Abstract

We show that a temporal effect equivalent to the spatial Talbot effect (self-imaging) applies to the reflection of periodic pulse trains from linearly chirped fiber gratings (LCFG’s). For specific input repetition periods the reflected signal is an exact replica of the input signal. Input repetition period values that give rise to this effect depend on the dispersion coefficient of the grating. We propose to use this effect as an alternative for dispersion measurement in LCFG’s. Furthermore, by using the properties of the temporal Talbot effect, we can design linear passive devices (LCFG’s) for use as frequency multipliers, able to multiply the repetition rate of a given pulse train.

© 1999 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. A. Papoulis, “Pulse compression, fiber communications, and diffraction: a unified approach,” J. Opt. Soc. Am. A 11, 3–13 (1994).
    [CrossRef]
  2. B. H. Kolner, “Space–time duality and the theory of temporal imaging,” IEEE J. Quantum. Electron. 30, 1951–1963 (1994).
    [CrossRef]
  3. T. Jannson, J. Jannson, “Temporal self-imaging effect in single-mode fibers,” J. Opt. Soc. Am. 71, 1373–1376 (1981).
  4. B. E. A. Saleh, M. I. Irshid, “Collet–Wolf equivalence theorem and propagation of a pulse in a single-mode optical fiber,” Opt. Lett. 7, 342–343 (1982).
    [CrossRef] [PubMed]
  5. T. Jannson, “Real-time Fourier transformation in dispersive optical fibers,” Opt. Lett. 8, 232–234 (1983).
    [CrossRef] [PubMed]
  6. B. H. Kolner, M. Nazarathy, “Temporal imaging with a time lens,” Opt. Lett. 14, 630–632 (1989).
    [CrossRef] [PubMed]
  7. B. H. Kolner, M. Nazarathy, “Temporal imaging with a time lens: erratum,” Opt. Lett. 15, 655 (1990).
    [CrossRef]
  8. M. T. Kauffman, A. A. Godil, B. A. Auld, W. C. Banyai, D. M. Bloom, “Applications of time lens optical systems,” Electron. Lett. 29, 268–269 (1993).
    [CrossRef]
  9. C. V. Bennet, R. P. Scott, B. H. Kolner, “Temporal magnification and reversal of 100 Gb/s optical data with an up-conversion time microscope,” Appl. Phys. Lett. 65, 2513–2515 (1994).
    [CrossRef]
  10. A. W. Lohmann, D. Mendlovic, “Temporal filtering with time lenses,” Appl. Opt. 31, 6212–6219 (1992).
    [CrossRef] [PubMed]
  11. M. A. Muriel, J. Azaña, A. Carballar, “Real-time Fourier transformer based on fiber gratings,” Opt. Lett. 24, 1–3 (1999).
    [CrossRef]
  12. 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]
  13. M. Mansuripur, “The Talbot effect,” Opt. Photon. News 8(4), 42–47 (1997).
  14. J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, New York, 1996).
  15. F. Mitschke, U. Morgner, “The temporal Talbot effect,” Opt. Photon. News 9(6), 45–47 (1998).
    [CrossRef]
  16. F. Ouellette, “Dispersion cancellation using linearly chirped Bragg grating filters in optical waveguides,” Opt. Lett. 12, 847–849 (1987).
    [CrossRef] [PubMed]
  17. M. A. Muriel, A. Carballar, J. Azaña, “Field distributions inside fiber gratings,” IEEE J. Quantum Electron. 35, 548–558 (1999).
    [CrossRef]
  18. K. Ennser, M. N. Zervas, R. I. Laming, “Optimization of apodized linearly chirped fiber gratings for optical communications,” IEEE J. Quantum Electron. 34, 770–778 (1998).
    [CrossRef]
  19. A. Papoulis, The Fourier Integral and its Applications (McGraw-Hill, New York, 1962).
  20. S. Barcelos, M. N. Zervas, R. I. Laming, D. N. Payne, L. Reekie, J. A. Tucknott, R. Kashyap, P. F. McKee, F. Sladen, B. Wojciechowicz, “High accuracy dispersion measurements of chirped fibre gratings,” Electron. Lett. 31, 1280–1282 (1995).
    [CrossRef]

