Abstract

Polarization-mode coupling in birefringent fiber gratings is analyzed. The general expression for coupling coefficient components is also derived. It indicates that the polarization-mode coupling between any two linearly polarized (LP) core modes is possible by appropriately adjusting the grating parameters such as the grating tilt angle, the grating length, the orientation of the grating plane, the grating period, the birefringence, and the birefringent axis. It is analytically found that the complete LP01x-to-LP01y mode coupling and LP01x-to-LP11y mode coupling occur when fiber is pressed periodically. The LP01x-to-LP11y mode coupling in the linearly birefringent gratings created by pressing a two-mode fiber with a groove plate with a period of 80 µm at a tilt angle between 81.5° and 83.5° has also been experimentally demonstrated. The resonant LP01x-to-LP11y mode coupling in the birefringent gratings and their experimental transmission spectra were reasonably well predicted by the coupled-mode analysis.

© 2002 Optical Society of America

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References

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  5. K. Hill, B. Malo, K. Vineberg, F. Bilodo, D. Johnson, I. Skinner, “Efficient mode conversion in telecommunication fibre using externally written gratings,” Electron. Lett. 26, 1270–1272 (1990).
    [CrossRef]
  6. H. Park, B. Kim, “Intermodal coupler using permanently photoinduced grating in two-mode optical fiber,” Electron. Lett. 25, 797–799 (1989).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  10. K. S. Lee, “Mode coupling in tilted planar waveguide gratings,” Appl. Opt. 39, 6144–6149 (2000).
    [CrossRef]
  11. K. S. Lee, T. Erdogan, “Mode coupling in spiral fibre gratings,” Electron. Lett. 37, 156–157 (2001).
    [CrossRef]
  12. C. Poole, C. Townsend, K. Nelson, “Helical grating two-mode fiber spatial-mode coupler,” J. Lightwave Technol. 9, 589–604 (1991).
    [CrossRef]
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    [CrossRef]
  17. H. Lefevre, “Single-mode fiber fractional wave devices and polarization controllers,” Electron. Lett. 16, 778–780 (1980).
    [CrossRef]
  18. F. Ouellette, D. Gagnon, M. Poirier, “Permanent photoinduced birefringence in a Ge-doped fiber,” Appl. Phys. Lett. 58, 1813–1815 (1991).
    [CrossRef]
  19. T. Erdogan, V. Mizrahi, “Characterization of UV-induced birefringence in photosensitive Ge-doped silica optical fibers,” J. Opt. Soc. Am. B 11, 2100–2105 (1994).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]

2001 (3)

K. S. Lee, T. Erdogan, “Mode coupling in spiral fibre gratings,” Electron. Lett. 37, 156–157 (2001).
[CrossRef]

K. S. Lee, “Coupling analysis of spiral fiber gratings,” Opt. Commun. 198, 317–324 (2001).
[CrossRef]

K. S. Lee, T. Erdogan, “Fiber mode conversion with tilted gratings in optical fiber,” J. Opt. Soc. Am. A 18, 1176–1185 (2001).
[CrossRef]

2000 (5)

1999 (1)

K. S. Lee, T. Erdogan, “Transmissive tilted gratings for LP01-to-LP11 mode coupling,” IEEE Photonics Technol. Lett. 11, 1286–1288 (1999).
[CrossRef]

1996 (2)

1994 (2)

T. Erdogan, V. Mizrahi, “Characterization of UV-induced birefringence in photosensitive Ge-doped silica optical fibers,” J. Opt. Soc. Am. B 11, 2100–2105 (1994).
[CrossRef]

C. Poole, J. Wiesenfeld, D. Digiovanni, A. Vengsarkar, “Optical fiber-based dispersion compensation using higher order modes near cutoff,” J. Lightwave Technol. 12, 1746–1758 (1994).
[CrossRef]

1991 (2)

C. Poole, C. Townsend, K. Nelson, “Helical grating two-mode fiber spatial-mode coupler,” J. Lightwave Technol. 9, 589–604 (1991).
[CrossRef]

F. Ouellette, D. Gagnon, M. Poirier, “Permanent photoinduced birefringence in a Ge-doped fiber,” Appl. Phys. Lett. 58, 1813–1815 (1991).
[CrossRef]

