Abstract

We propose the definition of a new family of complex-amplitude filters. These filters are optimal for mode conversion in single-mode fibers and give rise to a large variety of spatial designs. Among them, we focus on a few that are easy to manufacture and implement, in particular, in an array shape, and present some technical advantages such as tolerance to positioning. In the second part, we discuss their implementations by using various technologies and the effect of some technical constraints on filter shape and phase distribution. Finally, we illustrate their uses in the case of a dynamic wavelength blocker within the frame of wavelength-division-multiplexing network systems.

© 2004 Optical Society of America

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References

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  1. A. W. Snyder, “Excitation and scattering of modes on a dielectric or optical fiber,” IEEE Trans. Microwave Theory Tech. 17, 1138–1144 (1969).
    [CrossRef]
  2. G. Cohen, “Fiber optical attenuator,” Israel patentWO 02071133 (September12, 2002).
  3. G. Cohen, “Fiber optical gain equalizer,” Israel patentWO 03009054 (January30, 2003).
  4. N. Wolffer, B. Vinouze, P. Gravey, “Holographic switching between single mode fibers based on electrically ad-dressed nematic liquid crystal gratings with high deflection efficiency,” Opt. Commun. 160, 42–46 (1999).
    [CrossRef]
  5. M. Ikeda, K. Kitayama, “Transfer function of long spliced graded-index fibers with mode scramblers,” Appl. Opt. 17, 63–67 (1978).
    [CrossRef] [PubMed]
  6. T. Loukina, R. Chevallier, J. L. de Bougrenet de la Tocnaye, M. Barge, “Dynamic spectral equalizer using free-space dispersive optics combined with a polymer dispersed liquid crystal spatial light attenuator,” J. Lightwave Technol. 21, 2067–2073 (2003).
    [CrossRef]
  7. M. C. Parker, A. D. Cohen, R. J. Mears, “Dynamic digital holographic wavelength filtering,” J. Lightwave Technol. 16, 1259–1270 (1998).
    [CrossRef]
  8. J. F. Kenney, E. S. Keeping, “Quantiles,” in Mathematics of Statistics (Van Nostrand, Princeton, N.J., 1962), pp. 37–38.
  9. P. Chanclou, B. Vinouze, M. Roy, C. Cornu, H. Ramanitra, “Phase shifting VOA with polymer dispersed liquid crystal,” J. Lightwave Technol. 21 (2003).
    [CrossRef]
  10. V. Nourrit, J. L. de Bougrenet de la Tocnaye, P. Chanclou, “Propagation and diffraction of truncated Gaussian beams,” J. Opt. Soc. Am. A 18, 546–556 (2001).
    [CrossRef]
  11. J. Feinleib, S. Lipson, P. Cone, “Monolithic piezoelectric mirror for wavefront correction,” Appl. Phys. Lett. 25, 311–313 (1974).
    [CrossRef]
  12. R. Krishnamoorthy, T. G. Bifano, G. Sandri, “Statistical performance evaluation of electrostatic micro actuators for a deformable mirror,” in Microelectronic Structures and MEMS for Optical Processing II, M. E. Motamedi, W. Bailey, eds., Proc. SPIE2881, 35–44 (1996).
    [CrossRef]
  13. C. Dragone, C. A. Edwards, R. C. Kistler, “An N×N optical multiplexer using planar arrangement of two star couplers,” IEEE Photon. Technol. Lett. 3, 812–815 (1991).
    [CrossRef]

2003 (2)

2001 (1)

1999 (1)

N. Wolffer, B. Vinouze, P. Gravey, “Holographic switching between single mode fibers based on electrically ad-dressed nematic liquid crystal gratings with high deflection efficiency,” Opt. Commun. 160, 42–46 (1999).
[CrossRef]

1998 (1)

1991 (1)

C. Dragone, C. A. Edwards, R. C. Kistler, “An N×N optical multiplexer using planar arrangement of two star couplers,” IEEE Photon. Technol. Lett. 3, 812–815 (1991).
[CrossRef]

1978 (1)

1974 (1)

J. Feinleib, S. Lipson, P. Cone, “Monolithic piezoelectric mirror for wavefront correction,” Appl. Phys. Lett. 25, 311–313 (1974).
[CrossRef]

1969 (1)

A. W. Snyder, “Excitation and scattering of modes on a dielectric or optical fiber,” IEEE Trans. Microwave Theory Tech. 17, 1138–1144 (1969).
[CrossRef]

Barge, M.

