Abstract

We demonstrate theoretically and experimentally that two orthogonal polarizations of light see an unequal index contrast in a UV-induced fiber Bragg grating. We have found that in a Bragg grating the phase-velocity mismatch is a function of detuning, and hence we can introduce an effective birefringence. The description of the polarization dynamics that we developed was confirmed by a set of experiments with a fiber grating. We show theoretically that by adjusting the value of the unequal index contrast relative to the average birefringence it should be possible to eliminate the phase-velocity mismatch between polarizations at a given frequency.

© 2002 Optical Society of America

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References

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  1. C. M. de Sterke and J. E. Sipe, “Gap solitons,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1994), Vol. 33, pp. 203–260.
  2. S. Pereira and J. E. Sipe, “Light propagation through birefringent, nonlinear media with deep gratings,” Phys. Rev. E 62, 5745–5757 (2000).
    [CrossRef]
  3. T. Erdogan and V. Mizrahi, “Characterization of UV-induced birefringence in photosensitive Ge-doped silica optical fibers,” J. Opt. Soc. Am. B 11, 2100–2105 (1994).
    [CrossRef]
  4. A. M. Vengsarkar, Q. Zhong, D. Inniss, W. A. Reed, P. J. Lemaire, and S. G. Kosinski, “Birefringence reduction in side-written photoinduced fiber devices by a dual-exposure method,” Opt. Lett. 19, 1260–1262 (1994).
    [CrossRef] [PubMed]
  5. D. Taverner, N. G. R. Broderick, D. J. Richardson, M. Ibsen, and R. I. Laming, “All-optical AND gate based on coupled gap-soliton formation in a fiber Bragg grating,” Opt. Lett. 23, 259–261 (1998).
    [CrossRef]
  6. R. E. Slusher, S. Spälter, B. J. Eggleton, S. Pereira, and J. E. Sipe, “Bragg-grating-enhanced polarization instabilities,” Opt. Lett. 25, 749–751 (2000).
    [CrossRef]
  7. J. E. Sipe, L. Poladian, and C. M. de Sterke, “Propagation through nonuniform grating structures,” J. Opt. Soc. Am. A 11, 1307–1320 (1994).
    [CrossRef]
  8. C. R. Menyuk, “Stability of solitons in birefringent optical fibers. I: Equal propagation amplitudes,” Opt. Lett. 12, 614–616 (1987).
    [CrossRef] [PubMed]

2000

S. Pereira and J. E. Sipe, “Light propagation through birefringent, nonlinear media with deep gratings,” Phys. Rev. E 62, 5745–5757 (2000).
[CrossRef]

R. E. Slusher, S. Spälter, B. J. Eggleton, S. Pereira, and J. E. Sipe, “Bragg-grating-enhanced polarization instabilities,” Opt. Lett. 25, 749–751 (2000).
[CrossRef]

1998

1994

1987

Broderick, N. G. R.

de Sterke, C. M.

Eggleton, B. J.

Erdogan, T.

Ibsen, M.

Inniss, D.

Kosinski, S. G.

Laming, R. I.

Lemaire, P. J.

Menyuk, C. R.

Mizrahi, V.

Pereira, S.

S. Pereira and J. E. Sipe, “Light propagation through birefringent, nonlinear media with deep gratings,” Phys. Rev. E 62, 5745–5757 (2000).
[CrossRef]

R. E. Slusher, S. Spälter, B. J. Eggleton, S. Pereira, and J. E. Sipe, “Bragg-grating-enhanced polarization instabilities,” Opt. Lett. 25, 749–751 (2000).
[CrossRef]

Poladian, L.

Reed, W. A.

Richardson, D. J.

Sipe, J. E.

Slusher, R. E.

Spälter, S.

Taverner, D.

Vengsarkar, A. M.

Zhong, Q.

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Figures (6)

Fig. 1
Fig. 1

δni(z)/δn¯i for our experimental grating (labeled Initial Pulse). The apodization in the wings is Gaussian with a half-width at half-maximum of 0.25 cm. Dashed line (Second Pulse), profile of the uniform illumination meant to reduce the Fabry–Perot oscillations that occur in a raised-cosine grating. Imperfections in this nominal dc correction lead to a broadening of the transmission spectrum.

Fig. 2
Fig. 2

Normalized dispersion relations with ν=0.03 and M=0, 2; fx refers to solution (4), in which case δ represents δx; fy refers to solution (6), in which case δ represents δy. Horizontal arrows, wave-number mismatch between the two polarizations for M=2. Closer to the photonic bandgap the mismatch is larger. Furthermore, the vertical distance between the x and y bands is different for positive and negative detuning. This asymmetry is not seen for the M=0 dispersion relation.

