Abstract

We present a theoretical study of a new application of the phase-shifted Bragg grating (PSBG) as an optical spatial differentiator operating in reflection. We demonstrate that the PSBG allows to calculate the first-order spatial derivative at oblique incidence and the second-order derivative at normal incidence. As an example, the differentiator is numerically shown to be able to convert an input 2D Gaussian beam into a 2D Hermite–Gaussian mode. We expect the proposed application to be useful for all-optical data processing.

© 2014 Optical Society of America

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References

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    [CrossRef]

2013 (1)

2012 (1)

2011 (1)

2009 (2)

2008 (1)

2007 (2)

1995 (1)

1986 (1)

E. Popov, L. Mashev, and D. Maystre, Opt. Acta 33, 607 (1986).
[CrossRef]

Azaña, J.

Berger, N. K.

Boudreau, S.

Bykov, D. A.

Carballar, A.

Doskolovich, L. L.

Fischer, B.

Harper, P.

Kulishov, M.

LaRochelle, S.

Levit, B.

Li, H.

Li, M.

Mashev, L.

E. Popov, L. Mashev, and D. Maystre, Opt. Acta 33, 607 (1986).
[CrossRef]

Maystre, D.

E. Popov, L. Mashev, and D. Maystre, Opt. Acta 33, 607 (1986).
[CrossRef]

Neviere, M.

Ngo, N. Q.

Painchaud, Y.

Park, Y.

Plant, D. V.

Popov, E.

M. Neviere, E. Popov, and R. Reinisch, J. Opt. Soc. Am. A 12, 513 (1995).
[CrossRef]

E. Popov, L. Mashev, and D. Maystre, Opt. Acta 33, 607 (1986).
[CrossRef]

Preciado, M. A.

Reinisch, R.

Rivas, L. M.

Shu, X.

Slavík, R.

Soifer, V. A.

Sugden, K.

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Figures (6)

Fig. 1.
Fig. 1.

Geometry of the beam diffraction by a PSBG.

Fig. 2.
Fig. 2.

(a) Reflection spectra for PSBGs composed of 9, 13, and 17 layers (solid blue, dashed green, and dotted red curves, respectively). The spectra were computed for TE-polarization at θ0=30°, kx=0. (b) TF amplitudes (bottom of the plot, left vertical axis) and phases (top of the plot, right vertical axis) versus kx/k0 at λ=1500nm, |kx/k0|sin10° for PSBGs composed of 9, 13, and 17 layers (solid blue, dashed green, and dotted red curves, respectively).

Fig. 3.
Fig. 3.

Reflected beam amplitude for a 9-layer PSBG at zrefl=0 (solid-blue curve), zrefl=500μm (dashed-green curve), and zrefl=1mm (dotted-red curve). (Inset) incident Gaussian beam (red curve, right vertical axis), amplitude of the reflected beam (solid-blue curve, left vertical axis), and the modulus of the analytically computed derivative (dotted-blue curve).

Fig. 4.
Fig. 4.

Electromagnetic field (the amplitude of the y-component of the electric field) generated upon diffraction of a Gaussian beam on the 17-layer PSBG.

Fig. 5.
Fig. 5.

Incident Gaussian beam (blue curve, right vertical axis), the amplitude of the beam reflected at the PSBG (solid-red curve, left vertical axis) and the modulus of the analytically calculated second derivative (red dots, left vertical axis).

Fig. 6.
Fig. 6.

Amplitude of the beam reflected at PSBG for a wavelength of 1499.2 nm at zrefl=0 (solid-blue curve), zrefl=500μm (dashed-green curve), and zrefl=1mm (dotted-red curve). (Inset) Incident Gaussian beam (blue curve, right vertical axis), reflected beam amplitude at zrefl=0 (solid-blue curve, left vertical axis), and the modulus of the Hermite–Gaussian mode H2(x/σ) (blue dots, left vertical axis).

Equations (9)

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Φinc(xinc,zinc)=Ainc(kx)exp(ikxxincikzzinc)dkx,
Arefl(kx)=Ainc(kx)R(k˜x),
Φrefl(xrefl,zrefl)=A(kx)R(k˜x)exp(ikxxrefl+ikzzrefl)dkx.
H(kx)R(kxcosθ0+kx0),
H(kx)Hdiff(kx)=ikx,|kx|g,
n˜1h1=n˜2h2=λB/4,
R(k˜x;λ)rk˜xkzero(λ)k˜xkpole(λ),
H(kx)=ikxHerr(kx),
H(kx)=R(kx)(kx2kzero2(λ))Herr(kx2),

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