Abstract

Diffraction of a 3D optical beam on a multilayer phase-shifted Bragg grating (PSBG) is considered. It is shown that the PSBG enables optical computation of the spatial Laplace operator of the electromagnetic field components of the incident beam. The computation of the Laplacian is performed in reflection at normal incidence. As a special case, the parameters of the PSBG transforming the incident Gaussian beam into a Laguerre–Gaussian mode of order (1,0) are obtained. Presented numerical results demonstrate high quality of the Laplace operator computation and confirm the possibility of the formation of Laguerre–Gaussian mode. We expect the proposed applications to be useful for all-optical data processing.

© 2014 Optical Society of America

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References

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  1. A. Silva, F. Monticone, G. Castaldi, V. Galdi, A. Alù, and N. Engheta, “Performing mathematical operations with metamaterials,” Science 343(6167), 160–163 (2014).
    [Crossref] [PubMed]
  2. R. Slavík, Y. Park, M. Kulishov, and J. Azaña, “Terahertz-bandwidth high-order temporal differentiators based on phase-shifted long-period fiber gratings,” Opt. Lett. 34(20), 3116–3118 (2009).
    [Crossref] [PubMed]
  3. L. M. Rivas, S. Boudreau, Y. Park, R. Slavík, S. LaRochelle, A. Carballar, and J. Azaña, “Experimental demonstration of ultrafast all-fiber high-order photonic temporal differentiators,” Opt. Lett. 34(12), 1792–1794 (2009).
    [Crossref] [PubMed]
  4. M.A. Preciado, X. Shu, P. Harper, and K. Sugden, “Experimental demonstration of an optical differentiator based on a fiber Bragg grating in transmission,” Opt. Lett. 38(6), 917–919 (2013).
    [Crossref] [PubMed]
  5. N. K. Berger, B. Levit, B. Fischer, M. Kulishov, D. V. Plant, and J. Azaña, “Temporal differentiation of optical signals using a phase-shifted fiber Bragg grating,” Opt. Lett. 15(2), 371–381 (2007).
  6. M. Kulishov and J. Azaña, “Design of high-order all-optical temporal differentiators based on multiple-phase-shifted fiber Bragg gratings,” Opt. Express 15(10), 6152–6166 (2007).
    [Crossref] [PubMed]
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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
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    [Crossref]
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2014 (2)

A. Silva, F. Monticone, G. Castaldi, V. Galdi, A. Alù, and N. Engheta, “Performing mathematical operations with metamaterials,” Science 343(6167), 160–163 (2014).
[Crossref] [PubMed]

L. L. Doskolovich, D. A. Bykov, E. A. Bezus, and V. A. Soifer, “Spatial differentiation of optical beams using phase-shifted Bragg grating,” Opt. Lett. 39(5), 1278–1281 (2014).
[Crossref] [PubMed]

2013 (1)

2012 (1)

2011 (1)

2010 (1)

2009 (2)

2008 (2)

2007 (4)

N. Q. Ngo, “Design of an optical temporal integrator based on a phase-shifted fiber Bragg grating in transmission,” Opt. Lett. 32(20), 3020–3022 (2007).
[Crossref]

N. K. Berger, B. Levit, B. Fischer, M. Kulishov, D. V. Plant, and J. Azaña, “Temporal differentiation of optical signals using a phase-shifted fiber Bragg grating,” Opt. Lett. 15(2), 371–381 (2007).

M. Kulishov and J. Azaña, “Design of high-order all-optical temporal differentiators based on multiple-phase-shifted fiber Bragg gratings,” Opt. Express 15(10), 6152–6166 (2007).
[Crossref] [PubMed]

V. Lomakin and E. Michielssen, “Beam transmission through periodic subwavelength hole structures,” IEEE Trans. Antennas Propag. 55(6), 1564–1581 (2007).
[Crossref]

2006 (1)

2003 (1)

1996 (2)

Alù, A.