Fig. 3
Fig. 3

(a) Birefringence enhancement factor Fb(f) for M=0 (solid curves) and M=2.0 (dashed curves) for both positive and negative detunings. The M=0 values are symmetric about the horizontal axis; the M=2 values are not. For M=2, Fb(f)=0 for a particular positive detuning, as discussed in the text. (b) Group-velocity mismatch as a function of positive detuning for M=0.45 (solid curve), M=0.5 (dashed curve), and M=0.55 (dotted curve). The mismatch vanishes for specific values of detuning, as discussed in the text.

Fig. 4
Fig. 4

Birefringence enhancement factor for a 4-cm grating (solid curve with filled circles), a 20-cm grating (dotted curve), and an infinite system (solid curve). The 4- and 20-cm gratings are apodized, as discussed in the text. The 4-cm grating exhibits some oscillations in the birefringence enhancement as a result of its short length.

Fig. 5
Fig. 5

Schematic diagram of the experimental apparatus used to measure the polarization properties of birefringent fiber gratings. A Q-switched, mode-locked laser produces 80-ps pulses. An electro-optic pulse picker (PP) transmits one pulse from each Q-switched pulse train to the grating through an attenuator (ATT). The input polarization state is set relative to the principal axes of the fiber grating by use of a half-wave (LP1). The collimated light beam is focused onto the fiber core by a microscope object lens (L). A similar objective recollimates the beam after transmission through the grating (G) and focuses it onto a detector at a distance of 2 m. This long distance prevents the detection of light that was leaked through the fiber cladding. The output polarization state is rotated to the vertical direction by a half-wave plate (LP2) such that the light experiences a constant transmission coefficient as it propagates through a series of mirrors and lenses to the fast photodiode detector (D) and a sampling scope. Transmission measurements of the grating bandgap were made with a continuous-wave laser.

Fig. 6
Fig. 6

(a) Transmission spectrum of the experimental grating (symbols) and results of the numerical simulation (curves). The mismatch in photonic bandgaps on the low detuning side is fitted by M=0.38; the slope of the edges of the spectrum is explained by δNi(z). (b) Measured (symbols) and simulated (curves) phase lag for the experimental grating.

Equations (21)

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nx(z)=[n0+δnx(z)]+δnx(z)cos(2k0z),
ny(z)=[n0+δny(z)]+δny(z)cos(2k0z),
nx(z)=n¯x+δnx(z)cos(2k0z),
ny(z)=n¯y+δny(z)cos(2k0z),
nbn¯y-n¯x,
δnbδn¯y-δn¯x.
M12δnbnb.
δni(z)=δn¯i exp[-(ln 2)(z-3zhω)2/zhω2]0<z<3zhωδn¯i3zhω<z<(L-3zhω),δn¯i exp{-(ln 2)[z-(L-3zhω)]2/zhω2}(L-3zhω)<z<L
ni(z)=n¯i+δNi(z)+δni(z)cos(2k0z),
δNi(z)=-δN¯iπ(2.6zhω-z)0.4zhω0<z<2.6zhω,δN¯i sinπ(z-2.6zhω)0.4zhω2.6zhω<z<3zhω,03zhω<z<(L-3zhω),δN¯i sinπ[z-(L-3zhω)]0.4zhω(L-3zhω)<z<(L-2.6zhω),-δN¯iπ[z-(L-2.6zhω)]0.4zhω(L-2.6zhω)<z<L,
0=±iAi±(z; ω)z+σi(z)+n¯ic(ω-ω0i)Ai±(z; ω)+κi(z)Ai(z; ω),
κi(z)=12δni(z)n¯ik0,σi(z)=δNi(z)n¯ik0,
ni(z)=n¯i+(δn¯i)cos(2k0z),
fi=n¯xcκ¯x(ωi-ω0x),δi=ki-k0κ¯x.
fx(δx)=±δx2+1,
ν=2nbδn¯x,
fy(δy)=n¯xn¯y-ν±δy2+n¯xn¯y(1+Mν)21/2-ν±[δy2+(1+Mν)2]1/2,
Fb(f)=δy(f)-δx(f)ν=±1ν{[(f+ν)2-(1+Mν)2]1/2-f2-1},
fno-bi=ν2(M2-1)+M.
ωx=cn¯xf2-1f,
ωy=cn¯y[(f+ν)2+(1+Mν)2]1/2f+ν.

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