Abstract

This paper describes the principles and experimental implementation of a new optical technique based on Fourier analysis for second-order statistics of short coherent light pulses. Using only passive filtering operations, we obtained images of the autocorrelation functions of the optical fields of light pulses, derived from mode-locked Nd:YAG laser pulses, with 40-psec to 4-nsec duration. An application of the method to a shaped sequence of 40-psec pulses is reported.

© 1980 Optical Society of America

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References

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  1. M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1975), pp. 495–499.
  2. R. Bracewell, The Fourier Transform and Its Applications (McGraw-Hill, New York, 1965), pp. 268–272.
  3. H. P. Weber, J. Appl. Phys. 38, 2231 (1967).
    [CrossRef]
  4. J. A. Armstrong, Appl. Phys. Lett. 10, 16 (1967).
    [CrossRef]
  5. J. A. Giordmaine, P. M. Rentzepis, S. L. Shapiro, K. W. Wecht, Appl. Phys. Lett. 11, 216 (1967).
    [CrossRef]
  6. Ref. 2, p. 115.
  7. F. Gires, IEEE J. Quantum Electron. QE-9, 2 (1973).
  8. W. E. Martin, D. Milam, Appl. Opt. 15, 3054 (1976).
    [CrossRef] [PubMed]
  9. B. Colombeau, M. Vampouille, C. Froehly, Opt. Commun. 19, 201 (1976).
    [CrossRef]
  10. Ref. 1, p. 499.
  11. W. V. Houston, Phys. Rev. 29, 478 (1927).
    [CrossRef]
  12. E. Gehrcke, E. Lau, Z. Tech. Phys. 8, 157 (1927).
  13. Ref. 1, pp. 323–329.
  14. B. Colombeau, C. Froehly, M. Vampouille, Opt. Commun. 28, 35 (1979).
    [CrossRef]
  15. C. Froehly, A. Lacourt, J.-C. Viénot, Nouv. Rev. Opt. 4, 183 (1973).
    [CrossRef]

1979 (1)

B. Colombeau, C. Froehly, M. Vampouille, Opt. Commun. 28, 35 (1979).
[CrossRef]

1976 (2)

W. E. Martin, D. Milam, Appl. Opt. 15, 3054 (1976).
[CrossRef] [PubMed]

B. Colombeau, M. Vampouille, C. Froehly, Opt. Commun. 19, 201 (1976).
[CrossRef]

1973 (2)

F. Gires, IEEE J. Quantum Electron. QE-9, 2 (1973).

C. Froehly, A. Lacourt, J.-C. Viénot, Nouv. Rev. Opt. 4, 183 (1973).
[CrossRef]

1967 (3)

H. P. Weber, J. Appl. Phys. 38, 2231 (1967).
[CrossRef]

J. A. Armstrong, Appl. Phys. Lett. 10, 16 (1967).
[CrossRef]

J. A. Giordmaine, P. M. Rentzepis, S. L. Shapiro, K. W. Wecht, Appl. Phys. Lett. 11, 216 (1967).
[CrossRef]

1927 (2)

W. V. Houston, Phys. Rev. 29, 478 (1927).
[CrossRef]

E. Gehrcke, E. Lau, Z. Tech. Phys. 8, 157 (1927).

Armstrong, J. A.

J. A. Armstrong, Appl. Phys. Lett. 10, 16 (1967).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1975), pp. 495–499.

Bracewell, R.

R. Bracewell, The Fourier Transform and Its Applications (McGraw-Hill, New York, 1965), pp. 268–272.

Colombeau, B.

B. Colombeau, C. Froehly, M. Vampouille, Opt. Commun. 28, 35 (1979).
[CrossRef]

B. Colombeau, M. Vampouille, C. Froehly, Opt. Commun. 19, 201 (1976).
[CrossRef]

Froehly, C.

B. Colombeau, C. Froehly, M. Vampouille, Opt. Commun. 28, 35 (1979).
[CrossRef]

B. Colombeau, M. Vampouille, C. Froehly, Opt. Commun. 19, 201 (1976).
[CrossRef]

C. Froehly, A. Lacourt, J.-C. Viénot, Nouv. Rev. Opt. 4, 183 (1973).
[CrossRef]

Gehrcke, E.

E. Gehrcke, E. Lau, Z. Tech. Phys. 8, 157 (1927).

Giordmaine, J. A.

J. A. Giordmaine, P. M. Rentzepis, S. L. Shapiro, K. W. Wecht, Appl. Phys. Lett. 11, 216 (1967).
[CrossRef]

Gires, F.