1999 (2)

M. A. Muriel, J. Azaña, A. Carballar, “Real-time Fourier transformer based on fiber gratings,” Opt. Lett. 24, 1–3 (1999).
[CrossRef]

M. A. Muriel, A. Carballar, J. Azaña, “Field distributions inside fiber gratings,” IEEE J. Quantum Electron. 35, 548–558 (1999).
[CrossRef]

1998 (2)

K. Ennser, M. N. Zervas, R. I. Laming, “Optimization of apodized linearly chirped fiber gratings for optical communications,” IEEE J. Quantum Electron. 34, 770–778 (1998).
[CrossRef]

F. Mitschke, U. Morgner, “The temporal Talbot effect,” Opt. Photon. News 9(6), 45–47 (1998).
[CrossRef]

1997 (1)

M. Mansuripur, “The Talbot effect,” Opt. Photon. News 8(4), 42–47 (1997).

1995 (1)

S. Barcelos, M. N. Zervas, R. I. Laming, D. N. Payne, L. Reekie, J. A. Tucknott, R. Kashyap, P. F. McKee, F. Sladen, B. Wojciechowicz, “High accuracy dispersion measurements of chirped fibre gratings,” Electron. Lett. 31, 1280–1282 (1995).
[CrossRef]

1994 (3)

A. Papoulis, “Pulse compression, fiber communications, and diffraction: a unified approach,” J. Opt. Soc. Am. A 11, 3–13 (1994).
[CrossRef]

B. H. Kolner, “Space–time duality and the theory of temporal imaging,” IEEE J. Quantum. Electron. 30, 1951–1963 (1994).
[CrossRef]

C. V. Bennet, R. P. Scott, B. H. Kolner, “Temporal magnification and reversal of 100 Gb/s optical data with an up-conversion time microscope,” Appl. Phys. Lett. 65, 2513–2515 (1994).
[CrossRef]

1993 (1)

M. T. Kauffman, A. A. Godil, B. A. Auld, W. C. Banyai, D. M. Bloom, “Applications of time lens optical systems,” Electron. Lett. 29, 268–269 (1993).
[CrossRef]

1992 (1)

1990 (1)

1989 (1)

1987 (1)

1983 (1)

1982 (1)

1981 (1)

1965 (1)

Auld, B. A.

M. T. Kauffman, A. A. Godil, B. A. Auld, W. C. Banyai, D. M. Bloom, “Applications of time lens optical systems,” Electron. Lett. 29, 268–269 (1993).
[CrossRef]

Azaña, J.

M. A. Muriel, A. Carballar, J. Azaña, “Field distributions inside fiber gratings,” IEEE J. Quantum Electron. 35, 548–558 (1999).
[CrossRef]

M. A. Muriel, J. Azaña, A. Carballar, “Real-time Fourier transformer based on fiber gratings,” Opt. Lett. 24, 1–3 (1999).
[CrossRef]

Banyai, W. C.

M. T. Kauffman, A. A. Godil, B. A. Auld, W. C. Banyai, D. M. Bloom, “Applications of time lens optical systems,” Electron. Lett. 29, 268–269 (1993).
[CrossRef]

Barcelos, S.

S. Barcelos, M. N. Zervas, R. I. Laming, D. N. Payne, L. Reekie, J. A. Tucknott, R. Kashyap, P. F. McKee, F. Sladen, B. Wojciechowicz, “High accuracy dispersion measurements of chirped fibre gratings,” Electron. Lett. 31, 1280–1282 (1995).
[CrossRef]

Bennet, C. V.

