Abstract

The general concept of temporal Fourier transformation in dispersive media is analyzed. The real-time optical Fourier transformer is shown to be realizable by using dispersive single-mode fibers and chirping lasers.

© 1983 Optical Society of America

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References

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  1. See, for example, J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), Chap. 7.5.
  2. D. Marcuse, “Pulse distortion in single-mode fibers. 3 Chirped pulses,” Appl. Opt. 20, 3573–3579 (1981).
    [CrossRef] [PubMed]
  3. B. Steverding, “A chirping laser for photochemical applications,” Appl. Phys. Lett. 30, 231–233 (1977).
    [CrossRef]
  4. A brief account of this concept was presented at the 1982 Annual Meeting of the Optical Society of America, Tucson, Arizona, October 18–22; see T. Jannson, “Temporal Fourier transformation in dispersive media,” J. Opt. Soc. Am. 72, 1816 (A) (1982).
  5. T. Jannson, J. Jannson, “Temporal self-imaging effect in single-mode fibers,” J. Opt. Soc. Am. 71, 1373–1376 (1981).
  6. In general, it may be a vector wave connected with one arbitrary mode. Then this analysis holds for any mode if can be treated separately.
  7. Strictly speaking, the medium may be weakly dissipative, but, in such a case, the absorption coefficient cannot be dependent on frequency within the passband of the information-bearing signal.
  8. See also T. Jannson, J. Sochacki, “Primary aberrations of thin planar surface lenses,” J. Opt. Soc. Am. 70, 1079–1084 (1980), Sec. 1.
    [CrossRef]
  9. There are also other useful representations of the first-order dispersion factor D; see Ref. 5, Eq. (24). Also D = vg−1 = n˙g/c, where ng is the group-velocity refractive index.
  10. See, for example, Ref. 1, Chap. 5.2.
  11. R. C. Williamson, “Comparison of surface acoustic-wave and optical signal processing,” Proc. Soc. Photo-Opt. Instrum. Eng. 185, 74–84 (1979).
  12. In that case, a self-imaging effect (see Ref. 5) and a temporal imaging effect (see Ref. 4) can also exist.
  13. See also, L. Mertz, Transformations in Optics (Wiley, New York, 1965), p. 96.
  14. In that case, the dispersion relation has the form β = 2[(mπ/h) ω]1/2, where m is the mass of the particle and h is the Planck constant.
  15. Such a situation is realistic for some materials (see Ref. 13, p. 32). In order to obtain the values of the D factor from the data in Ref. 13, its group-velocity representation should be used; see Ref 9.

1981 (2)

T. Jannson, J. Jannson, “Temporal self-imaging effect in single-mode fibers,” J. Opt. Soc. Am. 71, 1373–1376 (1981).

D. Marcuse, “Pulse distortion in single-mode fibers. 3 Chirped pulses,” Appl. Opt. 20, 3573–3579 (1981).
[CrossRef] [PubMed]

1980 (1)

1979 (1)

R. C. Williamson, “Comparison of surface acoustic-wave and optical signal processing,” Proc. Soc. Photo-Opt. Instrum. Eng. 185, 74–84 (1979).

1977 (1)

B. Steverding, “A chirping laser for photochemical applications,” Appl. Phys. Lett. 30, 231–233 (1977).
[CrossRef]

Goodman, J. W.

See, for example, J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), Chap. 7.5.

Jannson, J.

T. Jannson, J. Jannson, “Temporal self-imaging effect in single-mode fibers,” J. Opt. Soc. Am. 71, 1373–1376 (1981).

Jannson, T.

T. Jannson, J. Jannson, “Temporal self-imaging effect in single-mode fibers,” J. Opt. Soc. Am. 71, 1373–1376 (1981).

See also T. Jannson, J. Sochacki, “Primary aberrations of thin planar surface lenses,” J. Opt. Soc. Am. 70, 1079–1084 (1980), Sec. 1.
[CrossRef]

Marcuse, D.

Mertz, L.

See also, L. Mertz, Transformations in Optics (Wiley, New York, 1965), p. 96.

Sochacki, J.

Steverding, B.

B. Steverding, “A chirping laser for photochemical applications,” Appl. Phys. Lett. 30, 231–233 (1977).
[CrossRef]

Williamson, R. C.

R. C. Williamson, “Comparison of surface acoustic-wave and optical signal processing,” Proc. Soc. Photo-Opt. Instrum. Eng. 185, 74–84 (1979).

