, equivalently a PER of 17 dB, suppressing the output error below the order of 10−3 °=s requires the separation of the two ESOPs’ resonance dips to be between 2.25 and π rad, while the misalignment of the coupler equals 1.8° (PER 30 dB), the requirement of the separation becomes between 0.9 and π rad. The corresponding temperature stability is 0.15 °C and 0.37 °C, respectively. 2. OFRR with a Single 90° Polarization Axis Rotated Splice Consider an OFRR with a single 90° polarization axis rotated splice. Same as before, the misalignment at splice is neglected. For simplicity, Eq. (14) is amended as S11cw S22cw S22ccw S11ccw ΔβΔL ; (19a) sin θcr exp −j 2 6712


S22cw S11cw S11ccw S22ccw ΔβΔL ; sin θcr exp j 2 S12cw −S21cw S21ccw −S12ccw ΔβΔL ; cos θcr exp j 2 S21cw −S12cw S12ccw −S21ccw ΔβΔL ; − cos θcr exp −j 2

(19b)

(19c)

(19d)

where LR and LL are lengths of two parts of the OFRR split by the splice and the coupler, and ΔL LR − LL . Compared to Eq. (14), only the difference ΔL affects the transfer matrices. Thus, the temperature sensitivity can be reduced by an amount proportional to L=ΔL. Figure (3a) shows the influence of ΔβL that corresponds to the temperature changes on the two ESOPs of the OFRR with a single 90° polarization axis rotated splice and 8° coupler misalignment. As can be found in Fig. 3(a), the distance of the two ESOPs’ resonance curves is relatively large compared to the OFRR with 0° in-loop splice, around π rad, which indicates that the resonance dips of the two ESOPs are located at the middle points of dips of each other. However, the relative PER of the two ESOPs in this configuration is relatively low, ranging from 0–1.2 dB, which indicates that the two ESOPs of the OFRR are equally excited. Figure 3(b) shows the gyro bias error induced by polarization fluctuation with misalignments of 8° (PER 17 dB) and 1.8° (PER 30 dB) at the coupler C1, respectively. The length and finesse of the OFRR are 10 m and 50. As can be seen in Fig. 3(b), the OFRR with a misalignment of 8°, the R-FOG can realize the detection accuracy of 0.01°=s in full temperature range. Nevertheless, a higher precision demands stronger temperature controlling. For instance, a rotation rate sensitivity of 0.001°=s needs ΔβL to be suppressed within 2.75 to π rad. On the

3.10

1.0

3.05

0.8

3.00 0.6 2.95 0.4

2.90 single 90 deg splice

0.2

2.85

0.0

2.80 0.0

0.5

1.0

1.5

2.0

2.5

3.0


1.2

relative PER of ESOPs / dB

distance of ESOPs / rad

0.01 3.15

1E-3

1E-4

1.8 degree 8 degree 1E-5 single 90 deg splice 1E-6 0.0

0.5

1.0

1.5

2.0

2.5

temperature change ( ∆β L)

temperature change ∆β L / rad

(a)

(b)

3.0

Fig. 3. (Color online) OFRR with a single 90° polarization axis rotated splice. (a) resonance point separation and PER between the two ESOPs. (b) output error of the R-FOG.

other hand, if the misalignment of 1.8° at coupler is reached, gyro detection accuracy of 0.001°=s can be realized in full temperature range. In conclusion, employing an OFRR with a single 90° polarization axis rotated splice, the polarization fluctuation of the R-FOG has been largely reduced. However, two ESOPs are equally excited by the input lightwave in this scheme, and the polarization crosstalk hence affects the detection accuracy more significantly. 3. OFRR with Twin 90° Polarization Axis Rotated Splices Another splicing scheme used to reduce the polarization fluctuation induced error is an OFRR with twin 90° polarization axis rotated splices. Two 90° rotation splices divide the whole OFRR into three parts with length LR , Lm , and LL . For each ESOP, its lightwave propagates along one polarization axis of the PMF in the first and third part of the OFRR, and along the other polarization axis in the second part. If LR LL equals Lm, the two ESOPs then go through the same length along the two orthogonal polarization axes. As in the previous section, length difference is defined as ΔL LR LL − Lm. By adjusting ΔL or Δβ, the separation of the two ESOPs’ resonance dips 2ξ can be controlled close to π rad, in other words, the largest value, and the polarizationinduced noise can be hence reduced. For simplification, angular misalignments at twin 90° polarization axis rotated splices are neglected. The transfer matrices can be described as Eq. (14) just by replacing L with ΔL. Thus, the relationship between the polarization-induced error and temperature change is the same as that shown in Fig. 3, but revising the phase difference to ΔβΔL. The temperature stability can be enhanced by L=ΔL times by the OFRR with twin 90° polarization axis rotated splices. 4. Additional Kerr Effect In this section, the additional Kerr effect due to the polarization fluctuation in the OFRR with a single and twin 90° polarization axis rotated splices are investigated. The output of the OFRR is the linear

