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Azimuthal modulation instability for a cylindrically polarized wave in a nonlinear Kerr medium

Joseph W. Haus, Zasim Mozumder, and Qiwen Zhan

Open Access Open Access

Abstract

Inhomogeneously polarized optical waves form a class of nonlinear vector wave propagation that has not been widely studied in the literature. We find a modulation instability only when the wave has nonzero ellipticity in a medium where the Kerr nonlinearity possesses opposite handness. Under the modulation instability the wave develops an azimuthally periodic shape with two or four peaks.

©2006 Optical Society of America

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Supplementary Material (5)

Media 1: AVI (505 KB)     
Media 2: AVI (2540 KB)     
Media 3: AVI (393 KB)     
Media 4: AVI (382 KB)     
Media 5: AVI (351 KB)     

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Figures (8)
Fig. 1.
Fig. 1. Scalar case showing donut initial profile (right) and final profile (left). The parameters are B=6 and E0=1.5.

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Fig. 2.
Fig. 2. ector wave propagation for the case B=0 and E0=1.5. The intensity remains rotationally symmetric and the intensity monotonically falls. Other parameters are θ=0, and φ=π/4.

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Fig. 3.
Fig. 3. Intensity versus amplitude after propagation a distance z=2 through the medium. Other parameters are: B=6 θ=0 and φ=π/4 (Movie).

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Fig. 4.
Fig. 4. Evolution of the initial beam on the axis for the following parameter values: B=6, E0=2, θ=0, φ=π/4 (Movie).

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Fig. 5.
Fig. 5. The maximum intensity relative to its initial value versus z. The initial amplitudes are near the critical power for the given retardance values. The critical power lies between the amplitude values 2.3, shown by a dotted line, and 2.4, shown by a dashed line. The parameters are B=6, θ=0, and φ=π/4.

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Fig. 6.
Fig. 6. Intensity profile as a function of the phase retardance angle φ for B=6, E0=2 and θ=0. The propagation distance is z=2 (Movie).

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Fig. 7.
Fig. 7. The component, S3 , of the Stokes vector defined in Eq. (7) after propagating to z=2; the change of S3 with retardation angle is shown. The parameter values are: B=6, E0=2 and θ=0 (Movie).

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Fig. 8.
Fig. 8. Intensity profile as a function of the cylindrical orientation angle θ for B=6, E0=2 φ=π/4. The propagation distance is z=2 (Movie).

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Equations (7)

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

∂ E α ∂ z = i 2 ∇ ⊥ 2 E α + i ( ∣ E x ∣ 2 + ∣ E y ∣ 2 ) E a + i B 2 ( E x 2 + E y 2 ) E α * ,
∂ E ± ∂ z = i 2 ∇ ⊥ 2 E ± + i ( ∣ E ± ∣ 2 + ( 1 + B ) ∣ E ∓ ∣ 2 ) E ± .
E x = E 0 ( cos ⁡ θ x + sin ⁡ θ y ) e − r 2 ,
E y = E 0 e i ϕ ( − sin ⁡ θ x + cos ⁡ θ y ) e − r 2 ,
E = E 0 r exp ( − r 2 ) ,
∂ E ∂ z = i 2 ∇ ⊥ 2 E + i ( 1 + B 2 ) ∣ E ∣ 2 E ,
S 3 = ∣ E + ∣ 2 − ∣ E − ∣ 2 .
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