1990 (1)

K. Hill, B. Malo, K. Vineberg, F. Bilodo, D. Johnson, I. Skinner, “Efficient mode conversion in telecommunication fibre using externally written gratings,” Electron. Lett. 26, 1270–1272 (1990).
[CrossRef]

1989 (1)

H. Park, B. Kim, “Intermodal coupler using permanently photoinduced grating in two-mode optical fiber,” Electron. Lett. 25, 797–799 (1989).
[CrossRef]

1981 (1)

1980 (2)

A. M. Smith, “Single-mode fiber pressure sensitivity,” Electron. Lett. 16, 773–774 (1980).
[CrossRef]

H. Lefevre, “Single-mode fiber fractional wave devices and polarization controllers,” Electron. Lett. 16, 778–780 (1980).
[CrossRef]

Bendow, B.

Bergano, N.

Bilodo, F.

K. Hill, B. Malo, K. Vineberg, F. Bilodo, D. Johnson, I. Skinner, “Efficient mode conversion in telecommunication fibre using externally written gratings,” Electron. Lett. 26, 1270–1272 (1990).
[CrossRef]

Davidson, C.

Digiovanni, D.

C. Poole, J. Wiesenfeld, D. Digiovanni, A. Vengsarkar, “Optical fiber-based dispersion compensation using higher order modes near cutoff,” J. Lightwave Technol. 12, 1746–1758 (1994).
[CrossRef]

Digonnet, M.

El-Sherif, M.

R. Gafsi, M. El-Sherif, “Analysis of induced-birefringence effects on fiber Bragg gratings,” Opt. Fiber Technol. 6, 299–323 (2000).
[CrossRef]

Erdogan, T.

Gafsi, R.

R. Gafsi, M. El-Sherif, “Analysis of induced-birefringence effects on fiber Bragg gratings,” Opt. Fiber Technol. 6, 299–323 (2000).
[CrossRef]

Gagnon, D.

F. Ouellette, D. Gagnon, M. Poirier, “Permanent photoinduced birefringence in a Ge-doped fiber,” Appl. Phys. Lett. 58, 1813–1815 (1991).
[CrossRef]

Gianino, P.

Hill, K.

K. Hill, B. Malo, K. Vineberg, F. Bilodo, D. Johnson, I. Skinner, “Efficient mode conversion in telecommunication fibre using externally written gratings,” Electron. Lett. 26, 1270–1272 (1990).
[CrossRef]

Johnson, D.

K. Hill, B. Malo, K. Vineberg, F. Bilodo, D. Johnson, I. Skinner, “Efficient mode conversion in telecommunication fibre using externally written gratings,” Electron. Lett. 26, 1270–1272 (1990).
[CrossRef]

Judkins, J.

Kim, B.

H. Park, B. Kim, “Intermodal coupler using permanently photoinduced grating in two-mode optical fiber,” Electron. Lett. 25, 797–799 (1989).
[CrossRef]

Kino, G.

Kogelnik, H.

H. Kogelnik, “Theory of optical waveguides,” in Guided-Wave Optics, T. Tamir, ed. (Springer-Verlag, New York, 1990), Chap. 2.

Lee, K. S.

K. S. Lee, T. Erdogan, “Fiber mode conversion with tilted gratings in optical fiber,” J. Opt. Soc. Am. A 18, 1176–1185 (2001).
[CrossRef]

K. S. Lee, “Coupling analysis of spiral fiber gratings,” Opt. Commun. 198, 317–324 (2001).
[CrossRef]

K. S. Lee, T. Erdogan, “Mode coupling in spiral fibre gratings,” Electron. Lett. 37, 156–157 (2001).
[CrossRef]

K. S. Lee, “Mode coupling in tilted planar waveguide gratings,” Appl. Opt. 39, 6144–6149 (2000).
[CrossRef]

K. S. Lee, T. Erdogan, “Fiber mode coupling in transmissive and reflective tilted fiber gratings,” Appl. Opt. 39, 1394–1404 (2000).
[CrossRef]

K. S. Lee, T. Erdogan, “Transmissive tilted gratings for LP01-to-LP11 mode coupling,” IEEE Photonics Technol. Lett. 11, 1286–1288 (1999).
[CrossRef]

Lefevre, H.

