Abstract

We present a theoretical analysis of the temporal behavior of double-diffraction setups. It applies, in particular, to Talbot and Montgomery interferometers, whose operation is based on the self-imaging effect. The use of both types of interferometer as temporal filters for optical and terahertz applications was recently suggested. We show that double-diffraction setups can be modeled as communications channels with dispersive behavior caused by diffraction. We develop mathematical expressions for the phase delay, the group velocity, and the group-velocity dispersion for both quasi-monochromatic and polychromatic case. Based on these results, the temporal impulse response of a double-diffraction setup is derived. Finally, a general description of its practical implementation are presented.

© 2004 Optical Society of America

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References

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  1. H. F. Talbot, “Facts relating to optical science,” Phil. Mag. 9, 401–407 (1836).
  2. E. Lau, “Beugungserscheinungen an Doppelrastern,” Ann. Phys. (Leipzig) 2, 417–423 (1948).
  3. A. W. Lohmann, D. E. Silva, “An interferometer based on the Talbot effect,” Opt. Commun. 2, 413–415 (1971).
    [CrossRef]
  4. J. Jahns, A. W. Lohmann, “The Lau effect (a diffraction experiment with incoherent illumination),” Opt. Commun. 28, 263–267 (1979).
    [CrossRef]
  5. J. Jahns, E. El Joudi, D. Hagedorn, S. Kinne, “Talbot interferometer as a time filter,” Optik (Stuttgart) 112, 295–298 (2001).
    [CrossRef]
  6. J. Jahns, H. Knuppertz, A. W. Lohmann, “Montgomery self-imaging effect using computer-generated diffractive optical elements,” Opt. Commun. 225, 13–17 (2003).
    [CrossRef]
  7. W. D. Montgomery, “Self-imaging objects of infinite aperture,” J. Opt. Soc. Am. 57, 772–778 (1967).
    [CrossRef]
  8. A. W. Lohmann, D. Mendlovic, “Temporal filtering with time lenses,” Appl. Opt. 31, 6212–6219 (1992).
    [CrossRef] [PubMed]
  9. B. E. A. Saleh, M. C. Teich, Fundamentals of Photonics (Wiley, New York, 1991).
    [CrossRef]
  10. F. Gires, P. Tournois, “Interféromètre utilisable pour la compression d’impulsions lumineuses modulées en fréquences,” C. R. Acad. Sci. 258, 6112–6115 (1964).
  11. E. B. Treacy, “Optical pulse compression with diffraction gratings,” IEEE J. Quantum Electron. QE-5, 454–458 (1969).
    [CrossRef]
  12. C. Froehly, B. Colombeau, M. Vampouille, “Shaping and analysis of picosecond light pulses,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1983), Vol. 20, pp. 63–153.
    [CrossRef]
  13. A. M. Weiner, J. P. Heritage, E. M. Kirschner, “Encoding and decoding of femtosecond pulses,” Opt. Lett. 13, 300–302 (1988).
    [CrossRef] [PubMed]
  14. P.-C. Sun, K. Oba, Y. T. Mazurenko, Y. Fainman, “Space-time processing with photorefractive volume holography,” Proc. IEEE 87, 2086–2097 (1999).
    [CrossRef]
  15. O. Bryngdahl, “Image formation using self-imaging techniques,” J. Opt. Soc. Am. 63, 416–419 (1973).
    [CrossRef]
  16. R. Ulrich, G. Ankele, “Self-imaging in homogeneous planar optical waveguides,” Appl. Phys. Lett. 27, 337–339 (1975).
    [CrossRef]
  17. A. W. Lohmann, J. Ojeda-Castañeda, E. E. Sicre, “Multiple interaction bandstop filters based on the Talbot effect,” Opt. Commun. 49, 388–392 (1984).
    [CrossRef]

2003 (1)

J. Jahns, H. Knuppertz, A. W. Lohmann, “Montgomery self-imaging effect using computer-generated diffractive optical elements,” Opt. Commun. 225, 13–17 (2003).
[CrossRef]