Appl. Opt. (1)

Appl. Phys. Lett. (1)

B. Steverding, “A chirping laser for photochemical applications,” Appl. Phys. Lett. 30, 231–233 (1977).
[CrossRef]

J. Opt. Soc. Am. (1)

T. Jannson, J. Jannson, “Temporal self-imaging effect in single-mode fibers,” J. Opt. Soc. Am. 71, 1373–1376 (1981).

J. Opt. Soc. Am. (1)

Proc. Soc. Photo-Opt. Instrum. Eng. (1)

R. C. Williamson, “Comparison of surface acoustic-wave and optical signal processing,” Proc. Soc. Photo-Opt. Instrum. Eng. 185, 74–84 (1979).

Other (10)

In that case, a self-imaging effect (see Ref. 5) and a temporal imaging effect (see Ref. 4) can also exist.

See also, L. Mertz, Transformations in Optics (Wiley, New York, 1965), p. 96.

In that case, the dispersion relation has the form β = 2[(mπ/h) ω]1/2, where m is the mass of the particle and h is the Planck constant.

Such a situation is realistic for some materials (see Ref. 13, p. 32). In order to obtain the values of the D factor from the data in Ref. 13, its group-velocity representation should be used; see Ref 9.

In general, it may be a vector wave connected with one arbitrary mode. Then this analysis holds for any mode if can be treated separately.

Strictly speaking, the medium may be weakly dissipative, but, in such a case, the absorption coefficient cannot be dependent on frequency within the passband of the information-bearing signal.

A brief account of this concept was presented at the 1982 Annual Meeting of the Optical Society of America, Tucson, Arizona, October 18–22; see T. Jannson, “Temporal Fourier transformation in dispersive media,” J. Opt. Soc. Am. 72, 1816 (A) (1982).

There are also other useful representations of the first-order dispersion factor D; see Ref. 5, Eq. (24). Also D = vg−1 = n˙g/c, where ng is the group-velocity refractive index.

See, for example, Ref. 1, Chap. 5.2.

See, for example, J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), Chap. 7.5.

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Equations (23)

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U ( t , z ) = 0 U ^ ( ω , z ) e - i ω t d ω ,
U ^ ( ω , z ) = U ^ ( ω , 0 ) exp ( i β z ) ,
U ( t , z ) = A ( t , z ) exp [ i ( β 0 z - ω 0 t ) ] ,
β = β 0 + β ˙ 0 ( ω - ω 0 ) + ( ½ ) β ¨ 0 ( ω - ω 0 ) 2 + ,
A ( t , z ) = - ω 0 G ( ω , z ) U ^ ( ω 0 + ω , 0 ) × exp [ - i ω ( t - z / v g ) ] d ω ,
G ( ω , z ) = exp { i [ β ¯ ( ω ) - β 0 ] z } × exp ( - i β ˙ 0 ω z ) ,
A ¯ ( t R ) = - + A ^ ( ω ) exp ( - i ω t R ) d ω ,
A ^ ( ω ) = G ( ω , z ) A ^ 0 ( ω ) ,
G ( ω , z ) = exp ( i ω 2 z β ¨ 0 2 ) .
A ¯ ( t R ) = exp ( i π / 4 ) ( 2 Π D z ) 1 / 2 × - + A 0 ( t 0 ) exp [ - i ( t R - t 0 ) 2 2 D z ] d t 0 ,
γ 2 A ¯ ( t R , z ) γ t R 2 - 2 i D γ A ¯ ( t R , z ) γ z = 0.
A ( t , 0 ) = A 0 ( t 0 ) = - B 0 ( t 0 ) exp ( i a t 0 2 2 ) ,
U 0 ( t 0 ) = B 0 ( t 0 ) ( exp [ i Φ ( t 0 ) ] ,
Φ ( t 0 ) = ω 0 t 0 - a t 0 2 2 .
ω 0 = d ϕ d t 0 = ω 0 - a t 0 .
A ¯ ( t R ) = exp ( i π / 4 ) ( 2 π D z ) 1 / 2 exp ( - i t R 2 2 D z ) × - + B 0 ( t 0 ) exp [ i t 0 2 2 ( a - 1 D z ) ] exp ( i t R t 0 D z ) d t 0 .
z = F = 1 D a .
A ¯ ( t R ) = exp ( i π / 4 ) ( 2 π D z ) 1 / 2 × exp ( - i t R 2 2 D z ) B ^ 0 ( - t R D z ) ,
B ^ 0 ( ω F ) = 1 2 π - + B 0 ( t 0 ) exp ( - i ω F t 0 ) d t 0
ω F = - t R D F .
2 t R max = 2 × D × F × ω F max .
Δ ω 0 = a × T .
2 t R max = 2 T ω F max Δ ω 0 .

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