H. Lefevre, “Single-mode fiber fractional wave devices and polarization controllers,” Electron. Lett. 16, 778–780 (1980).
[CrossRef]

Lemaire, P.

Malo, B.

K. Hill, B. Malo, K. Vineberg, F. Bilodo, D. Johnson, I. Skinner, “Efficient mode conversion in telecommunication fibre using externally written gratings,” Electron. Lett. 26, 1270–1272 (1990).
[CrossRef]

Mizrahi, V.

Nelson, K.

C. Poole, C. Townsend, K. Nelson, “Helical grating two-mode fiber spatial-mode coupler,” J. Lightwave Technol. 9, 589–604 (1991).
[CrossRef]

Ouellette, F.

F. Ouellette, D. Gagnon, M. Poirier, “Permanent photoinduced birefringence in a Ge-doped fiber,” Appl. Phys. Lett. 58, 1813–1815 (1991).
[CrossRef]

Park, H.

H. Park, B. Kim, “Intermodal coupler using permanently photoinduced grating in two-mode optical fiber,” Electron. Lett. 25, 797–799 (1989).
[CrossRef]

Pedrazzani, R.

Poirier, M.

F. Ouellette, D. Gagnon, M. Poirier, “Permanent photoinduced birefringence in a Ge-doped fiber,” Appl. Phys. Lett. 58, 1813–1815 (1991).
[CrossRef]

Poole, C.

C. Poole, J. Wiesenfeld, D. Digiovanni, A. Vengsarkar, “Optical fiber-based dispersion compensation using higher order modes near cutoff,” J. Lightwave Technol. 12, 1746–1758 (1994).
[CrossRef]

C. Poole, C. Townsend, K. Nelson, “Helical grating two-mode fiber spatial-mode coupler,” J. Lightwave Technol. 9, 589–604 (1991).
[CrossRef]

Savin, S.

Shaw, H.

Sipe, J.

Skinner, I.

K. Hill, B. Malo, K. Vineberg, F. Bilodo, D. Johnson, I. Skinner, “Efficient mode conversion in telecommunication fibre using externally written gratings,” Electron. Lett. 26, 1270–1272 (1990).
[CrossRef]

Smith, A. M.

A. M. Smith, “Single-mode fiber pressure sensitivity,” Electron. Lett. 16, 773–774 (1980).
[CrossRef]

Stegall, D.

Townsend, C.

C. Poole, C. Townsend, K. Nelson, “Helical grating two-mode fiber spatial-mode coupler,” J. Lightwave Technol. 9, 589–604 (1991).
[CrossRef]

Vengsarkar, A.

A. Vengsarkar, R. Pedrazzani, J. Judkins, P. Lemaire, N. Bergano, C. Davidson, “Long-period fiber-grating-based gain equalizers,” Opt. Lett. 21, 339–337 (1996).
[CrossRef]

C. Poole, J. Wiesenfeld, D. Digiovanni, A. Vengsarkar, “Optical fiber-based dispersion compensation using higher order modes near cutoff,” J. Lightwave Technol. 12, 1746–1758 (1994).
[CrossRef]

Vineberg, K.

K. Hill, B. Malo, K. Vineberg, F. Bilodo, D. Johnson, I. Skinner, “Efficient mode conversion in telecommunication fibre using externally written gratings,” Electron. Lett. 26, 1270–1272 (1990).
[CrossRef]

Wiesenfeld, J.

C. Poole, J. Wiesenfeld, D. Digiovanni, A. Vengsarkar, “Optical fiber-based dispersion compensation using higher order modes near cutoff,” J. Lightwave Technol. 12, 1746–1758 (1994).
[CrossRef]

Appl. Opt. (3)

Appl. Phys. Lett. (1)

F. Ouellette, D. Gagnon, M. Poirier, “Permanent photoinduced birefringence in a Ge-doped fiber,” Appl. Phys. Lett. 58, 1813–1815 (1991).
[CrossRef]

Electron. Lett. (5)

K. Hill, B. Malo, K. Vineberg, F. Bilodo, D. Johnson, I. Skinner, “Efficient mode conversion in telecommunication fibre using externally written gratings,” Electron. Lett. 26, 1270–1272 (1990).
[CrossRef]