A. Silva, F. Monticone, G. Castaldi, V. Galdi, A. Alù, and N. Engheta, “Performing mathematical operations with metamaterials,” Science 343(6167), 160–163 (2014).
[Crossref] [PubMed]

Azaña, J.

Berger, N. K.

N. K. Berger, B. Levit, B. Fischer, M. Kulishov, D. V. Plant, and J. Azaña, “Temporal differentiation of optical signals using a phase-shifted fiber Bragg grating,” Opt. Lett. 15(2), 371–381 (2007).

Bezus, E. A.

Boudreau, S.

Bykov, D. A.

Carballar, A.

Castaldi, G.

A. Silva, F. Monticone, G. Castaldi, V. Galdi, A. Alù, and N. Engheta, “Performing mathematical operations with metamaterials,” Science 343(6167), 160–163 (2014).
[Crossref] [PubMed]

Doskolovich, L. L.

Engheta, N.

A. Silva, F. Monticone, G. Castaldi, V. Galdi, A. Alù, and N. Engheta, “Performing mathematical operations with metamaterials,” Science 343(6167), 160–163 (2014).
[Crossref] [PubMed]

Fan, S.

Fischer, B.

N. K. Berger, B. Levit, B. Fischer, M. Kulishov, D. V. Plant, and J. Azaña, “Temporal differentiation of optical signals using a phase-shifted fiber Bragg grating,” Opt. Lett. 15(2), 371–381 (2007).

Galdi, V.

A. Silva, F. Monticone, G. Castaldi, V. Galdi, A. Alù, and N. Engheta, “Performing mathematical operations with metamaterials,” Science 343(6167), 160–163 (2014).
[Crossref] [PubMed]

Guo, C.-S.

Harper, P.

Joannopoulos, J. D.

Kulishov, M.

Landry, G. D.

LaRochelle, S.

Levit, B.

N. K. Berger, B. Levit, B. Fischer, M. Kulishov, D. V. Plant, and J. Azaña, “Temporal differentiation of optical signals using a phase-shifted fiber Bragg grating,” Opt. Lett. 15(2), 371–381 (2007).

Li, D.

Li, L.

Lomakin, V.

V. Lomakin and E. Michielssen, “Beam transmission through periodic subwavelength hole structures,” IEEE Trans. Antennas Propag. 55(6), 1564–1581 (2007).
[Crossref]

Lu, L.-L.

Maldonado, T. A.

Michielssen, E.

V. Lomakin and E. Michielssen, “Beam transmission through periodic subwavelength hole structures,” IEEE Trans. Antennas Propag. 55(6), 1564–1581 (2007).
[Crossref]

Monticone, F.

A. Silva, F. Monticone, G. Castaldi, V. Galdi, A. Alù, and N. Engheta, “Performing mathematical operations with metamaterials,” Science 343(6167), 160–163 (2014).
[Crossref] [PubMed]

Ngo, N. Q.

Park, Y.

Plant, D. V.

N. K. Berger, B. Levit, B. Fischer, M. Kulishov, D. V. Plant, and J. Azaña, “Temporal differentiation of optical signals using a phase-shifted fiber Bragg grating,” Opt. Lett. 15(2), 371–381 (2007).

Preciado, M.A.

Rivas, L. M.

Ryle, J. P.

Sepke, S. M.

Sheridan, J. T.

Shu, X.

Siegman, A. E.

A. E. Siegman, Lasers (University Science Books, 1986).

Silva, A.

A. Silva, F. Monticone, G. Castaldi, V. Galdi, A. Alù, and N. Engheta, “Performing mathematical operations with metamaterials,” Science 343(6167), 160–163 (2014).
[Crossref] [PubMed]

Slavík, R.

Soifer, V. A.

Sugden, K.

Suh, W.

Umstadter, D. P.

Wei, G.-X.

Yue, Q.-Y.

Yue, S.-J.

Zhou, G.