combination of two independent resonance curves corresponding to two ESOPs. Their amplitude xj and yj are projections of the input lightwave E0j onto the two ESOPs, V 1j and V 2j , respectively. In previous sections, the input lightwave is assumed to be linearly polarized; thus xj and yj should be of the same initial phase. However, the changing of the polarization state of the input lightwave, e.g., from linearly polarized to circularly polarized, had different effects on these two projections. The input lightwave E0j can be expressed as E0j aj · x^ bj expjθ · y^ xj V1j · cosθ=2 yj V2j · sinθ=2;

(20)

where θ is the phase difference of two projections of the input lightwave. The amplitudes of two projections both change as a sinusoidal function of θ. For simplification, (ΔβΔL) of two splicing schemes are assumed to be 2mπ π, which make two ESOPs π-radian separated. The PER of the input lightwave and the coupler are both 20 dB. As can be seen in Fig. 4, the resonant power change is much more serious in the OFRR with a single 90° polarization axis rotated splice than in the OFRR with twin 90° polarization axis rotated splices. The maximum resonant power changes about 12% and 1% in the OFRR with single and twin splices, respectively. According to the analysis in [13], these resonant power variations produce an additional Kerr effect error as high as 6 × 10−4 and 5 × 10−5 rad=s considering the launching power of the OFRR as 0.1 mW, respectively. This additional Kerr effect error cannot be treated as the normal Kerr effect caused by the fluctuations of the input power. It cannot be overcome using the conventional countermeasures [14]. The OFRR with twin 90° polarization axis rotated splices is preferred to reduce the additional Kerr effect error. B. Influence of the Coupler Polarization-Dependent Losses

It is understood from the above discussion that the polarization fluctuation induced noise is minimized 1 October 2012 / Vol. 51, No. 28 / APPLIED OPTICS

6713

0.55

0.8

Normalized intensity / a.u.

Normalized intensity / a.u.

1.0

ESOP1with two 90 degree splices ESOP1with one 90 degree splice

0.6

0.4

ESOP1with one 90 degree splice 0.50

0.45

0.40

0.35 0

2

4

6

8

phase difference of input light / rad

21

22

23

24

25

26

temperature of TP / deg

(a)

(b)

Fig. 4. (Color online) Results of the power variation with change of the phase difference θ. (a) simulation results. (b) experimental results.

when the phase separation between the two ESOPs’ resonances is π. In this case, the P-ESOP in the OFRR with twin 90° polarization axis rotated splices is at resonance, and the S-ESOP is in a nonresonant state. The influence of the S-ESOP on the error is minimized. The interference component becomes important when the coupler is polarization dependent. G. A. Sanders et al. pointed out that the gyro performance was limited by residual errors due to polarization-dependent properties of the OFRR with once 90° polarization axis rotated splicing [9,15]. In this section, further analysis will be conducted on the performance comparisons between the reflector and transmitter OFRR, both with twin 90° polarization axis rotated splices. 1. Influence of the Polarization-Dependent Losses on a Reflector OFRR The practical OFRR should consider the polarization-dependent losses (PDL) in the coupler. The presence of the polarization dependency in the fiber loop and the splices are all neglected for simplicity. Introduce two parameters αcx and αcy , which are defined as the excess losses of coupler along the fast axis and slow axis, respectively. Equation (15) should be amended as I 4j jaj j2 jU 1j j2 k2 e−Δαc e−Δαc jY 01j j2 jbj j2 jU 2j j2 k2 e−Δαc jX 02j j2 e−Δαc 2k2 e−Δαc − eΔαc Reaj bj U 1j U 2j X 02j ; αc αcx αcy =2; Δαc αcx − αcy ;

6714

(21) (22a) (22b)

jU 1j j2 e−αc f1 − ρj Γj βL − ξg;

(22c)

jU 2j j2 e−αc f1 − ρj Γj βL ξg;

(22d)

λ1j C2 U 1j e−αc =2 Cbar cross ; Cb ar 1 − λ1j

(22e)


−αc =2

U 2j e

C2cross λ2j Cbar : Cb ar 1 − λ2j

(22f)