C. V. Bennet, R. P. Scott, B. H. Kolner, “Temporal magnification and reversal of 100 Gb/s optical data with an up-conversion time microscope,” Appl. Phys. Lett. 65, 2513–2515 (1994).
[CrossRef]

Bloom, D. M.

M. T. Kauffman, A. A. Godil, B. A. Auld, W. C. Banyai, D. M. Bloom, “Applications of time lens optical systems,” Electron. Lett. 29, 268–269 (1993).
[CrossRef]

Carballar, A.

M. A. Muriel, A. Carballar, J. Azaña, “Field distributions inside fiber gratings,” IEEE J. Quantum Electron. 35, 548–558 (1999).
[CrossRef]

M. A. Muriel, J. Azaña, A. Carballar, “Real-time Fourier transformer based on fiber gratings,” Opt. Lett. 24, 1–3 (1999).
[CrossRef]

Ennser, K.

K. Ennser, M. N. Zervas, R. I. Laming, “Optimization of apodized linearly chirped fiber gratings for optical communications,” IEEE J. Quantum Electron. 34, 770–778 (1998).
[CrossRef]

Godil, A. A.

M. T. Kauffman, A. A. Godil, B. A. Auld, W. C. Banyai, D. M. Bloom, “Applications of time lens optical systems,” Electron. Lett. 29, 268–269 (1993).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, New York, 1996).

Irshid, M. I.

Jannson, J.

Jannson, T.

Kashyap, R.

S. Barcelos, M. N. Zervas, R. I. Laming, D. N. Payne, L. Reekie, J. A. Tucknott, R. Kashyap, P. F. McKee, F. Sladen, B. Wojciechowicz, “High accuracy dispersion measurements of chirped fibre gratings,” Electron. Lett. 31, 1280–1282 (1995).
[CrossRef]

Kauffman, M. T.

M. T. Kauffman, A. A. Godil, B. A. Auld, W. C. Banyai, D. M. Bloom, “Applications of time lens optical systems,” Electron. Lett. 29, 268–269 (1993).
[CrossRef]

Kolner, B. H.

C. V. Bennet, R. P. Scott, B. H. Kolner, “Temporal magnification and reversal of 100 Gb/s optical data with an up-conversion time microscope,” Appl. Phys. Lett. 65, 2513–2515 (1994).
[CrossRef]

B. H. Kolner, “Space–time duality and the theory of temporal imaging,” IEEE J. Quantum. Electron. 30, 1951–1963 (1994).
[CrossRef]

B. H. Kolner, M. Nazarathy, “Temporal imaging with a time lens: erratum,” Opt. Lett. 15, 655 (1990).
[CrossRef]

B. H. Kolner, M. Nazarathy, “Temporal imaging with a time lens,” Opt. Lett. 14, 630–632 (1989).
[CrossRef] [PubMed]

Laming, R. I.

K. Ennser, M. N. Zervas, R. I. Laming, “Optimization of apodized linearly chirped fiber gratings for optical communications,” IEEE J. Quantum Electron. 34, 770–778 (1998).
[CrossRef]

S. Barcelos, M. N. Zervas, R. I. Laming, D. N. Payne, L. Reekie, J. A. Tucknott, R. Kashyap, P. F. McKee, F. Sladen, B. Wojciechowicz, “High accuracy dispersion measurements of chirped fibre gratings,” Electron. Lett. 31, 1280–1282 (1995).
[CrossRef]

Lohmann, A. W.

Mansuripur, M.

M. Mansuripur, “The Talbot effect,” Opt. Photon. News 8(4), 42–47 (1997).

McKee, P. F.

S. Barcelos, M. N. Zervas, R. I. Laming, D. N. Payne, L. Reekie, J. A. Tucknott, R. Kashyap, P. F. McKee, F. Sladen, B. Wojciechowicz, “High accuracy dispersion measurements of chirped fibre gratings,” Electron. Lett. 31, 1280–1282 (1995).
[CrossRef]

Mendlovic, D.