Bifano, T. G.

R. Krishnamoorthy, T. G. Bifano, G. Sandri, “Statistical performance evaluation of electrostatic micro actuators for a deformable mirror,” in Microelectronic Structures and MEMS for Optical Processing II, M. E. Motamedi, W. Bailey, eds., Proc. SPIE2881, 35–44 (1996).
[CrossRef]

Chanclou, P.

P. Chanclou, B. Vinouze, M. Roy, C. Cornu, H. Ramanitra, “Phase shifting VOA with polymer dispersed liquid crystal,” J. Lightwave Technol. 21 (2003).
[CrossRef]

V. Nourrit, J. L. de Bougrenet de la Tocnaye, P. Chanclou, “Propagation and diffraction of truncated Gaussian beams,” J. Opt. Soc. Am. A 18, 546–556 (2001).
[CrossRef]

Chevallier, R.

Cohen, A. D.

Cohen, G.

G. Cohen, “Fiber optical attenuator,” Israel patentWO 02071133 (September12, 2002).

G. Cohen, “Fiber optical gain equalizer,” Israel patentWO 03009054 (January30, 2003).

Cone, P.

J. Feinleib, S. Lipson, P. Cone, “Monolithic piezoelectric mirror for wavefront correction,” Appl. Phys. Lett. 25, 311–313 (1974).
[CrossRef]

Cornu, C.

P. Chanclou, B. Vinouze, M. Roy, C. Cornu, H. Ramanitra, “Phase shifting VOA with polymer dispersed liquid crystal,” J. Lightwave Technol. 21 (2003).
[CrossRef]

de Bougrenet de la Tocnaye, J. L.

Dragone, C.

C. Dragone, C. A. Edwards, R. C. Kistler, “An N×N optical multiplexer using planar arrangement of two star couplers,” IEEE Photon. Technol. Lett. 3, 812–815 (1991).
[CrossRef]

Edwards, C. A.

C. Dragone, C. A. Edwards, R. C. Kistler, “An N×N optical multiplexer using planar arrangement of two star couplers,” IEEE Photon. Technol. Lett. 3, 812–815 (1991).
[CrossRef]

Feinleib, J.

J. Feinleib, S. Lipson, P. Cone, “Monolithic piezoelectric mirror for wavefront correction,” Appl. Phys. Lett. 25, 311–313 (1974).
[CrossRef]

Gravey, P.

N. Wolffer, B. Vinouze, P. Gravey, “Holographic switching between single mode fibers based on electrically ad-dressed nematic liquid crystal gratings with high deflection efficiency,” Opt. Commun. 160, 42–46 (1999).
[CrossRef]

Ikeda, M.

Keeping, E. S.

J. F. Kenney, E. S. Keeping, “Quantiles,” in Mathematics of Statistics (Van Nostrand, Princeton, N.J., 1962), pp. 37–38.

Kenney, J. F.

J. F. Kenney, E. S. Keeping, “Quantiles,” in Mathematics of Statistics (Van Nostrand, Princeton, N.J., 1962), pp. 37–38.

Kistler, R. C.

C. Dragone, C. A. Edwards, R. C. Kistler, “An N×N optical multiplexer using planar arrangement of two star couplers,” IEEE Photon. Technol. Lett. 3, 812–815 (1991).
[CrossRef]

Kitayama, K.

Krishnamoorthy, R.

R. Krishnamoorthy, T. G. Bifano, G. Sandri, “Statistical performance evaluation of electrostatic micro actuators for a deformable mirror,” in Microelectronic Structures and MEMS for Optical Processing II, M. E. Motamedi, W. Bailey, eds., Proc. SPIE2881, 35–44 (1996).
[CrossRef]

Lipson, S.

J. Feinleib, S. Lipson, P. Cone, “Monolithic piezoelectric mirror for wavefront correction,” Appl. Phys. Lett. 25, 311–313 (1974).
[CrossRef]

Loukina, T.

Mears, R. J.

Nourrit, V.

Parker, M. C.

Ramanitra, H.

P. Chanclou, B. Vinouze, M. Roy, C. Cornu, H. Ramanitra, “Phase shifting VOA with polymer dispersed liquid crystal,” J. Lightwave Technol. 21 (2003).
[CrossRef]

Roy, M.