As the equations above indicate, the output lightwave intensity of the OFRR is composed of three parts: the resonant curves of the two independent ESOPs and the interference component of them. Since the PDL of the coupler is not neglected, the eΔαc factor is relatively large and the interference component takes a large proportion as a consequence. As the interference component varies, the polarization fluctuation is induced. There are two reasons for the interference component variation; one is the changes of the two ESOPs themselves, and the other is the change of the polarization state of the input lightwave. To better understand the polarization-dependent induced errors, decompose the output lightwave of the OFRR into two components: a straight-through component and a cross-through component. The straight-through component travels through the coupler from the input port to the output port directly, without being looped inside the OFRR, and its two SOPs are denoted by Etx and Ety ; the cross-through component travels through the coupler from the input port to the internal port, cycling in the OFRR, and then being coupled to the output port, and its two SOPs are denoted by Ecx and Ecy . The output of the OFRR should be the sum of these four components. Assume that the two ESOPs are adjusted to be πradian separated, and the frequency of the input beam is at the center of a P-ESOP resonance. For the P-ESOP, its two components Etx and Ecx are of same magnitude and opposite phase. They interfere at the output port and form the resonance dip. For the S-ESOP, it is in the nonresonant state, and have Ety ≫ Ecy ≈ 0. As the input lightwave is highly polarized, we have the relation Etx ≈ Ecx ≫ Ety ≫ Ecy ≈ 0 in magnitude respectively. As the polarization-dependent losses exist, the fluctuations of these four components mentioned above will finally lead to the polarization-induced error. Considering that Etx and Ety are isolated, only enabling Ecx and Ecy to be coupled out, the polarizationinduced error caused by the polarization-dependent

2. Influence of the Polarization-Dependent Loss on a Transmitter OFRR The schematic of a transmitter OFRR is shown in Fig. 1(b). It is composed of two couplers connected with each other by two in-loop splices. As the conclusion of section 3.1, these two splices are all with 90° polarization axis rotated to lower the effects of the coupler polarization crosstalk and additional Kerr effect. As shown in Fig. 1(b), the entire fiber loop consists of four parts, which are of the length of L1, L2 , L3 , and L4 . If we have ΔβΔL 2mπ πΔL L1 L4 − L2 − L3 , the two ESOPs of the OFRR are π-radian separated. For simplification, coupler C1 and coupler C2 are assumed to be identical. The output lightwave of the OFRR, denoted by E4, can be regarded as a part of E3, which is cycling in the OFRR. The output lightwaves propagating along the CW and CCW directions can be expressed as E4cw Ct E3cw Ct Scw Ct Cl T tcw E1cw ; E4ccw Ct E3ccw Ct Sccw Ct Cl T tcw E1ccw :

(23a) (23b)

Considering the polarization-dependent losses at couplers C1 and C2 , the output light intensity can be expressed as I oj jaj j2 jU 1j j2 k2 e−Δαc eΔαc jY 01j j2 jbj j2 jU 2j j2 k2 e−Δαc jX 02j j2 eΔαc 2k2 e−Δαc − eΔαc Reaj bj U 1j U 2j X 02j ; jU 1j j2

(24)

C4cross e−αc q2 Γj βL − ξ; (25a) 1 − tf jS11j j2 jS12j j2

1


loss would be greatly suppressed. This proposition can be realized by a transmitter OFRR. The separation of an input coupler and an output coupler isolates most power of the nonresonant light, and it will be thoroughly discussed in the next section.

PDL: 0 dB PDL: 0.1dB

0.1 0.01 1E-3 1E-4 1E-5 0.5

1.0

1.5

2.0

temperature change

2.5

∆β∆ L / rad

3.0

Fig. 5. (Color online) Influence of the PDL on the polarizationinduced error in an R-FOG with a transmitter OFRR.

P-ESOP resonance. The S-ESOP is in the nonresonant state and its output intensity is hence almost zero, as is the U 2j factor (for its peak-like curve). On the contrary, the U 2j factor in Eq. (22) 1 when the SESOP is in the nonresonant state for its valleylike curve. Therefore, a transmitter OFRR is more conducive to the inhibition of the polarization-induced error in the R-FOG compared with a reflector OFRR. It is expected that the polarization fluctuation induced noise can be reduced by 1 to 2 orders of magnitude. Figure 5 shows the polarization-induced noise in the transmitter OFRR with twin 90° polarization axis rotated splices. The angular misalignment at two couplers is set to be 8°. As can be seen in Fig. 5, with the influence of the coupler polarizationdependent losses, the polarization-induced noise is getting louder. Figure 6 shows the polarization-induced error as a function of the phase difference of the input lightwave in a transmitter OFRR with twin 90° polarization axis rotated splices. The curves are given with different ESOP distances. The coupler angular misalignment of 8° and coupler PDL of 0.1 dB are 0.006