Appl. Opt. (1)

IEEE Trans. Antennas Propag. (1)

V. Lomakin and E. Michielssen, “Beam transmission through periodic subwavelength hole structures,” IEEE Trans. Antennas Propag. 55(6), 1564–1581 (2007).
[Crossref]

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

Opt. Express (2)

Opt. Lett. (10)

D. A. Bykov, L. L. Doskolovich, and V. A. Soifer, “Temporal differentiation of optical signals using resonant gratings,” Opt. Lett. 36(17), 3509–3511 (2011).
[Crossref] [PubMed]

L. L. Doskolovich, D. A. Bykov, E. A. Bezus, and V. A. Soifer, “Spatial differentiation of optical beams using phase-shifted Bragg grating,” Opt. Lett. 39(5), 1278–1281 (2014).
[Crossref] [PubMed]

R. Slavík, Y. Park, M. Kulishov, and J. Azaña, “Terahertz-bandwidth high-order temporal differentiators based on phase-shifted long-period fiber gratings,” Opt. Lett. 34(20), 3116–3118 (2009).
[Crossref] [PubMed]

L. M. Rivas, S. Boudreau, Y. Park, R. Slavík, S. LaRochelle, A. Carballar, and J. Azaña, “Experimental demonstration of ultrafast all-fiber high-order photonic temporal differentiators,” Opt. Lett. 34(12), 1792–1794 (2009).
[Crossref] [PubMed]

M.A. Preciado, X. Shu, P. Harper, and K. Sugden, “Experimental demonstration of an optical differentiator based on a fiber Bragg grating in transmission,” Opt. Lett. 38(6), 917–919 (2013).
[Crossref] [PubMed]

N. K. Berger, B. Levit, B. Fischer, M. Kulishov, D. V. Plant, and J. Azaña, “Temporal differentiation of optical signals using a phase-shifted fiber Bragg grating,” Opt. Lett. 15(2), 371–381 (2007).

S. M. Sepke and D. P. Umstadter, “Exact analytical solution for the vector electromagnetic field of Gaussian, flattened Gaussian, and annular Gaussian laser modes,” Opt. Lett. 31(10), 1447–1449 (2006).
[Crossref] [PubMed]

N. Q. Ngo, “Design of an optical temporal integrator based on a phase-shifted fiber Bragg grating in transmission,” Opt. Lett. 32(20), 3020–3022 (2007).
[Crossref]

C.-S. Guo, Q.-Y. Yue, G.-X. Wei, L.-L. Lu, and S.-J. Yue, “Laplacian differential reconstruction of in-line holograms recorded at two different distances,” Opt. Lett. 33(17), 1945–1947 (2008).
[Crossref] [PubMed]

J. P. Ryle, D. Li, and J. T. Sheridan, “Dual wavelength digital holographic Laplacian reconstruction,” Opt. Lett. 35(18), 3018–3020 (2010).
[Crossref] [PubMed]

Science (1)

A. Silva, F. Monticone, G. Castaldi, V. Galdi, A. Alù, and N. Engheta, “Performing mathematical operations with metamaterials,” Science 343(6167), 160–163 (2014).
[Crossref] [PubMed]

Other (2)

V. A. Soifer, Diffractive Nanophotonics (CRC Press, 2014).
[Crossref]

A. E. Siegman, Lasers (University Science Books, 1986).

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Figures (4)

Fig. 1
Fig. 1

A phase-shifted Bragg grating. Transverse field distributions of the incident (left) and reflected (right) beams are shown.

Fig. 2
Fig. 2

Amplitudes (bottom plots, left vertical axis) and phases (top plots, right vertical axis) of the TFs HTE,TM (α, 0) = RTE,TM(α) versus α/k0 at |α/k0| ≤ sin 10° for PSBGs consisting of 13, 17, and 21 layers. The TFs HTE (α, 0) (solid lines) and HTM(α, 0) (dotted lines) are shown. Dashed lines correspond to “ideal” TFs performing exact computation of the Laplace operator.

Fig. 3
Fig. 3

Transverse distributions (bottom) and profiles (top) of the Ex component magnitude of the incident Gaussian beam (a) and the reflected beam (b). Circles in (b) show exact Laplacian of the incident beam.