2001 (1)

J. Jahns, E. El Joudi, D. Hagedorn, S. Kinne, “Talbot interferometer as a time filter,” Optik (Stuttgart) 112, 295–298 (2001).
[CrossRef]

1999 (1)

P.-C. Sun, K. Oba, Y. T. Mazurenko, Y. Fainman, “Space-time processing with photorefractive volume holography,” Proc. IEEE 87, 2086–2097 (1999).
[CrossRef]

1992 (1)

1988 (1)

1984 (1)

A. W. Lohmann, J. Ojeda-Castañeda, E. E. Sicre, “Multiple interaction bandstop filters based on the Talbot effect,” Opt. Commun. 49, 388–392 (1984).
[CrossRef]

1979 (1)

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

1975 (1)

R. Ulrich, G. Ankele, “Self-imaging in homogeneous planar optical waveguides,” Appl. Phys. Lett. 27, 337–339 (1975).
[CrossRef]

1973 (1)

1971 (1)

A. W. Lohmann, D. E. Silva, “An interferometer based on the Talbot effect,” Opt. Commun. 2, 413–415 (1971).
[CrossRef]

1969 (1)

E. B. Treacy, “Optical pulse compression with diffraction gratings,” IEEE J. Quantum Electron. QE-5, 454–458 (1969).
[CrossRef]

1967 (1)

1964 (1)

F. Gires, P. Tournois, “Interféromètre utilisable pour la compression d’impulsions lumineuses modulées en fréquences,” C. R. Acad. Sci. 258, 6112–6115 (1964).

1948 (1)

E. Lau, “Beugungserscheinungen an Doppelrastern,” Ann. Phys. (Leipzig) 2, 417–423 (1948).

1836 (1)

H. F. Talbot, “Facts relating to optical science,” Phil. Mag. 9, 401–407 (1836).

Ankele, G.

R. Ulrich, G. Ankele, “Self-imaging in homogeneous planar optical waveguides,” Appl. Phys. Lett. 27, 337–339 (1975).
[CrossRef]

Bryngdahl, O.

Colombeau, B.

C. Froehly, B. Colombeau, M. Vampouille, “Shaping and analysis of picosecond light pulses,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1983), Vol. 20, pp. 63–153.
[CrossRef]

El Joudi, E.

J. Jahns, E. El Joudi, D. Hagedorn, S. Kinne, “Talbot interferometer as a time filter,” Optik (Stuttgart) 112, 295–298 (2001).
[CrossRef]

Fainman, Y.

P.-C. Sun, K. Oba, Y. T. Mazurenko, Y. Fainman, “Space-time processing with photorefractive volume holography,” Proc. IEEE 87, 2086–2097 (1999).
[CrossRef]

Froehly, C.

C. Froehly, B. Colombeau, M. Vampouille, “Shaping and analysis of picosecond light pulses,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1983), Vol. 20, pp. 63–153.
[CrossRef]

Gires, F.

F. Gires, P. Tournois, “Interféromètre utilisable pour la compression d’impulsions lumineuses modulées en fréquences,” C. R. Acad. Sci. 258, 6112–6115 (1964).

Hagedorn, D.

J. Jahns, E. El Joudi, D. Hagedorn, S. Kinne, “Talbot interferometer as a time filter,” Optik (Stuttgart) 112, 295–298 (2001).
[CrossRef]

Heritage, J. P.

Jahns, J.

J. Jahns, H. Knuppertz, A. W. Lohmann, “Montgomery self-imaging effect using computer-generated diffractive optical elements,” Opt. Commun. 225, 13–17 (2003).
[CrossRef]

J. Jahns, E. El Joudi, D. Hagedorn, S. Kinne, “Talbot interferometer as a time filter,” Optik (Stuttgart) 112, 295–298 (2001).
[CrossRef]

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

Kinne, S.

