Abstract

We provide a rigorous, closed-form mathematical model of pulsed diffraction in both time and frequency domains and interpret its anomalies. The customary continuous-wave (cw) approximation is corrected for the regime below 3 fs in pulse width, the present state of the art. The time-differentiated aspect of diffraction is linked with old theory and the conservation of energy. Convolution of the pulse with differentiated aperture edges produces traveling waves in the focal plane. Their collision near the focal point corresponds to the cw case. (This is generalized for a Gaussian beam.) Spectral sampling depicts a mode-locked laser of extremely broad spectrum, validating the nonintuitive phenomena. The square-modulus tool is validated for these pulses. Ultrashort pulses are related to data transmission rates above 100 THz. Diffraction anomalies cause confusion and loss of information in the sidelobes. Anomalous diffraction may provide new diagnostics. Diffracted energy (salient sidelobes versus continuum) can measure transform-limited pulse width.

© 1998 Optical Society of America

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References

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  1. Z. Jiang, R. Jacquemin, W. Eberhardt, “Time dependence of Fresnel diffraction of ultrashort laser pulses by a circular aperture,” Appl. Opt. 36, 4358–4361 (1997).
    [CrossRef] [PubMed]
  2. J. M. Anderson, Diffraction of an Optical Pulse of Extremely Short Duration, M.S. thesis (University of Connecticut, Storrs, Conn., 1997).
  3. M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980), Secs. 8.2 and 8.3, pp. 370–382.
  4. A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986), pp. 638–647.
  5. H. A. Haus, Waves and Fields in Optoelectronics (Prentice-Hall, Englewood Cliffs, N.J., 1984), pp. 3, 109, 114.
  6. A. Lacourt, J.-C. Viénot, J.-P. Goedgebuer, “Reassessing basic landmarks on space–time optics,” Opt. Commun. 19, 68–71 (1976).
    [CrossRef]
  7. J.-C. Viénot, J.-P. Goedgebuer, A. Lacourt, “Space and time variables in optics and holography: recent experimental aspects,” Appl. Opt. 16, 454–461 (1977).
    [CrossRef] [PubMed]
  8. A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986), pp. 635, 716, 723–725 and references therein.
  9. J. B. Keller, “Diffraction by an aperture,” J. Appl. Phys. 28, 426–444 (1957).
    [CrossRef]
  10. J. B. Keller, R. M. Lewis, B. D. Seckler, “Diffraction by an aperture II,” J. Appl. Phys. 28, 570–579 (1957).
    [CrossRef]
  11. National Bureau of Standards, Handbook of Mathematical Functions (U.S. Government Printing Office, Washington, D.C., 1964), Chap. 7, pp. 297–300.
  12. M. Gu, X. S. Gan, “Fresnel diffraction by circular and serrated apertures illuminated with an ultrashort pulsed-laser beam,” J. Opt. Soc. Am. A 13, 771–778 (1996).
    [CrossRef]

1997 (1)

1996 (1)

1977 (1)

1976 (1)

A. Lacourt, J.-C. Viénot, J.-P. Goedgebuer, “Reassessing basic landmarks on space–time optics,” Opt. Commun. 19, 68–71 (1976).
[CrossRef]

1957 (2)

J. B. Keller, “Diffraction by an aperture,” J. Appl. Phys. 28, 426–444 (1957).
[CrossRef]

J. B. Keller, R. M. Lewis, B. D. Seckler, “Diffraction by an aperture II,” J. Appl. Phys. 28, 570–579 (1957).
[CrossRef]

Anderson, J. M.

J. M. Anderson, Diffraction of an Optical Pulse of Extremely Short Duration, M.S. thesis (University of Connecticut, Storrs, Conn., 1997).

Born, M.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980), Secs. 8.2 and 8.3, pp. 370–382.

Eberhardt, W.

Gan, X. S.

Goedgebuer, J.-P.

J.-C. Viénot, J.-P. Goedgebuer, A. Lacourt, “Space and time variables in optics and holography: recent experimental aspects,” Appl. Opt. 16, 454–461 (1977).
[CrossRef] [PubMed]

A. Lacourt, J.-C. Viénot, J.-P. Goedgebuer, “Reassessing basic landmarks on space–time optics,” Opt. Commun. 19, 68–71 (1976).
[CrossRef]

Gu, M.

Haus, H. A.

H. A. Haus, Waves and Fields in Optoelectronics (Prentice-Hall, Englewood Cliffs, N.J., 1984), pp. 3, 109, 114.

Jacquemin, R.

Jiang, Z.

Keller, J. B.

J. B. Keller, “Diffraction by an aperture,” J. Appl. Phys. 28, 426–444 (1957).
[CrossRef]

J. B. Keller, R. M. Lewis, B. D. Seckler, “Diffraction by an aperture II,” J. Appl. Phys. 28, 570–579 (1957).
[CrossRef]

Lacourt, A.

J.-C. Viénot, J.-P. Goedgebuer, A. Lacourt, “Space and time variables in optics and holography: recent experimental aspects,” Appl. Opt. 16, 454–461 (1977).
[CrossRef] [PubMed]

A. Lacourt, J.-C. Viénot, J.-P. Goedgebuer, “Reassessing basic landmarks on space–time optics,” Opt. Commun. 19, 68–71 (1976).
[CrossRef]

Lewis, R. M.

J. B. Keller, R. M. Lewis, B. D. Seckler, “Diffraction by an aperture II,” J. Appl. Phys. 28, 570–579 (1957).
[CrossRef]

Seckler, B. D.

J. B. Keller, R. M. Lewis, B. D. Seckler, “Diffraction by an aperture II,” J. Appl. Phys. 28, 570–579 (1957).
[CrossRef]

Siegman, A. E.