F. Gires, IEEE J. Quantum Electron. QE-9, 2 (1973).

Houston, W. V.

W. V. Houston, Phys. Rev. 29, 478 (1927).
[CrossRef]

Lacourt, A.

C. Froehly, A. Lacourt, J.-C. Viénot, Nouv. Rev. Opt. 4, 183 (1973).
[CrossRef]

Lau, E.

E. Gehrcke, E. Lau, Z. Tech. Phys. 8, 157 (1927).

Martin, W. E.

Milam, D.

Rentzepis, P. M.

J. A. Giordmaine, P. M. Rentzepis, S. L. Shapiro, K. W. Wecht, Appl. Phys. Lett. 11, 216 (1967).
[CrossRef]

Shapiro, S. L.

J. A. Giordmaine, P. M. Rentzepis, S. L. Shapiro, K. W. Wecht, Appl. Phys. Lett. 11, 216 (1967).
[CrossRef]

Vampouille, M.

B. Colombeau, C. Froehly, M. Vampouille, Opt. Commun. 28, 35 (1979).
[CrossRef]

B. Colombeau, M. Vampouille, C. Froehly, Opt. Commun. 19, 201 (1976).
[CrossRef]

Viénot, J.-C.

C. Froehly, A. Lacourt, J.-C. Viénot, Nouv. Rev. Opt. 4, 183 (1973).
[CrossRef]

Weber, H. P.

H. P. Weber, J. Appl. Phys. 38, 2231 (1967).
[CrossRef]

Wecht, K. W.

J. A. Giordmaine, P. M. Rentzepis, S. L. Shapiro, K. W. Wecht, Appl. Phys. Lett. 11, 216 (1967).
[CrossRef]

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1975), pp. 495–499.

Appl. Opt. (1)

Appl. Phys. Lett. (2)

J. A. Armstrong, Appl. Phys. Lett. 10, 16 (1967).
[CrossRef]

J. A. Giordmaine, P. M. Rentzepis, S. L. Shapiro, K. W. Wecht, Appl. Phys. Lett. 11, 216 (1967).
[CrossRef]

IEEE J. Quantum Electron. (1)

F. Gires, IEEE J. Quantum Electron. QE-9, 2 (1973).

J. Appl. Phys. (1)

H. P. Weber, J. Appl. Phys. 38, 2231 (1967).
[CrossRef]

Nouv. Rev. Opt. (1)

C. Froehly, A. Lacourt, J.-C. Viénot, Nouv. Rev. Opt. 4, 183 (1973).
[CrossRef]

Opt. Commun. (2)

B. Colombeau, C. Froehly, M. Vampouille, Opt. Commun. 28, 35 (1979).
[CrossRef]

B. Colombeau, M. Vampouille, C. Froehly, Opt. Commun. 19, 201 (1976).
[CrossRef]

Phys. Rev. (1)

W. V. Houston, Phys. Rev. 29, 478 (1927).
[CrossRef]

Z. Tech. Phys. (1)

E. Gehrcke, E. Lau, Z. Tech. Phys. 8, 157 (1927).

Other (5)

Ref. 1, pp. 323–329.

Ref. 1, p. 499.

Ref. 2, p. 115.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1975), pp. 495–499.

R. Bracewell, The Fourier Transform and Its Applications (McGraw-Hill, New York, 1965), pp. 268–272.

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Figures (12)

Fig. 1
Fig. 1

Two optical methods providing spatial images of autocorrelation functions of optical fields: (A) The illumination recorded by photographic film R1 varies as the real part of the autocorrelation function C(τ) of the analytic signal V(t). (B) The Fourier transform of the spatially recorded power spectrum (P.S.) is the autocorrelation function C(τ) of V(t).

Fig. 2
Fig. 2

High-resolution spectrograph used to record the spectrum of IR and harmonic light pulses with ~200 spectral elements in the free spectral range; P: polarizer; Q: λ/4 plate; S: spectral plane; e1 = 5 mm; e2 = 60 mm; f = 500 mm.

Fig. 3
Fig. 3

Diffraction of a plane monochromatic wave by the Fabry-Perot spectrogram. of the pulse, masked by an annular sector selecting the area of interest.

Fig. 4
Fig. 4

(A) Beams diffracted by an axially symmetrical pupil S′ with the same radial transmissivity T(r2) as the Fabry-Perot spectrogram of the pulse. (B) The pupil S′ being located in the first focal plane of a lens L, the complex amplitude of the field diffracted along the axis behaves as the Fourier transform of the radial transmissivity T(r2), originated at the second focus F′.