Mitschke, F.

F. Mitschke, U. Morgner, “The temporal Talbot effect,” Opt. Photon. News 9(6), 45–47 (1998).
[CrossRef]

Morgner, U.

F. Mitschke, U. Morgner, “The temporal Talbot effect,” Opt. Photon. News 9(6), 45–47 (1998).
[CrossRef]

Muriel, M. A.

M. A. Muriel, A. Carballar, J. Azaña, “Field distributions inside fiber gratings,” IEEE J. Quantum Electron. 35, 548–558 (1999).
[CrossRef]

M. A. Muriel, J. Azaña, A. Carballar, “Real-time Fourier transformer based on fiber gratings,” Opt. Lett. 24, 1–3 (1999).
[CrossRef]

Nazarathy, M.

Ouellette, F.

Papoulis, A.

Payne, D. N.

S. Barcelos, M. N. Zervas, R. I. Laming, D. N. Payne, L. Reekie, J. A. Tucknott, R. Kashyap, P. F. McKee, F. Sladen, B. Wojciechowicz, “High accuracy dispersion measurements of chirped fibre gratings,” Electron. Lett. 31, 1280–1282 (1995).
[CrossRef]

Reekie, L.

S. Barcelos, M. N. Zervas, R. I. Laming, D. N. Payne, L. Reekie, J. A. Tucknott, R. Kashyap, P. F. McKee, F. Sladen, B. Wojciechowicz, “High accuracy dispersion measurements of chirped fibre gratings,” Electron. Lett. 31, 1280–1282 (1995).
[CrossRef]

Saleh, B. E. A.

Scott, R. P.

C. V. Bennet, R. P. Scott, B. H. Kolner, “Temporal magnification and reversal of 100 Gb/s optical data with an up-conversion time microscope,” Appl. Phys. Lett. 65, 2513–2515 (1994).
[CrossRef]

Sladen, F.

S. Barcelos, M. N. Zervas, R. I. Laming, D. N. Payne, L. Reekie, J. A. Tucknott, R. Kashyap, P. F. McKee, F. Sladen, B. Wojciechowicz, “High accuracy dispersion measurements of chirped fibre gratings,” Electron. Lett. 31, 1280–1282 (1995).
[CrossRef]

Tucknott, J. A.

S. Barcelos, M. N. Zervas, R. I. Laming, D. N. Payne, L. Reekie, J. A. Tucknott, R. Kashyap, P. F. McKee, F. Sladen, B. Wojciechowicz, “High accuracy dispersion measurements of chirped fibre gratings,” Electron. Lett. 31, 1280–1282 (1995).
[CrossRef]

Winthrop, J. T.

Wojciechowicz, B.

S. Barcelos, M. N. Zervas, R. I. Laming, D. N. Payne, L. Reekie, J. A. Tucknott, R. Kashyap, P. F. McKee, F. Sladen, B. Wojciechowicz, “High accuracy dispersion measurements of chirped fibre gratings,” Electron. Lett. 31, 1280–1282 (1995).
[CrossRef]

Worthington, C. R.

Zervas, M. N.

K. Ennser, M. N. Zervas, R. I. Laming, “Optimization of apodized linearly chirped fiber gratings for optical communications,” IEEE J. Quantum Electron. 34, 770–778 (1998).
[CrossRef]

S. Barcelos, M. N. Zervas, R. I. Laming, D. N. Payne, L. Reekie, J. A. Tucknott, R. Kashyap, P. F. McKee, F. Sladen, B. Wojciechowicz, “High accuracy dispersion measurements of chirped fibre gratings,” Electron. Lett. 31, 1280–1282 (1995).
[CrossRef]

Appl. Opt. (1)

Appl. Phys. Lett. (1)

C. V. Bennet, R. P. Scott, B. H. Kolner, “Temporal magnification and reversal of 100 Gb/s optical data with an up-conversion time microscope,” Appl. Phys. Lett. 65, 2513–2515 (1994).
[CrossRef]

