Abstract

We propose an analytical model of the self-imaging of Gaussian beams and discuss the impact of the finite aperture and the beam truncation on the quality of the self-image reconstruction. Extension to polychromatic operation is then presented in the context of wavelength division multiplexing systems. From the first point we derive conditions for good self-imaging, compatible with the requirements of the telecommunication environment. A basic optical setup is then proposed to implement both wavelength demultiplexing and routing in a lensless configuration.

© 2002 Optical Society of America

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References

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  1. T. Ito, K. Fukuchi, T. Kasamatsu, “Enabling technologies for 10 Tbits/s transmission capacity and beyond,” in Proceedings of the 27th European Conference on Optical Communication (Nexus Media, Swanley, UK, 2001), pp. 598–601.
  2. J. P. Laude, Wavelength Division Multiplexing (Prentice-Hall, Englewood Cliffs, N.J., 1993).
  3. K. Patorski, “The self-imaging phenomenon and its applications,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1989), Vol. 27, pp. 1–108.
  4. J. T. Winthrop, C. R. Worthington, “Theory of Fresnel images. I. Plane periodic objects in monochromatic light,” J. Opt. Soc. Am. 55, 373–381 (1965).
    [CrossRef]
  5. J. Jahns, A. W. Lohmann, “The Lau effect (a diffraction experiment with incoherent illumination),” Opt. Commun. 28, 263–267 (1979).
    [CrossRef]
  6. B. Packross, R. Eschbach, O. Bryngdhal, “Achromatization of the self-imaging (Talbot) effect,” Opt. Commun. 50, 205–209 (1984).
    [CrossRef]
  7. D. Marcuse, “Loss analysis of single mode fiber splices,” Bell Syst. Tech. J. 56, 703–718 (1977).
    [CrossRef]
  8. 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]
  9. J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, Singapore, 1996), Chap. 4.
  10. I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series and Products (Academic, London, 1980), pp. 307 and 391.
  11. A. P. Smirnov, “Fresnel images of periodic transparencies of finite dimensions,” Opt. Spectrosc. 44, 208–212 (1978).
  12. W. D. Montgomery, “Self-imaging objects of infinite aperture,” J. Opt. Soc. Am. 57, 772–778 (1967).
    [CrossRef]
  13. H. Haman, J.-L. de Bougrenet de la Tocnaye, “Efficient Fresnel-transform algorithm based on fractional Fresnel diffraction,” J. Opt. Soc. Am. A 12, 1920–1931 (1995).
    [CrossRef]
  14. J. P. Laude, “A new athermal very dense wavelength division multiplexing,” in Proceedings of the 26th European Conference on Optical Communication (VDE Verlag, Berlin, 2000), pp. 181–182.
  15. P. Chanclou, M. Thual, J. Lostec, D. Pavy, M. Gadonna, A. Poudoulec, “Collective micro-optics on fiber ribbon for optical interconnecting devices,” J. Lightwave Technol. 17, 924–928 (1999).
    [CrossRef]

2001

1999

1995

1984

B. Packross, R. Eschbach, O. Bryngdhal, “Achromatization of the self-imaging (Talbot) effect,” Opt. Commun. 50, 205–209 (1984).
[CrossRef]

1979

J. Jahns, A. W. Lohmann, “The Lau effect (a diffraction experiment with incoherent illumination),” Opt. Commun. 28, 263–267 (1979).
[CrossRef]

1978

A. P. Smirnov, “Fresnel images of periodic transparencies of finite dimensions,” Opt. Spectrosc. 44, 208–212 (1978).

1977

D. Marcuse, “Loss analysis of single mode fiber splices,” Bell Syst. Tech. J. 56, 703–718 (1977).
[CrossRef]

1967

1965

Bryngdhal, O.

B. Packross, R. Eschbach, O. Bryngdhal, “Achromatization of the self-imaging (Talbot) effect,” Opt. Commun. 50, 205–209 (1984).
[CrossRef]

Chanclou, P.

de Bougrenet de la Tocnaye, J.-L.

Eschbach, R.

B. Packross, R. Eschbach, O. Bryngdhal, “Achromatization of the self-imaging (Talbot) effect,” Opt. Commun. 50, 205–209 (1984).
[CrossRef]

Fukuchi, K.