U 1j C2cross e−αc =2 =1 − λ1j ;

(25c)

U 2j C2cross e−αc =2 =1 − λ2j :

(25d)

As seen in Eqs. (24) and (25), the output of a transmitter OFRR is composed of three parts, independent resonance outputs of the two ESOPs and the interference component of them. Since the two ESOPs are fully departed (π-radian separated) and the input frequency is adjusted at the center of a

polarizatioin error / deg/s

0.003

C4cross e−αc jU 2j j2 q2 Γj βL ξ; (25b) 1 − tf jS11j j2 jS12j j2

0.000 -0.003

ESOPs distance: 1.9 ESOPs distance: 2.5 ESOPs distance: 3.1

-0.006 -0.009 -0.012 0

1

2

3

4

5

6

phase different of input light / rad

Fig. 6. (Color online) Polarization-induced error as a function of the phase difference of the input lightwave with different ESOPs distances. 1 October 2012 / Vol. 51, No. 28 / APPLIED OPTICS

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Fig. 7. (Color online) Schematic for simultaneously measuring the bias stabilities of the R-FOG with reflector and transmitter OFRRs.

presumed. Figure 7 shows the schematic for simultaneously measuring the bias stabilities of the R-FOG based on a reflector and transmitter OFRRs. The frequency of the laser beam is adjusted at the center of a CCW resonance of the OFRR by a feedback loop with a demodulated signal from PD3. The demodulated signal from PD2 is the open-loop output for the RFOG with the reflector OFRR, and the demodulated signal from PD1 is the open-loop output for the transmitter one. Figure 8 shows the gyro outputs at 6000 s duration. The integral time is 3 s. It can be seen, in Fig. 8, that the bias stability of the R-FOG with the reflector OFRR is about 0.14°=s (biased at −1.18°=s), and the bias stability of that of the transmitter OFRR is about 0.018°=s (biased at 0.09°=s). The approximate 8× improvement is expected using the transmitter OFRR configuration. This measurement shows good agreement with that theoretical expected value. Compared with the reflector OFRR, the transmitter OFRR isolates the errors induced by polarizationdependent losses of the input coupler. However, the polarization-dependent losses of the output coupler still exist. As Fig. 6 shows, with a coupler misalignment of 8° and a coupler PDL of 0.1 dB, the peak-topeak fluctuation of the polarization-induced error is approximately 6 × 10−3 °=s when the two ESOPs are 2.5 rad separated. It can be reduced by controlling the temperature of the input fiber connected to the 0.2

Bias output / deg/s

0.0 -0.2 up: transmitter OFRR down: reflector OFRR

-0.4 -0.6

4. Conclusion

The polarization characteristics are modeled based on the transfer matrix method, by which the polarization-induced noises of typical OFRR are theoretically analyzed and compared. Considering the polarization-dependent losses of couplers, we find that in a reflector OFRR, the straight-through component of the output lightwave would introduce a large portion of the polarization-induced noise; meanwhile, in a transmitter OFRR with the same parameters, the straight-through signal is isolated and the polarization-induced noise is therefore largely reduced. Experimental results show that the bias stabilities of the R-FOG with a transmitter OFRR and reflector OFRR are 0.018°=s and 0.14°=s over 6000 seconds, respectively. It is a nearly 8× improvement in bias stability, which is in accordance with the theoretical expected value. Considering the additional Kerr effect due to the polarization fluctuation–induced noise, the OFRR with twin 90° polarization axis rotated splices is the best. Therefore, it is reasonable to conclude that an R-FOG based on a transmitter OFRR with twin 90° polarization axis rotated splices is better for temperature stability and achieving high-performance. References

-0.8 -1.0 -1.2 -1.4 -1.6 0

1000

2000

3000

4000

5000

6000

time / s Fig. 8. (Color online) Bias outputs of the R-FOG based on a reflector and a transmitter OFRR, respectively. 6716

input coupler, but high precision temperature control is needed for a high-performance R-FOG. There are another two practical solutions to reduce the polarization-induced error; one is constructing the transmitter OFRR with two couplers with the same polarization dependency, which can be compensated by each other; the other is replacing the fiber couplers with the integrated couplers, which has lower polarization crosstalk and PDL.


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