Abstract

Using the new beam tracing approach of dividing a fiber grating into a pile of infinitesimally thin films and calculating the multiple beam interference of the Fresnel reflections from the thin-film interfaces, a tilted fiber grating based polarizer is theoretically investigated. On the Gaussian fiber mode model, new analytical formulas for expressing the characteristics of the 45° tilted fiber grating polarizer are obtained as the explicit functions of the fiber grating and fiber mode parameters. The present approach provides a clear understanding and fine perspective of all the properties of the tilted fiber grating polarizer, as well as making their quantitative evaluation much easier, as compared with previous works. Our numerical result is shown to be in good agreement with the numerical solution by the volume current method. The achievable polarization performance with different grating parameters is numerically examined and discussed.

© 2012 Optical Society of America

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References

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1996

1977

T. Yoshino and M. Takeda, “Study on transmission characteristics of Gaussian beams through tilted etalons using angular-spectrum representations,” Jpn. J. Opt. 6, 113–119 (1977).

Bennion, I.

Brown, T. G.

Chen, X.

Erdogan, T.

Feng, S.

Froggatt, M.

Grobnic, D.

Huang, W.

Jeunhomme, L. B.

L. B. Jeunhomme, Single-Mode Fiber Optics (Marcel Dekker, 1983).

Jian, S.

Li, Y.

Lu, P.

Lu, S.

Lu, Y.

Mihailov, S. J.

Mou, C.

Simpson, G.

Sipe, J. P.

Takeda, M.

T. Yoshino and M. Takeda, “Study on transmission characteristics of Gaussian beams through tilted etalons using angular-spectrum representations,” Jpn. J. Opt. 6, 113–119 (1977).

Walker, R. B.

Xu, O.

Yan, Z.

Yoshino, T.

T. Yoshino and M. Takeda, “Study on transmission characteristics of Gaussian beams through tilted etalons using angular-spectrum representations,” Jpn. J. Opt. 6, 113–119 (1977).

Zhang, L.

Zhou, K.

J. Lightwave Technol.

J. Opt. Soc. Am. A

J. Opt. Soc. Am. B

Jpn. J. Opt.

T. Yoshino and M. Takeda, “Study on transmission characteristics of Gaussian beams through tilted etalons using angular-spectrum representations,” Jpn. J. Opt. 6, 113–119 (1977).

Opt. Express

Opt. Lett.

Other

L. B. Jeunhomme, Single-Mode Fiber Optics (Marcel Dekker, 1983).

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Figures (4)

Fig. 1.
Fig. 1.

Schematic diagram for tilted fiber grating polarizer.

Fig. 2.
Fig. 2.

Dependence of polarization extinction ratio Γ0 of 45° tilted fiber grating polarizer on grating length Lg with n0=1.46, λR=1.55μm and w0=3λR, for different degrees of refractive index modulation: (a) κ=3×103, (b) κ=2.05×103 (dashed curve from Zhou et al. [7]), (c) κ=1.5×103, (d) κ=1.0×103, (e) κ=6×104, (f) κ=3×104.

Fig. 3.
Fig. 3.

Dependence of polarization extinction ratio Γ0 of 45° tilted fiber grating on refractive index modulation degree κ for different grating lengths Lg with n0=1.46, λR=1.55μm and w0=3λR.

Fig. 4.
Fig. 4.

Input wavelength λ dependence of polarization extinction ratio Γ of 45° tilted fiber gating for different resonance wavelengths λR, keeping w0/λR=3 with n0=1.46, κ=103, and Lg=50mm.

Equations (43)

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n(x,z)=n0[1+κsin(2πxtanθ+zΛg)]
nm=n0[1+κsin(2πmM)].
n0sinθ=nmsinθm=nm+1sinθm+1.
sinθmsinθκsin(2πmM)sinθ,
cosθmcosθ+κsin(2πmM)sin2θcosθ.
rs,mnmcosθmnm+1cosθm+1nmcosθm+nm+1cosθm+1κcos[2π(m/M)]2cos2θ+2κsin[2π(m/M)]·2πM,
rp,mnm+1cosθmnmcosθm+1nm+1cosθm+nmcosθm+1κcos[2π(m/M)]cos2θ2cos2θ+2κsin[2π(m/M)]·2πM.
δm=θmθ,
δm=κsin(2πmM)tanθ,
|rs,m/rp,m|=1cos2θ.
rs,m=κcos(2πmΔzΛg)(2πΔzΛg)rm,
rp,m=0.
Em(y,zP)=amrmf(xzPmΔz,y)exp(iβmΔz),(1mNM),
E(y,zP)=m=1NMEm(y,zp),
=m=1NMamrmf(zPmΔz,y)exp(iβkmΔz).
E(y,zP)=a(zP)r(z)f(zPz,y)exp(iβz)dz,
r(z)=κπΛg[exp(i2πΛgz)+exp(i2πΛgz)],
E(y,zP)=a(zP)κπΛgf(zPz,y){exp[i(β2πΛg)z]+exp[i(β+2πΛg)z]}dz,
R(zP)α=|E(y,zP)|2dya2(zP)|f(x,y)|2dxdy.
f(x,y)=exp(x2+y2w02),
f(zPz,y)=exp[(zPz)2+y2w02].
E(y,zP)=a(zP)κπΛgexp[(zPz)2+y2w02]{exp[i(β2πΛg)z]+exp[i(β+2πΛg)z]}dz,
E(y,zP)=a(zP)w0πκπΛgexp(y2w02)×{exp[(β2πΛg)2(w02)2]exp[i(β2πΛg)zp]+exp[(β+2πΛg)2(w02)2]exp[i(β+2πΛg)ZP]}.
|E(y,zP)|2=κ2a2(zP)π3w02Λg2exp(2y2w02)exp[w022(β2πΛg)2],
|E(y,zP)|2dy=κ2a2(zP)π2π3w03Λg2exp[w022(β2πΛg)2].
α=κ2π22πw0Λg2exp[w022(β2πΛg)2],
γ=exp(αLg),
Γ=10log10γ[dB]=4.34αLg[dB].
Γ=4.34κ2π22πw0Λg2Lgexp[w022(β2πΛg)2][dB].
Γ0=4.34κ2π22πw0Λg2Lg[dB],
β=2πΛg,
β=2πλn0(λ,light wavelength in vacuum),
λR=n0Λg.
Γ=4.34κ2π22πn02w0λR2Lgexp{2[πn0w(1λ1λR)]2}[dB],
Γ0=4.34κ2π22πn02w0λR2Lg[dB].
Γ=Γ0exp[2(πn0w0ΔλλR2)2][dB].
ΔλHλR=2ln2πn0·λRwR.
Γ=4.34π22πn02(w0λR)(LgλR)κ2exp{2[πn0w(1/λ1/λR)]2}[dB].
Er(x,y,z)=E0δ(xz)f(x,y)exp(ikz).
Ar(α,β,γ)Er(x,y,z)exp[ik(αx+βy+λz)]dxdydz,(α2+β2+γ2=1)=E0δ(xz)f(x,y)⁢ exp(kz)exp[ik(αx+βy+λz)]dxdydz,=E0f(z,y)exp{ik[(α+γ1)z+βy]}dydz.
Ar(α,β,λ)=E0exp(z2+y2w02)exp{ik[(α+γ1)z+βy]}dydz,=E0πw02exp[(kw02β)2]exp{[kw02(α+γ1)]2}.
α+γ1=α21α+1+γβ2+γ22+γ,(α+γ1)2γ2.
Ar(α,β,λ)=E0πw02exp[(kw02β)2]exp[(kw02γ)2],

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