T. Ito, K. Fukuchi, T. Kasamatsu, “Enabling technologies for 10 Tbits/s transmission capacity and beyond,” in Proceedings of the 27th European Conference on Optical Communication (Nexus Media, Swanley, UK, 2001), pp. 598–601.

Gadonna, M.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, Singapore, 1996), Chap. 4.

Gradshteyn, I. S.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series and Products (Academic, London, 1980), pp. 307 and 391.

Haman, H.

Ito, T.

T. Ito, K. Fukuchi, T. Kasamatsu, “Enabling technologies for 10 Tbits/s transmission capacity and beyond,” in Proceedings of the 27th European Conference on Optical Communication (Nexus Media, Swanley, UK, 2001), pp. 598–601.

Jahns, J.

J. Jahns, A. W. Lohmann, “The Lau effect (a diffraction experiment with incoherent illumination),” Opt. Commun. 28, 263–267 (1979).
[CrossRef]

Kasamatsu, T.

T. Ito, K. Fukuchi, T. Kasamatsu, “Enabling technologies for 10 Tbits/s transmission capacity and beyond,” in Proceedings of the 27th European Conference on Optical Communication (Nexus Media, Swanley, UK, 2001), pp. 598–601.

Laude, J. P.

J. P. Laude, “A new athermal very dense wavelength division multiplexing,” in Proceedings of the 26th European Conference on Optical Communication (VDE Verlag, Berlin, 2000), pp. 181–182.

J. P. Laude, Wavelength Division Multiplexing (Prentice-Hall, Englewood Cliffs, N.J., 1993).

Lohmann, A. W.

J. Jahns, A. W. Lohmann, “The Lau effect (a diffraction experiment with incoherent illumination),” Opt. Commun. 28, 263–267 (1979).
[CrossRef]

Lostec, J.

Marcuse, D.

D. Marcuse, “Loss analysis of single mode fiber splices,” Bell Syst. Tech. J. 56, 703–718 (1977).
[CrossRef]

Montgomery, W. D.

Nourrit, V.

Packross, B.

B. Packross, R. Eschbach, O. Bryngdhal, “Achromatization of the self-imaging (Talbot) effect,” Opt. Commun. 50, 205–209 (1984).
[CrossRef]

Patorski, K.

K. Patorski, “The self-imaging phenomenon and its applications,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1989), Vol. 27, pp. 1–108.

Pavy, D.

Poudoulec, A.

Ryzhik, I. M.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series and Products (Academic, London, 1980), pp. 307 and 391.

Smirnov, A. P.

A. P. Smirnov, “Fresnel images of periodic transparencies of finite dimensions,” Opt. Spectrosc. 44, 208–212 (1978).

Thual, M.

Winthrop, J. T.

Worthington, C. R.

Bell Syst. Tech. J.

D. Marcuse, “Loss analysis of single mode fiber splices,” Bell Syst. Tech. J. 56, 703–718 (1977).
[CrossRef]

J. Lightwave Technol.

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Opt. Commun.

J. Jahns, A. W. Lohmann, “The Lau effect (a diffraction experiment with incoherent illumination),” Opt. Commun. 28, 263–267 (1979).
[CrossRef]

B. Packross, R. Eschbach, O. Bryngdhal, “Achromatization of the self-imaging (Talbot) effect,” Opt. Commun. 50, 205–209 (1984).
[CrossRef]

Opt. Spectrosc.

A. P. Smirnov, “Fresnel images of periodic transparencies of finite dimensions,” Opt. Spectrosc. 44, 208–212 (1978).

Other

J. P. Laude, “A new athermal very dense wavelength division multiplexing,” in Proceedings of the 26th European Conference on Optical Communication (VDE Verlag, Berlin, 2000), pp. 181–182.

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, Singapore, 1996), Chap. 4.

I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series and Products (Academic, London, 1980), pp. 307 and 391.

T. Ito, K. Fukuchi, T. Kasamatsu, “Enabling technologies for 10 Tbits/s transmission capacity and beyond,” in Proceedings of the 27th European Conference on Optical Communication (Nexus Media, Swanley, UK, 2001), pp. 598–601.