Fig. 5
Fig. 5

Whole diffractometer—the source is a cw laser. The complex amplitude on screen E along the axis of L varies as the complex autocorrelation function C(τ) of the optical field of the pulse. The time delay τ is originated at F′.

Fig. 6
Fig. 6

(a) The function A0(σ) defined over the free spectral range δσ is one period of |R(σ)|2, R(σ) being the transfer function of the two Fabry-Perot interferometers in series. (b) Fourier transform α(τ) of A0(σ). This function fixes an upper limit to the duration of the pulses to which the method can be applied.

Fig. 7
Fig. 7

Experimental device used to make the shaped sequence of 40-psec mode-locked laser pulses with 1 nsec time interval between two consecutive pulses.

Fig. 8
Fig. 8

Positive photograph of the power spectrum of the 40-psec pulse sequence (magnification 9×). The dashed line limits the effective area selected for further optical Fourier transformation.

Fig. 9
Fig. 9

Spatial image of the autocorrelation function of the optical field of the pulse train resulting from monochromatic axial diffraction by the spectrogram of Fig. 8. A time scale is given, and the various terms are numbered.

Fig. 10
Fig. 10

Relative energies of the first three terms of the autocorrelation function as numbered in Fig. 9 (central term not considered): + denote experimental values; x are relative to calculated values, taking into account the function α(τ) drawn in Fig. 6(b).

Fig. 11
Fig. 11

Experimental setup used to measure the relative phases of the various terms in the autocorrelation function C(T). S is the spectrogram, and S0 is the filtered image of the spectrogram.

Fig. 12
Fig. 12

Photographs of the images of the spectrogram obtained with three different filtering screens, showing that the phase varies linearly vs time in the train of 40-psec pulses.

Equations (19)

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C ( τ ) = - + V ( t ) V * ( t - τ ) d t .
I t / I i = 1 1 + F sin 2 ( ϕ / 2 ) · 1 1 + F sin 2 ( ϕ / 2 ) ,
d E σ 0 ( r 2 ) = [ - + III p 0 ( r 2 - r 2 ) A 0 ( r 2 ) d r 2 ] B ( σ 0 ) d σ = [ III p 0 ( r 2 ) A 0 ( r 2 ) ] B ( σ 0 ) d σ .
III p 0 ( r 2 ) = m = - + δ ( r 2 - m p 0 ) ;
F ( r 2 ) = III p 0 ( r 2 ) A 0 ( r 2 ) ;             E σ 0 ( r 2 ) = d E σ 0 ( r 2 ) d σ ,
E σ 0 ( r 2 ) = F ( r 2 ) · B ( σ 0 ) .
d E σ ( r 2 ) = [ III p ( r 2 - h ) A ( r 2 ) ] B ( σ ) d σ ,
p = f 2 e σ = p 0             and             A ( r 2 ) = A 0 ( r 2 ) .
e σ 0 ( 1 - r 0 n 2 2 f 2 ) = n = e σ ( 1 - r n 2 2 f 2 ) ,             n = int e g e r , h = r n 2 - r 0 n 2 = 2 σ - σ 0 σ 0 f 2 .
E σ ( r 2 ) = d E σ ( r 2 ) d σ = F ( r 2 - h ) · B ( σ ) .
ξ ( r 2 ) = 0 + E σ ( r 2 ) d σ = σ 0 2 f 2 0 F ( r 2 - h ) B ( σ 0 h 2 f 2 ) d h .
ξ ( r 2 ) = III p 0 ( r 2 ) A 0 ( r 2 ) B 1 ( r 2 ) ,
T ( r 2 ) = 1 - K [ A 0 ( r 2 ) B 1 ( r 2 ) ] ,
a ( M ) = - j ρ [ exp ( - j 2 π λ z ) ] { F [ F ( q ) ] } ,
F ( q ) = 1 Δ θ 0 Δ θ F ( r 2 , θ ) d θ .
a ( M ) = - j 2 λ z [ exp ( - j 2 π λ z ) ] { F [ T ( r 2 ) ] } .
F z ¯ = - f 2 O z ¯ ,
a ( z ) F { 1 - K [ A 0 ( r 2 ) B 1 ( r 2 ) ] } ,
a ( z ) ( δ ( z ) - K F [ A 0 ( r 2 ) ] } · { F [ B 1 ( r 2 ) ] } ) .

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