Electron. Lett. (2)

S. Barcelos, M. N. Zervas, R. I. Laming, D. N. Payne, L. Reekie, J. A. Tucknott, R. Kashyap, P. F. McKee, F. Sladen, B. Wojciechowicz, “High accuracy dispersion measurements of chirped fibre gratings,” Electron. Lett. 31, 1280–1282 (1995).
[CrossRef]

M. T. Kauffman, A. A. Godil, B. A. Auld, W. C. Banyai, D. M. Bloom, “Applications of time lens optical systems,” Electron. Lett. 29, 268–269 (1993).
[CrossRef]

IEEE J. Quantum Electron. (2)

M. A. Muriel, A. Carballar, J. Azaña, “Field distributions inside fiber gratings,” IEEE J. Quantum Electron. 35, 548–558 (1999).
[CrossRef]

K. Ennser, M. N. Zervas, R. I. Laming, “Optimization of apodized linearly chirped fiber gratings for optical communications,” IEEE J. Quantum Electron. 34, 770–778 (1998).
[CrossRef]

IEEE J. Quantum. Electron. (1)

B. H. Kolner, “Space–time duality and the theory of temporal imaging,” IEEE J. Quantum. Electron. 30, 1951–1963 (1994).
[CrossRef]

J. Opt. Soc. Am. (2)

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

Opt. Lett. (6)

Opt. Photon. News (2)

M. Mansuripur, “The Talbot effect,” Opt. Photon. News 8(4), 42–47 (1997).

F. Mitschke, U. Morgner, “The temporal Talbot effect,” Opt. Photon. News 9(6), 45–47 (1998).
[CrossRef]

Other (2)

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, New York, 1996).

A. Papoulis, The Fourier Integral and its Applications (McGraw-Hill, New York, 1962).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (4)

Fig. 1
Fig. 1

Reflectivity delay (left-hand axis) and reflection group delay (right-hand axis) of the LCFG as a function of the optical frequency.

Fig. 2
Fig. 2

(a) Top plot, Gaussian pulse train incident on the LCFG. n.u., normalized units. The repetition period (141.42 ps) satisfies the Talbot condition. Bottom plot, pulse train reflected from the LCFG. The reflected signal is practically a distortionless replica of the input signal. (b) Top plot, input Gaussian pulse train. The repetition period (125 ps) does not satisfy the Talbot condition. Bottom plot, reflected pulse train. The signal undergoes a strong distortion.

Fig. 3
Fig. 3

Cross-correlation coefficient between the input and reflected signals in the LCFG as a function of the input repetition period. The Talbot periods T T1 = 141.42 ps, T T2 = 100 ps, and T T3 = 81.65 ps are marked on the figure (dashed lines).

Fig. 4
Fig. 4

Diagram of an integral frequency multiplier based on a LCFG. The repetition period (200 ps) of the input signal satisfies the sub-Talbot condition.

Equations (12)

Equations on this page are rendered with MathJax. Learn more.

hdx=hspace exp-j πλd x2,
rω=A exp-jΦ0exp-jτg0ω-ω0×exp-j Φ¨2ω-ω02,
r͡ω=rω0+ω=A exp-jΦ0exp-jτg0ωexp-j 12 Φ¨ω2.
h͡rt=htime expj 12Φ¨t-τg02,
h͡rtR=htime expj 12Φ¨ tR2.
x  tR,λd  -2πΦ¨.
dT=Λ2/λ.
T2=TTs2=2π|Φ¨|s,  s=1, 2, 3,,
C=-+ PitPrtdt-+ Pi2tdt -+ Pr2tdt1/2,
|Φ¨|=TTs2TTs+122πTTs2-TTs+12.
T2=4π|Φ¨|2s-1,  s=1, 2, 3,.
T2=m 2π|Φ¨|2s-1,  s=1, 2, 3,.

Metrics