J. P. Laude, Wavelength Division Multiplexing (Prentice-Hall, Englewood Cliffs, N.J., 1993).

K. Patorski, “The self-imaging phenomenon and its applications,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1989), Vol. 27, pp. 1–108.

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Figures (9)

Fig. 1
Fig. 1

Different cases and their analytical description.

Fig. 2
Fig. 2

Modulus of NTG (dotted curve) multiplied by a factor of 20 and modulus of the sum (solid curve) in Eq. (15). Curves are plotted for N=1, z=zT, ω0=10 μm, τ=4, and λ=1.55 μm. Units for the vertical axis are dimensionless.

Fig. 3
Fig. 3

Same as Fig. 2 but for τ=8.

Fig. 4
Fig. 4

Modulus of Eq. (24) for z=0 (solid curve) and z=zT (dotted curve), and modulus of Eq. (15) for z=zT (dashed curve). The curves are plotted for ω0=30 μm, τ=4.2, L/ω0=1.4, N=20, and λ=1.55 μm. Units for the vertical axis are dimensionless.

Fig. 5
Fig. 5

Values of ηi (boxes) for 1nN with N=20, ω0=30 μm, τ=4.2, L/ω0=1.4, z=zT, and λ=1.55 μm.

Fig. 6
Fig. 6

Modulus (solid curve) and argument (dashed curve) of the last self-imaged pattern with respect to the SFOR for z=zT, and modulus of Eq. (15) for z=0 (dotted curve). Curves are plotted for =-0.2 dB, ω0=30 μm, τ=4.2, L/ω0=1.4, N=20, and λ=1.55 μm. Units for the vertical axis are radians (dashed curve) or dimensionless (dotted and solid curves).

Fig. 7
Fig. 7

Modulus of a propagated slit array (dotted curve) with respect to the original (solid curve). Slit width is 50 μm, d=150 μm, N=14, z=zT, and λ=1.55 μm. Units for the vertical axis are dimensionless.

Fig. 8
Fig. 8

Wavelength demultiplexing optical system based on the self-imaging phenomenon (three dimensions).

Fig. 9
Fig. 9

Wavelength demultiplexing optical system based on the self-imaging phenomenon (two dimensions).

Tables (6)

Tables Icon

Table 1 Influence of the Distance of Propagation and the Number of Gaussian Beams on the SFOR in the Case (Subsection 5.A) Where  = -0.1 dB, d = 4ω0 = 40 μm, and λ = 1.55 μm

Tables Icon

Table 2 Influence of the Compression Ratio, the Distance of Propagation, and the Number of Gaussian Beams on the SFOR in the Case (Subsection 5.A) Where z = zT,  = -0.2dB, λ = 1.55 μm, and ω0 = 10 μm

Tables Icon

Table 3 Maximum Absolute Value of the Difference between Moduli of h(x, z) from Eqs. (15 ) and (19 ) Due to L, Where ω0 = 10 μm, λ = 1.55 μm, and z = zT

Tables Icon

Table 4 Maximum Absolute Value of the Difference between Moduli of h(x, z) from Eqs. (15 ) and (24 ), Where ω0 = 10 μm, z = zT, λ = 1.55 μm, and L = 1.4ω0

Tables Icon

Table 5 Maximum Absolute Value of the Difference between Moduli of h(x, z) from Eqs. (15 ) and (24 ), Where ω0 = 10 μm, z = zT, λ = 1.55 μm, and L = 2ω0

Tables Icon

Table 6 Design Frame of an Axial Demultiplexing for 2N + 1 Fibers SFOR, Where L = 42.5 μm, λ = 1.55 μm, Δλ = 0.8 nm,  = -0.2dB

Equations (55)

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gd(x)=exp-(x-d)2ω02,g-d(x)=exp-(x+d)2ω02.
 