J. Jahns, E. El Joudi, D. Hagedorn, S. Kinne, “Talbot interferometer as a time filter,” Optik (Stuttgart) 112, 295–298 (2001).
[CrossRef]

Kirschner, E. M.

Knuppertz, H.

J. Jahns, H. Knuppertz, A. W. Lohmann, “Montgomery self-imaging effect using computer-generated diffractive optical elements,” Opt. Commun. 225, 13–17 (2003).
[CrossRef]

Lau, E.

E. Lau, “Beugungserscheinungen an Doppelrastern,” Ann. Phys. (Leipzig) 2, 417–423 (1948).

Lohmann, A. W.

J. Jahns, H. Knuppertz, A. W. Lohmann, “Montgomery self-imaging effect using computer-generated diffractive optical elements,” Opt. Commun. 225, 13–17 (2003).
[CrossRef]

A. W. Lohmann, D. Mendlovic, “Temporal filtering with time lenses,” Appl. Opt. 31, 6212–6219 (1992).
[CrossRef] [PubMed]

A. W. Lohmann, J. Ojeda-Castañeda, E. E. Sicre, “Multiple interaction bandstop filters based on the Talbot effect,” Opt. Commun. 49, 388–392 (1984).
[CrossRef]

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

A. W. Lohmann, D. E. Silva, “An interferometer based on the Talbot effect,” Opt. Commun. 2, 413–415 (1971).
[CrossRef]

Mazurenko, Y. T.

P.-C. Sun, K. Oba, Y. T. Mazurenko, Y. Fainman, “Space-time processing with photorefractive volume holography,” Proc. IEEE 87, 2086–2097 (1999).
[CrossRef]

Mendlovic, D.

Montgomery, W. D.

Oba, K.

P.-C. Sun, K. Oba, Y. T. Mazurenko, Y. Fainman, “Space-time processing with photorefractive volume holography,” Proc. IEEE 87, 2086–2097 (1999).
[CrossRef]

Ojeda-Castañeda, J.

A. W. Lohmann, J. Ojeda-Castañeda, E. E. Sicre, “Multiple interaction bandstop filters based on the Talbot effect,” Opt. Commun. 49, 388–392 (1984).
[CrossRef]

Saleh, B. E. A.

B. E. A. Saleh, M. C. Teich, Fundamentals of Photonics (Wiley, New York, 1991).
[CrossRef]

Sicre, E. E.

A. W. Lohmann, J. Ojeda-Castañeda, E. E. Sicre, “Multiple interaction bandstop filters based on the Talbot effect,” Opt. Commun. 49, 388–392 (1984).
[CrossRef]

Silva, D. E.

A. W. Lohmann, D. E. Silva, “An interferometer based on the Talbot effect,” Opt. Commun. 2, 413–415 (1971).
[CrossRef]

Sun, P.-C.

P.-C. Sun, K. Oba, Y. T. Mazurenko, Y. Fainman, “Space-time processing with photorefractive volume holography,” Proc. IEEE 87, 2086–2097 (1999).
[CrossRef]

Talbot, H. F.

H. F. Talbot, “Facts relating to optical science,” Phil. Mag. 9, 401–407 (1836).

Teich, M. C.

B. E. A. Saleh, M. C. Teich, Fundamentals of Photonics (Wiley, New York, 1991).
[CrossRef]

Tournois, P.

F. Gires, P. Tournois, “Interféromètre utilisable pour la compression d’impulsions lumineuses modulées en fréquences,” C. R. Acad. Sci. 258, 6112–6115 (1964).

Treacy, E. B.

E. B. Treacy, “Optical pulse compression with diffraction gratings,” IEEE J. Quantum Electron. QE-5, 454–458 (1969).
[CrossRef]

Ulrich, R.

R. Ulrich, G. Ankele, “Self-imaging in homogeneous planar optical waveguides,” Appl. Phys. Lett. 27, 337–339 (1975).
[CrossRef]

Vampouille, M.