H. Park, B. Kim, “Intermodal coupler using permanently photoinduced grating in two-mode optical fiber,” Electron. Lett. 25, 797–799 (1989).
[CrossRef]

K. S. Lee, T. Erdogan, “Mode coupling in spiral fibre gratings,” Electron. Lett. 37, 156–157 (2001).
[CrossRef]

A. M. Smith, “Single-mode fiber pressure sensitivity,” Electron. Lett. 16, 773–774 (1980).
[CrossRef]

H. Lefevre, “Single-mode fiber fractional wave devices and polarization controllers,” Electron. Lett. 16, 778–780 (1980).
[CrossRef]

IEEE Photonics Technol. Lett. (1)

K. S. Lee, T. Erdogan, “Transmissive tilted gratings for LP01-to-LP11 mode coupling,” IEEE Photonics Technol. Lett. 11, 1286–1288 (1999).
[CrossRef]

J. Lightwave Technol. (2)

C. Poole, C. Townsend, K. Nelson, “Helical grating two-mode fiber spatial-mode coupler,” J. Lightwave Technol. 9, 589–604 (1991).
[CrossRef]

C. Poole, J. Wiesenfeld, D. Digiovanni, A. Vengsarkar, “Optical fiber-based dispersion compensation using higher order modes near cutoff,” J. Lightwave Technol. 12, 1746–1758 (1994).
[CrossRef]

J. Opt. Soc. Am. A (3)

J. Opt. Soc. Am. B (1)

Opt. Commun. (1)

K. S. Lee, “Coupling analysis of spiral fiber gratings,” Opt. Commun. 198, 317–324 (2001).
[CrossRef]

Opt. Fiber Technol. (1)

R. Gafsi, M. El-Sherif, “Analysis of induced-birefringence effects on fiber Bragg gratings,” Opt. Fiber Technol. 6, 299–323 (2000).
[CrossRef]

Opt. Lett. (2)

Other (1)

H. Kogelnik, “Theory of optical waveguides,” in Guided-Wave Optics, T. Tamir, ed. (Springer-Verlag, New York, 1990), Chap. 2.

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Figures (9)

Fig. 1
Fig. 1

Configuration of a birefringent fiber grating whose grating plane is tilted by θ from the x axis and rotated by Δϕ with respect to the x axis around the fiber axis.

Fig. 2
Fig. 2

Schematic view of a linear birefringent fiber grating created by applying a periodic force F at tilt angle θ.

Fig. 3
Fig. 3

Coupled powers of LP01x and LP01y modes after a birefringent grating with Λ=8.3 mm created by a periodic pressing with F=3700 N/m and ψ=22.5° over (a) L=9.9 mm and (b) L=29.7 mm.

Fig. 4
Fig. 4

Coupled powers of LP01x and LP11y modes after a birefringent fiber grating tilted by θ=88.5° with Λ=489 µm, L=10.5 mm, ψ=45°, and a=5.5 µm (n1=1.46 and n2=1.4527).

Fig. 5
Fig. 5

Transmission spectra of the tilted birefringent gratings formed by the periodic pressing with different forces (θ=85°, Λ=489 µm, and L=22 mm).

Fig. 6
Fig. 6

Transmission spectra of the tilted birefringent gratings with different tilt angles (F=5000 N/m, Λ=489 µm and L=20 mm).

Fig. 7
Fig. 7

Experimental setup for demonstrating the LP01x-to-LP11y mode conversion in a linearly birefringent fiber grating formed by pressing a two-mode fiber (TMF) and the intensity patterns of the (a) LP01x mode and (b) LP11y mode after polarizers (POL). Here a tunable laser diode (TLD) was used as a source and a single-mode fiber polarizer (SMFP) was employed to excite a pure LP01x mode.

Fig. 8
Fig. 8

Transmission spectra of the linearly birefringent gratings created with two different line forces F of ∼5800 N/m (dashed curve) and ∼9800 N/m (solid curve) (θ83.4°, L=20 mm, Λ=696 nm), (a) Experimental transmission spectra and (b) theoretical transmission spectra.