h(x, z)=[gd(x)+g-d(x)]*Nf(x)
=-exp-(t-d)2ω02exp[-iK(x-t)2]dt+-exp-(t+d)2ω02×exp[-iK(x-t)2]dt,
β=ω02411+iKω02=ω0241-iKω021+K2ω04,
γd=-2dω02+iKx,α=exp(-ikz)-iλz,
gd(x)*Nf(x)=α-exp-(t-d)2ω02-iK(x-t)2dt
=2απβexp(βγd2)exp-d2ω02-iKx2
=2απβexp-K(x-d)2Kω02+i1+K2ω04.
NTG=2απβexp(-iKx2+βγ2),
h(x, z)=NTGexp-π2ω02d2λ2z2-iπd2λz11+K2ω04×expi2πdxλz+2π2ω02dxλ2z211+K2ω04+exp-i2πdxλz-2π2ω02dxλ2z211+K2ω04.
h(x, z)=NTGexp-π2ω02d2λ2z2-iπd2λz×expi2πdxλz+2π2ω02dxλ2z2+exp-i2πdxλz-2π2ω02dxλ2z2.
h(x, z)=NTG2 cos2πdxλzexp-iπd2λz.
h(x, z)=n=-NNδ(x-nd)*g(x)*Nf(x)
=n=-NNexp-x-ndω02*Nf(x)
=n=-NN2απβexp-K(x-nd)21+K2ω04 (Kω02+i)
=NTGn=-NNexp-K(-2ndx+n2d2)1+K2ω04 ×(Kω02+i)
=NTGn=-NNexp-iπn2d2λz+i2πndxλz-π2ω02n2d2λ2z2+2π2ω02ndxλ2z21+π2ω04λ2z2-1.
hinf(x, z)=n=-δ(x-nd)*g(x)*Nf(x).
hinf(x, z)=ω0πdn=-expiπλzn2d2+2iπnxd-π2ω02n2d2exp(-ikz)
n=-NNδ(x-nd)*g(x)2L*Nf(x).
h(x, z)=n=-NN12NTGnd[erf(Xn+)+erf(Xn-)],
NTGnd=2απβexp(βγn2)exp-(nd)2ω02-iKx2,
β=ω02411+iKω02=ω0241-iKω021+K2ω04,
γn=-2ndω02+iKx,
Xn±=±γnβ+L2β,
h(x, z)=n=-NNδ(x-nd)*[g(x)2L(x)]*Nf(x)
=n=-NNexp-(x-nd)2ω02×2L(x-nd)*Nf(x)
=αn=-NNnd-Lnd+Lexp-(t-nd)2ω02×exp[-iK(x-t)2]dt
=n=-NNαπβexpβγn2-n2d2ω02-iKx2×erfγnβ+nd+L2β+erf-γnβ+L-nd2β
=n=-NN12NTGnd[erf(Xn+)+erf(Xn-)]
=αn=-NNδ(x-nd)*exp-x2ω022L(x)*Nf(x),
NTGnd=2απβexp(βγn2)exp-(nd)2ω02-iKx2,
β=ω02411+iKω02=ω0241-iKω021+K2ω04,
γn=-2ndω02+iKx,
Xn±=±γnβ+±nd+L2β.
ηi=gig0*2|gi|2|g0|2,
gi(x)=hx-i d2, z for-d/2xd/2,
g0(x)=h(x, 0)for-d/2xd/2.
zT(p/q)=pq2d2λ,
Δ(zT(m/2))=md2Δλλ2
=zT(m/2)Δλλ,
zT(m/2)(λ3)-zT(m/2)(λ1)=2Δ(zT(m/2))ξ,
Δ(zT(m/2))ξ/2,
mτ2ω02Δλλ2ξ/2.
λ=1.55 μm,Δλ=0.8 nm,
H(u, z)=n=--[δ(x-nd)*g(x)*Nf(x)]×exp(-2iπux)dx,
H(u, z)=αn=--iλzω0πd δu-nd×exp(-π2ω02u2)exp(iπλzu2).
h(x, z)=αn=--H(u, z)exp(2iπux)dx,
h(x, z)=ω0πdn=-expiπλzn2d2exp2iπnxd×exp-π2ω02n2d2exp(-ikz).
erf(X±)=1-exp(-X±2)πX±,
|X±|=Lω0ndω02+K2ω02(Lx)2-1/2×(1+K2ω04)-1/4,
ϕ±=arg(X±)=arctanLω0ndω0sin θ+Kω0(Lx)cos θLω0ndω0cos θ-Kω0(Lx)sin θ
ϕ±=-π+arctanLω0ndω0sin θ+Kω0(Lx)cos θLω0ndω0cos θ-Kω0(Lx)sin θ
z2<π2ω042λ2-L±xLnd.

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