Fig. 4
Fig. 4

(a) Transverse distribution (bottom) and profile (top) of the Ex component magnitude of the reflected beam (circles show the profile of the Laguerre–Gaussian mode). (b) Evolution of the beam profile during the propagation: z = 0 (solid line), z = 100μm (dashed line), z = 200μm (dotted line).

Equations (19)

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Φ ( r ) = Φ TE , TM ( α , β ) exp ( i p r ) ,
Φ TM ( α , β ) = 1 α 2 + β 2 [ α γ β γ α 2 + β 2 k 0 β ε k 0 α ε 0 ] , Φ TE ( α , β ) = 1 α 2 + β 2 [ k 0 β k 0 α 0 α γ β γ α 2 + β 2 ] ,
A inc ( r ) = A inc , TE ( r ) + A inc , TM ( r ) = G TE ( α , β ) Φ TE ( α , β ) exp ( i p r ) d α d β + + G TM ( α , β ) Φ TM ( α , β ) exp ( i p r ) d α d β = G inc ( α , β ) exp ( i p r ) d α d β ,
G inc ( α , β ) = G TE ( α , β ) Φ TE ( α , β ) + G TM ( α , β ) Φ TM ( α , β ) .
{ E x , inc ( x , y , 0 ) = G x ( α , β ) exp ( i ( α x + β y ) ) d α d β ; E y , inc ( x , y , 0 ) 0 ,
{ α γ G TM ( α , β ) β k 0 G TE ( α , β ) = G x ( α , β ) α 2 + β 2 ; β γ G TM ( α , β ) + α k 0 G TE ( α , β ) = 0 .
G TE ( α , β ) = β G x ( α , β ) k 0 α 2 + β 2 , G TM ( α , β ) = α G x ( α , β ) γ α 2 + β 2 .
G x ( α , β ) = σ 2 2 π exp { σ 2 2 ( α 2 + β 2 ) } .
A refl ( r ) = A refl , TE ( r ) + A refl , TM ( r ) = G TE ( α , β ) Φ TE ( α , β ) R TE ( α , β ) exp ( i p r ) d α d β + + G TM ( α , β ) Φ TM ( α , β ) R TM ( α , β ) exp ( i p r ) d α d β ,
R TE ( α , β ) = R TE ( α 2 + β 2 , 0 ) ; R TM ( α , β ) = R TM ( α 2 + β 2 , 0 ) .
H TE , TM ( α , β ) ~ H Δ ( α , β ) = ( i α ) 2 + ( i β ) 2 , α 2 + β 2 < δ ,
A refl , TE ( x , y , 0 ) ~ Δ A inc , TE ( x , y , 0 ) , A refl , TM ( x , y , 0 ) ~ Δ A inc , TM ( x , y , 0 ) .
n ˜ 1 h 1 = n ˜ 2 h 2 = λ B / 4 ,
R ( α ) r + j = 1 M b j α p j ,
R ( α ) r α 2 α 0 2 α 2 p 2 = r r ( α 0 2 p 2 ) 2 p ( 1 α p 1 α + p ) .
H TE , TM ( α , β ) = r TE , TM α 2 + β 2 α 2 + β 2 p TE , TM 2 = r TE , TM p TE , TM 2 H Δ ( α , β ) H err ( α , β ) ,
A refl ( r ) = G inc ( α , β ) R TE ( α , β ) exp ( i p r ) d α d β ,
E x , LG ( x , y ) = exp ( ρ 2 2 σ 2 ) L 1 ( ρ 2 σ 2 ) , ρ 2 = x 2 + y 2 ,
E x , LG ( x , y ) = σ 2 [ 1 σ 2 exp ( ρ 2 2 σ 2 ) + Δ exp ( ρ 2 2 σ 2 ) ] = σ 2 [ 1 σ 2 + 1 σ 2 ( 2 ρ 2 σ 2 ) ] exp ( ρ 2 2 σ 2 ) .

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