Fig. 9
Fig. 9

Grating periods required for polarization-mode coupling at different wavelengths (a=5.125 µm, n11.452, n2=1.447). The circles are the measured values and are best fitted to the solid curve. The dashed curve stands for the theoretical values predicted by the coupled mode analysis under the assumption that the fiber has a simple step-index profile.

Equations (67)

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dAμ(z)dz=iω4-dxdyP·eμ*(x, y)exp(-iβμz),
dBμ(z)dz=-iω4-dxdyP·e-μ*(x, y)exp(iβμz),
P=Δ(x, y, z)E,
E=Et+Ez=μ[Aμ exp(iβμ z)+Bμ exp(-iβμ z)]eμt++Δ×μ[Aμ exp(iβμ z)-Bμ exp(-iβμ z)]eμz,
12-dx-dy(eμt×hνt*)·z=Pμδμν,
PxPyPz=ΔxxΔxy0ΔyxΔyy000ΔzzExEyEz,
P·e±μ*=ν[Aν exp(iβν z)+Bν exp(-iβν z)]×(Δxxeνxeμx*+Δxyeνy eμx*)+ν[Aν exp(iβν z)+Bν exp(-iβν z)]×(Δyxeνxeμy*+Δyyeνyeμy*)±ν[Aν exp(iβν z)-Bν exp(-iβν z)][/(+Δzz)]Δzzeνzeμz*,
dAμ(z)/dz=iνAν(Kνxμx+Kνxμy+Kνyμx+Kνyμy+Kνzμz)exp[i(βν-βμ)z]+iνBν(Kνxμx+Kνxμy+Kνyμx+Kνyμy-Kνzμz)exp[-i(βν+βμ)z],
dBμ(z)/dz=-iνAν(Kνxμx+Kνxμy+Kνyμx+Kνyμy-Kνzμz)exp[i(βν+βμ)z]-iνBν(Kνxμx+Kνxμy+Kνyμx+Kνyμy+Kνzμz)exp[-i(βν-βμ)z],
Kνjμiω4dxdyΔijeνjeμi*(forj=x, yandi=x, y),
Kνzμzω4dxdy[/(+Δzz)]Δzzeνzeμz*,
Δij(x, y, z)=Δij¯ f(z),
f(z)=f1(z)=1+V0 cos(2Kg z)
f(z)=f2(z)=1,MΛg<z<(M+1/2)Λg(M=integer)0,otherwise=12+2πN=11(2N-1)sin 2Kg(2N-1)z,
Δijeνjeμi*=Δij¯ eνjeμi*1+V02exp(2iKg z)+V02exp(-2iKg z),
Kνjμi(z)=gνjμi+ exp(2iKg z cos θ)+gνjμi- ×exp(-2iKg z cos θ)+fνjμi,
gνjμi±=w8V0 Δij¯02πdϕ0ardr eνjeμi*×exp[2iKgr cos(ϕ-Δϕ)sin θ],
fνjμi=w4Δij¯02πdϕ0ardr eνjeμi*(forj=x, yandi=x, y),
eui=EuiJu(uuir)cos uϕsin uϕ(fori=x, yandu=μ, ν),
Eui=4z0buiuπna2|Ju-1(uuia)Ju+1(uuia)|1/2.
gνj-01i±=πw4Δij¯V0EνE01(i)l cos lΔϕ0ardrJl(uνr)×J0(u0r)Jl(2Kgr sin θ)
gνj-01i±=πw4Δij¯V0EνE01(i)l sin lΔϕ×0ardrJl(uνr)J0(u0r)Jl(2Kgr sin θ)
Δijeνjeμi*=Δij¯ eνjeμi*12+1iπN=112N-1×{exp[i2(2N-1)Kg z]-exp[-i2(2N-1)Kg z]} .
Kνjμi(z)=N=1{gνjμi+N exp[i2(2N-1)Kg z cos θ]-gνjμi-N exp[-i2(2N-1)Kg z cos θ]}+fνjμi/2,