P. Chanclou, B. Vinouze, M. Roy, C. Cornu, H. Ramanitra, “Phase shifting VOA with polymer dispersed liquid crystal,” J. Lightwave Technol. 21 (2003).
[CrossRef]

Sandri, G.

R. Krishnamoorthy, T. G. Bifano, G. Sandri, “Statistical performance evaluation of electrostatic micro actuators for a deformable mirror,” in Microelectronic Structures and MEMS for Optical Processing II, M. E. Motamedi, W. Bailey, eds., Proc. SPIE2881, 35–44 (1996).
[CrossRef]

Snyder, A. W.

A. W. Snyder, “Excitation and scattering of modes on a dielectric or optical fiber,” IEEE Trans. Microwave Theory Tech. 17, 1138–1144 (1969).
[CrossRef]

Vinouze, B.

P. Chanclou, B. Vinouze, M. Roy, C. Cornu, H. Ramanitra, “Phase shifting VOA with polymer dispersed liquid crystal,” J. Lightwave Technol. 21 (2003).
[CrossRef]

N. Wolffer, B. Vinouze, P. Gravey, “Holographic switching between single mode fibers based on electrically ad-dressed nematic liquid crystal gratings with high deflection efficiency,” Opt. Commun. 160, 42–46 (1999).
[CrossRef]

Wolffer, N.

N. Wolffer, B. Vinouze, P. Gravey, “Holographic switching between single mode fibers based on electrically ad-dressed nematic liquid crystal gratings with high deflection efficiency,” Opt. Commun. 160, 42–46 (1999).
[CrossRef]

Appl. Opt. (1)

Appl. Phys. Lett. (1)

J. Feinleib, S. Lipson, P. Cone, “Monolithic piezoelectric mirror for wavefront correction,” Appl. Phys. Lett. 25, 311–313 (1974).
[CrossRef]

IEEE Photon. Technol. Lett. (1)

C. Dragone, C. A. Edwards, R. C. Kistler, “An N×N optical multiplexer using planar arrangement of two star couplers,” IEEE Photon. Technol. Lett. 3, 812–815 (1991).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (1)

A. W. Snyder, “Excitation and scattering of modes on a dielectric or optical fiber,” IEEE Trans. Microwave Theory Tech. 17, 1138–1144 (1969).
[CrossRef]

J. Lightwave Technol. (3)

J. Opt. Soc. Am. A (1)

Opt. Commun. (1)

N. Wolffer, B. Vinouze, P. Gravey, “Holographic switching between single mode fibers based on electrically ad-dressed nematic liquid crystal gratings with high deflection efficiency,” Opt. Commun. 160, 42–46 (1999).
[CrossRef]

Other (4)

R. Krishnamoorthy, T. G. Bifano, G. Sandri, “Statistical performance evaluation of electrostatic micro actuators for a deformable mirror,” in Microelectronic Structures and MEMS for Optical Processing II, M. E. Motamedi, W. Bailey, eds., Proc. SPIE2881, 35–44 (1996).
[CrossRef]

G. Cohen, “Fiber optical attenuator,” Israel patentWO 02071133 (September12, 2002).

G. Cohen, “Fiber optical gain equalizer,” Israel patentWO 03009054 (January30, 2003).

J. F. Kenney, E. S. Keeping, “Quantiles,” in Mathematics of Statistics (Van Nostrand, Princeton, N.J., 1962), pp. 37–38.

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Figures (12)

Fig. 1
Fig. 1

Different filtering configurations: (a) in contact, (b) in a 4f imaging system used in reflection.

Fig. 2
Fig. 2

Filter array devices: (a) in a fiber array attenuator configuration, (b) in a spectral array equalization configuration.

Fig. 3
Fig. 3

Arrayed waveguide planar device configuration.

Fig. 4
Fig. 4

Examples of the 1D quantile.

Fig. 5
Fig. 5

Modulus and argument of coupling function η.

Fig. 6
Fig. 6

Examples of the bidimensional (2D) quantile.

Fig. 7
Fig. 7

Tolerance to magnification effect: attenuation dynamic versus support size for the third quartile.

Fig. 8
Fig. 8

Tolerance to positioning.