C. Froehly, B. Colombeau, M. Vampouille, “Shaping and analysis of picosecond light pulses,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1983), Vol. 20, pp. 63–153.
[CrossRef]

Weiner, A. M.

Ann. Phys. (Leipzig) (1)

E. Lau, “Beugungserscheinungen an Doppelrastern,” Ann. Phys. (Leipzig) 2, 417–423 (1948).

Appl. Opt. (1)

Appl. Phys. Lett. (1)

R. Ulrich, G. Ankele, “Self-imaging in homogeneous planar optical waveguides,” Appl. Phys. Lett. 27, 337–339 (1975).
[CrossRef]

C. R. Acad. Sci. (1)

F. Gires, P. Tournois, “Interféromètre utilisable pour la compression d’impulsions lumineuses modulées en fréquences,” C. R. Acad. Sci. 258, 6112–6115 (1964).

IEEE J. Quantum Electron. (1)

E. B. Treacy, “Optical pulse compression with diffraction gratings,” IEEE J. Quantum Electron. QE-5, 454–458 (1969).
[CrossRef]

J. Opt. Soc. Am. (2)

Opt. Commun. (4)

A. W. Lohmann, J. Ojeda-Castañeda, E. E. Sicre, “Multiple interaction bandstop filters based on the Talbot effect,” Opt. Commun. 49, 388–392 (1984).
[CrossRef]

J. Jahns, H. Knuppertz, A. W. Lohmann, “Montgomery self-imaging effect using computer-generated diffractive optical elements,” Opt. Commun. 225, 13–17 (2003).
[CrossRef]

A. W. Lohmann, D. E. Silva, “An interferometer based on the Talbot effect,” Opt. Commun. 2, 413–415 (1971).
[CrossRef]

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

Opt. Lett. (1)

Optik (Stuttgart) (1)

J. Jahns, E. El Joudi, D. Hagedorn, S. Kinne, “Talbot interferometer as a time filter,” Optik (Stuttgart) 112, 295–298 (2001).
[CrossRef]

Phil. Mag. (1)

H. F. Talbot, “Facts relating to optical science,” Phil. Mag. 9, 401–407 (1836).

Proc. IEEE (1)

P.-C. Sun, K. Oba, Y. T. Mazurenko, Y. Fainman, “Space-time processing with photorefractive volume holography,” Proc. IEEE 87, 2086–2097 (1999).
[CrossRef]

Other (2)

B. E. A. Saleh, M. C. Teich, Fundamentals of Photonics (Wiley, New York, 1991).
[CrossRef]

C. Froehly, B. Colombeau, M. Vampouille, “Shaping and analysis of picosecond light pulses,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1983), Vol. 20, pp. 63–153.
[CrossRef]

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Figures (6)

Fig. 1
Fig. 1

Optical double-diffraction setup with masks M1 and M2. The optical output is observed in the zeroth order behind M2.

Fig. 2
Fig. 2

Paths of the different orders in the double-diffraction experiment. The zeroth order behind M2 is composed of the combined orders (0/0), (1,-1), etc., where the first digit is the index of the diffraction order generated by M1. A m and B m are the amplitudes of the respective diffraction orders.

Fig. 3
Fig. 3

Schematic of a tapped-delay-line filter. Here constant delays of duration τ are shown. The weights in the branches are denoted a m .

Fig. 4
Fig. 4

Version 1 of a double-diffraction setup based on a 4f imaging system: a, transmissive system with two refractive lenses; b, refractive-reflective version obtained by placing a mirror in the Fourier plane.

Fig. 5
Fig. 5

Version 2 of a double-diffraction setup: purely reflective 4-f systems with a, two parabolic mirrors and b, one parabolic mirror and a planar mirror in the Fourier plane.

Fig. 6
Fig. 6

Version 3 of a double-diffraction setup based on 2f-2f imaging systems: a, a system that uses a refractive lens in transmission and b, a purely reflective system.