gνjμi±N=w4Δij¯1i(2N-1)π02πdϕ0ardr×exp[i2(2N-1)Kgr cos(ϕ-Δϕ)sin θ]eνjeμi*.
=R-1(ψ)X000Y000iR(ψ)=X-Δl sin2 ψxy0yxY+Δl sin2 ψ000i,
xy=yx=(Δl/2)sin 2ψ,Δl=X-Y, X=0nX2,Y=0nY2,R(ψ)
Δ=Δxx¯Δxy¯0Δyx¯Δyy¯0000,
Δxy¯=Δ¯yx=(Δl/2)sin 2ψ,
Δxx¯=ΔX-Δl sin2 ψ,
Δyy¯=ΔY+Δl sin2 ψ,
ΔX=0(nX2-ni2)20niΔnX,
ΔY=0(nY2-ni2)20niΔnY,
Δl=0(nX2-nY2)20niBl.
Bl=(2ni3/πYm)(1+σ)(P12-P11)F/r0,
σ=0.160.17,P11=0.12,P12=0.27, ni=1.46,Ym=7.6×1010 (N/m2),
ΔnX3.71×10-12 (F/r0),
ΔnY-0.74×10-12 (F/r0),
Δxy=Δyx=20ni4πYm(1+σ)(P12-P11)(F/r0)×sin 2ψf(z).
ΔnY=(ni3/4)(P11-2σP12)(r0/R)2,
ΔnX=(ni3/4)(P12-σP12-σP11)(r0/R)2.
Bl=ni34(P12-P11)(1+σ)r0R2,
ΔnY0.022(r0/R)2,ΔnX0.158(r0/R)2, Bl=0.136(r0/R)2.
Δxy=Δyx=0ni44(P12-P11)(1+σ)r0R2×sin 2ψf(z).
dA01x/dz=iA01y g01y-01x+ exp(-2iδ01x-01ytz),
dA01y/dz=iA01x g01x-01y- exp(2iδ01x-01ytz),
g01y-01x+V0π2λΔxy¯0b0ni1+J0J12,
g01x-01y-V0π2λΔyx¯0b0ni1+J0J12,
g0(V0π/2λ)Bl sin 2ψb0[1+(J0/J1)2],
dA01x/dz=iA01yg 0 exp(-2iδ01x-01ytz)
dA01y/dz=iA01x g0 exp(2iδ01x-01ytz).
Ax(z)=cos(γz)+(iδ/γ)sin(γz),
Ay(z)=-i(g0/γ)sin(γz),
Lλ/{V0Bl sin(2ψ)b0[1+(J0/J1)2]}.
dA01x/dz=iA01yG0 exp(-2iδ01x-01ytz),
dA01y/dz=iA01xG0* exp(2iδ01x-01ytz).
dA01x/dz=iAνygνy-01x+ exp(-2iδ01x-νytz),
dAνy/dz=iA01xg01x-νy- exp(2iδ01x-νytz),
gνy-01x±=(g01x-νy±)*=(πw/4)Δxy¯V0EνE01(i)l×0ardrJl(uνr)J0(u0r)Jl(2Kgr sin θ)
A01x(z)=cos γz+(iδ/γ)sin γz,
Aνy(z)=-i(gν)* sin(γz),
Ax(z)=(C0/gν*)(γ cosh γz-iδ sinh γz),
By(z)=C0 sinh γz,
Γ=sinh2 γLcosh2 γL-δ2/|gν|2
gνjμi±=πω8ΔijV0EνEμ[(i)l2-l1 cos(l2-l1)Δϕ×0ardrJl2(uνr)Jl1(uμr)Jl2-l1(2Kgr sin θ)+(i)l2+l1 cos(l2+l1)Δϕ0ardrJl2(uνr)×Jl1(uμr)Jl2+l1(2Kgr sin θ)
gνjμi±=πω8ΔijV0EνEμ(i)l2-l1 cos(l2-l1)Δϕ×0ardrJl2(uνr)Jl1(uμr)Jl2-l1(2Kgr sin θ)-(i)l2+l1 cos(l2+l1)Δϕ0ardrJl2(uνr)×Jl1(uμr)Jl2+l1(2Kgr sin θ)
gνjμi±=πω8ΔijV0EνEμ(i)l2-l1×sin(l2-l1)Δϕ0ardrJl2(uνr)Jl1(uμr)×Jl2-l1(2Kgr sin θ)+(i)l2+l1×sin(l2+l1)Δϕ0ardrJl2(uνr)×Jl1(uμr)Jl2+l1(2Kgr sin θ)

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