Fig. 9
Fig. 9

Quantile filters in an array form: (a) third quartile, (b) median, (c) quantile combination, (d) median (another configuration).

Fig. 10
Fig. 10

Effect of the residual attenuation on the decoupling dynamic.

Fig. 11
Fig. 11

Linear and quadratic (parabolic) profiles and their combination with binary profiles.

Fig. 12
Fig. 12

Top and cross section of three filters: (a) parabolic deformable membrane, (b) electro-optic modulator, (c) electro-optic modulator with passived interpixels.

Equations (42)

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g(x, y)=exp-x2+y2w02.
η=-+h(x, y)g*(x, y)dxdy2-+|h(x, y)|2dxdy-+|g(x, y)|2dxdy,
h(x, y)=exp-x2+y2w02f(x, y).
f(x, y)=exp[iΔϕ(x, y)],
Δϕ(x, y)=ϕ0(x, y)D0(x, y)D,
η=2w02π22h(x, y)g(x, y)dxdy2=2w02π22exp-2x2+y2w02×exp[iΔϕ(x, y)]dxdy2=2w02π22exp-2x2+y2w02dxdy+[exp(iϕ0)-1]Dexp-2x2+y2w02dxdy2.
I=2exp-2x2+y2w02dxdy.
Dexp-2x2+y2w02dxdy=kI,0<k<1.
η=2w02π2|I{1+k[exp(iϕ0)1]}|2=2w02π2I2[k2+(1+k)2>0+2k(1+k)>0cosϕ0].
η=2w02π2I2[k2+(1-k)2-2k(1-k)]=4w02π2I2k-122.
Dexp-2x2+y2w02dxdy=122exp-2x2+y2w02dxdy=12w0π21/22
Dx2w0πexp-2x2w02dxDy2w0πexp-2y2w02dy=12,
Dxexp(-x2/2)2πdxDyexp(-y2/2)2πdy=12.
-0exp(-x2/2)2πdx=12.
-qqexp(-x2/2)2πdx=20qexp(-x2/2)2πdx=2-qexp(-x2/2)2πdx-12=12.
-qexp(-x2/2)2πdx=34,
-a-bexp(-x2/2)2πdx+baexp(-x2/2)2πdx
=2baexp(-x2/2)2πdx
=2-aexp(-x2/2)2πdx--bexp(-x2/2)2πdx=12,
-aexp(-x2/2)2πdx--bexp(-x2/2)2πdx=14.
f(x, y)=A(x, y)exp[Δϕ(x, y)]
A(x, y)=A(x, y)D0(x, y)D.
η(k, A, ϕ0)=1-2k+k2(1+A2)+2kA(1-k)cos ϕ01+k(A2-1).
ηk=12, A, ϕ0=121+2A1+A2cos ϕ0.
ηk=11+A, A, ϕ0=2A(1+A)2(1+cos ϕ0).
Δϕ(x)=ϕ(1-|x/l|),for|x|l
η=8πw02exp(iϕ)0aexp-2x2w02-ixϕadx+a+exp-2x2w02dx2.
u+exp-x24β-γxdx=πβ exp(βγ2)1-erfγβ+u2β,
η=erf2aw0+exp-ϕ2w028a2+iϕ×erf2aw0+iϕw02a2-erfiϕw02a22.
-exp-2x2+y2w02exp[iΔϕ(x, y)]dxdy2
=02π0exp-2r2w02exp[iΔϕ(r)]rdrdθ2
=4π2exp(iϕ)0Rexp-r22w02+iϕR2rdrdθ+Rexp-2r2w02rdrdθ2=4π2-β2exp(iϕ)exp-2r2β0R+w024exp-2r2w02R2=|πβ|2-exp(iϕ)exp-2R2β-1+w022βexp-2R2w022=|πβ|2exp(iϕ)+iϕw022R2exp-2R2w022.
η=1+(ϕ/X)2 exp(-2X)+2(ϕ/X)exp(-X)sin ϕ1+(ϕ/X)2.
exp-2R2w02=2ϕ0R2w02.
1q(z)=1w02(z)+iπλR(z).
1d01,
q(z)=1·q(z)+d0·q(z)+1,
w0=w01+λdπw022,
R0=d1+πw02λd2.
w0w0=1.0002,
R0=2174.4µm,
1λR0=4.6×10-4μm-2.

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