A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986), pp. 635, 716, 723–725 and references therein.

A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986), pp. 638–647.

Viénot, J.-C.

J.-C. Viénot, J.-P. Goedgebuer, A. Lacourt, “Space and time variables in optics and holography: recent experimental aspects,” Appl. Opt. 16, 454–461 (1977).
[CrossRef] [PubMed]

A. Lacourt, J.-C. Viénot, J.-P. Goedgebuer, “Reassessing basic landmarks on space–time optics,” Opt. Commun. 19, 68–71 (1976).
[CrossRef]

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980), Secs. 8.2 and 8.3, pp. 370–382.

Appl. Opt. (2)

J. Appl. Phys. (2)

J. B. Keller, “Diffraction by an aperture,” J. Appl. Phys. 28, 426–444 (1957).
[CrossRef]

J. B. Keller, R. M. Lewis, B. D. Seckler, “Diffraction by an aperture II,” J. Appl. Phys. 28, 570–579 (1957).
[CrossRef]

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

Opt. Commun. (1)

A. Lacourt, J.-C. Viénot, J.-P. Goedgebuer, “Reassessing basic landmarks on space–time optics,” Opt. Commun. 19, 68–71 (1976).
[CrossRef]

Other (6)

A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986), pp. 635, 716, 723–725 and references therein.

National Bureau of Standards, Handbook of Mathematical Functions (U.S. Government Printing Office, Washington, D.C., 1964), Chap. 7, pp. 297–300.

J. M. Anderson, Diffraction of an Optical Pulse of Extremely Short Duration, M.S. thesis (University of Connecticut, Storrs, Conn., 1997).

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980), Secs. 8.2 and 8.3, pp. 370–382.

A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986), pp. 638–647.

H. A. Haus, Waves and Fields in Optoelectronics (Prentice-Hall, Englewood Cliffs, N.J., 1984), pp. 3, 109, 114.

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Figures (5)

Fig. 1
Fig. 1

Diffraction geometry of the Huygens–Fresnel model: a cross section representing a variety of aperture shapes.

Fig. 2
Fig. 2

Amplitude diffraction pattern of a one-cycle pulse (imaginary part). Instants of (a) peak at temporal center, (b) next peak, (c) second zero, and (d) fourth peak after temporal centers.

Fig. 3
Fig. 3

Unidimensional diffracted energy density from slit for a cw (solid curve), a three-cycle pulse (dashed curve), a two-cycle pulse (dotted curve), and a one-cycle pulse (dashed-dotted curve). (a) Central pattern, (b) magnified sidelobes (b).

Fig. 4
Fig. 4

Section of amplitude pattern, along the ξ axis, of a one-cycle repeated pulse (imaginary part) at (a) temporal center (μ=0), and instants later, in terms of carrier periods, by (b) μ=2, (c) μ=4, and (d) μ=5.

Fig. 5
Fig. 5

Amplitude time function for a single-cycle pulsed Gaussian beam. Sampled spectrum at (a) ρ=0, (b) ρ=3, (c) ρ=5, (d) ρ=9.

Equations (16)

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A=jfλ-aa-bb expjωxxˆfc+yyˆfcdyˆdxˆ.
A(x, y, t)=12πfc-aa-bb ddt+xxˆfc+yyˆfc×ψt+xxˆfc+yyˆfc×expjωct+xxˆfc+yyˆfcdyˆdxˆ.
A(x, y, t)=12πfc-aa -bb×-2 t+xxˆfc+yyˆfcτ2+jωc×exp -t+xxˆfc+yyˆfc2τ2×expjωct+xxˆfc+yyˆfcdyˆdxˆ.
A(x, y, t)=12ππ2fcτxyexp-ωc2τ24×erftτ+axfcτ+byfcτ-j ωcτ2-erftτ-axfcτ+byfcτ-j ωcτ2-erftτ+axfcτ-byfcτ-j ωcτ2+erftτ-axfcτ-byfcτ-j ωcτ2.
ωc=2πcλc,ωcτ=2πn,ωct=2πμ,
ωcaxfc=2πξ,ωcbyfc=2πη,ωcarfc=2πρ,
N=2n.
A(ξ, η, μ)=4abfλ1(2πξ)(2πη)π3/2n exp(-π2n2)×14erf1n(μ+ξ+η-jπn2)-erf1n(μ-ξ+η-jπn2)-erf1n(μ+ξ-η-jπn2)+erf1n(μ-ξ-η-jπn2).
A(ξ, η=0, μ)=4abfλ1(2πξ)×12exp-1n2(μ+ξ)2×exp[j2π(μ+ξ)]-exp-1n2(μ-ξ)2×exp[j2π(μ-ξ)].
4abfλc-μπn2+jexp-μ2n2exp(j2πμ).
A(x, y, ω)=(4ab) jω2πfcτ2exp-(ω-ωc)2τ24×sinω axfcω axfcsinω byfcω byfc.
Anorm(ξ, η, μ)=πp jpnP2exp-pP-12π2n2×sin2π pPξ2π pPξ×sin2π pPη2π pPηexpj2π pPμ.
Anorm(ξ, η, μ)=πp p200exp-p10-12 π24×sin2π p10ξ2π p10ξ×sin2π p10η2π p10ηcos2π p10μ.
Anorm(ξ, η=0, μ)=12πp p100exp-p10-12 π24×sin2π p10ξ2π p10ξcos2π p10μ.
Anorm(ρ, μ)=πp p200exp-p10-12 π24×2J1(2πp10ρ)2πp10ρcos2π p10μ.
Anorm(ρ, μ)=πp p200exp-p10-12 π24×exp-2 p10ρ2cos2π p10μ.

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