Equations (52)

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M1x= Aνexp2πiνxdν, Aν=M˜1ν,
M2x= Bνexp2πiνxdν, Bν=M˜2ν.
M1x=m Am exp2πimν0x,M˜1ν=-N/2N/2-1 Amδν-mν0,
M2x=n Bn exp2πinν0x and M˜2ν=-N/2N/2-1 Bnδν-nν0.
M˜1ν=A0+m>0 Amδν-mν0+m<0 Amδν-|m|ν0,
M˜2ν=B0+n>0 Bnδν-nν0+n<0 Bnδν-|n|ν0.
Sint=exp-2πiνtt.
ux, z=+0, t=SintM1x= M˜1νexp2πiνx-νttdν.
ux, z>0, t= M˜1νexp2πiνx+βz-νttdν.
Δxz-1c2t2ux, z, t=0.
2πi2  M˜1νν2+β2-νtc2exp2πiνx+βz-νttdν=0.
β2=νt/c2-ν2.
ux, z=z0-0, t= M˜1νexp2πiνx+βz0-νttdν,
ux, z=z0+0, t = M˜1ν1M˜2ν2exp2πiν1+ν2x+βz0-νttdν1dν2.
ux, z0+2f, t= ux, z0+0, texp-2πixx/λfdx= M˜1ν1M˜2ν2exp2πixν1+ν2-xνt/cf+βz0-νttdν1dν2dx.
ux, z0+2f, t= M˜1νM˜2ν-ν×exp2πiβz0-νttdν.
uext=u0, z0+2f, t= M˜1νM˜2-νexp2πiβz0-νttdν.
β=vtc2-ν21/2=νtc-νtc-νtc2-ν21/2,
uext= M˜1νM˜2-νexp2πiνtz0c-t-z0νtc-νtc2-ν21/2dν= M˜1νM˜2-νexp2πiνtz0c-t-iφDνdν.
φDν=2πz0νtc-νtc2-ν21/2
β=νtc-φD2πz0.
1vg=βνt=1c-12πz0φDνt.
D=2βνt2.
νtc2-ν21/2=νtc1-νcνt21/2νtc1-12νcνt2,
φDν-π cz0νtν2.
1νt1ν¯t2-νtν¯t;
φDν-2π cz0ν¯t1-νt2ν¯tν2.
vg12cνt22νt2+cν2,
D1=-4νt3cν2.
uext=|Am|2 expi2πνtz0c-t-φD,mν,
φD,mν=-π cz0νtmν02.
φD,mν-2π cz0ν¯t1-νt2ν¯tmν02.
vg22cνt22νt2+mcν02,
D2=-4νt3cmν02.
vg2c1-m νpν02, νp=c/p.
φD,mν=-π cz0νt mν02-2π cz0ν¯t1-νt2ν¯tmν02.
vg3=2cνt22νt2+mcν02c1-mνpνt2,
D3=-4νt3cmν02.
Ut= S˜νtuext; νtdνt.
uext=|Am|2  S˜νt-ν¯texp2πiβmνtz0-νt-ν¯ttdνt,
βm=νt-ν¯tc-cν¯t1-νtν¯tρm2,ρm2=m2ν02case 2mν02case 3.
=ν¯tt-ν¯tz0c-cρm2z0ν¯t-νtt-z0c-cρm2z02ν¯t2.
 S˜νt-ν¯texp-2πiνtt-z0c-cρm2z02ν¯t2dνt=exp-2πiν¯tt-z0c-τmSt-z0c-τm,
τm=m2τ1case 2mτ1case 3,
τ1=cν02z02ν¯t2.
Ut= |Am|2 exp-2πiν¯tτmSt-τm.
 S˜νtdνt= exp-2πiνtt-τmdνt=δt-τm.
Uδt= |Am|2 exp2πiρm2ν¯tc z0δt-τm.
τ1=z0zTλ¯c.
Uδt= |Am|2δt-τm,
τm=m2τ1 case 2,
τm=mτ1 case 3.

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