Abstract

A convenient approximate formula is proposed for the study of free-space propagation of spatial and temporal pulses with an identifiable carrier frequency. It does not contain a time derivative operation on the pulse’s temporal envelope explicitly. It is shown that once a short (for example, picosecond or subpicosecond) pulse is created with a spatial and a temporal structure, it does not last forever. The approximation discussed is valid over a certain distance as dictated by the wave equation. Beyond this distance, the spatial and temporal characteristics begin to influence each other significantly. Two examples are presented. The first example is that of a pulse with a factored form of a spatial envelope times a temporal envelope. The second example is that of a clear aperture with a grating, by which pulse stretching or temporal distortion is examined and the result is in agreement with that found in the literature.

© 1994 Optical Society of America

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References

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  1. N. Streibl, M. E. Prise, “Digital optics,” Phys. Status Solidi B 150, 447–454 (1988).
    [Crossref]
  2. E. B. Treacy, “Optical pulse compression with diffraction gratings,” IEEE J. Quantum Electron. QE-5454–458 (1969); “Measurement of picosecond pulse substructure using compression techniques,” Appl. Phys. Lett. 14, 112–114 (1969); “Compression of picosecond light pulses,” Phys. Lett. A 28, 34–35 (1968).
    [Crossref]
  3. C. Froehly, B. Colombeau, M. Vampouille, “Shaping and analysis of picosecond light pulses,” Prog. Opt. 20, 63–153 (1982).
    [Crossref]
  4. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), pp. 48–54.
  5. A. S. Marathay, Elements of Optical Coherence Theory (Wiley, New York, 1982), pp. 140–156, 174–178.
  6. H. J. Caulfield, J. E. Ludman, “Time-frequency interrelationships in optical information processing,” in Transformations in Optical Signal Processing, W. T. Rhodes, ed., Proc. Soc. Photo-Opt. Instrum. Eng.373, 63–67 (1981).
  7. J. Shamir, “Fundamental speed limitations on parallel processing,” Appl. Opt. 26, 1567 (1987).
    [Crossref] [PubMed]
  8. A. W. Lohmann, A. S. Marathay, “Globality and speed of optical parallel processor,” Appl. Opt. 28, 3838–3842 (1989).
    [Crossref] [PubMed]
  9. O. E. Martinez, “Matrix formalism for pulse compressors,” IEEE J. Quantum Electron. 24, 2530–2536 (1988); “Grating and prism compressors in the case of finite beam size,” J. Opt. Soc. Am. B 3, 929–934 (1986); O. E. Martinez, J. P. Gordon, R. L. Fork, “Negative group-velocity dispersion using refraction,” J. Opt. Soc. Am. A1003–1006 (1984); R. L. Fork, O. E. Martinez, J. P. Gordon, “Negative dispersion using pairs of prisms,” Opt. Lett. 9, 150–152 (1984); J. P. Gordon, R. L. Fork, “Optical resonator with negative dispersion,” Opt. Lett. 9, 153–155 (1984); O. E. Martinez, R. L. Fork, J. P. Gordon, “Theory of passively mode-locked lasers including self-phase modulation and group-velocity dispersion,” Opt. Lett. 9, 156–158 (1984).
    [Crossref] [PubMed]
  10. P. M. Morse, H. Feshback, Methods of Theoretical Physics (McGraw-Hill, New York, 1953), Part 1, Sec. 7.3, pp. 834–857.
  11. M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1980), pp. 375–378.
  12. A. Sommerfeld, Optics, Vol. 4 of Lectures on Theoretical Physics (Academic, New York, 1964), Vol. 4, pp. 195–207.
  13. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), Sec. 3.5, p. 47, Eq. (3-33).
  14. W. B. Davenport, W. L. Root, Random Signals and Noise (McGraw-Hill, New York, 1958), p. 129, Eqs. (8-22) and (8-23).
  15. J. W. Goodman, Statistical Optics (Wiley, New York, 1985), Sec. 2.7.2(c), p. 38.

1989 (1)

1988 (2)

N. Streibl, M. E. Prise, “Digital optics,” Phys. Status Solidi B 150, 447–454 (1988).
[Crossref]

O. E. Martinez, “Matrix formalism for pulse compressors,” IEEE J. Quantum Electron. 24, 2530–2536 (1988); “Grating and prism compressors in the case of finite beam size,” J. Opt. Soc. Am. B 3, 929–934 (1986); O. E. Martinez, J. P. Gordon, R. L. Fork, “Negative group-velocity dispersion using refraction,” J. Opt. Soc. Am. A1003–1006 (1984); R. L. Fork, O. E. Martinez, J. P. Gordon, “Negative dispersion using pairs of prisms,” Opt. Lett. 9, 150–152 (1984); J. P. Gordon, R. L. Fork, “Optical resonator with negative dispersion,” Opt. Lett. 9, 153–155 (1984); O. E. Martinez, R. L. Fork, J. P. Gordon, “Theory of passively mode-locked lasers including self-phase modulation and group-velocity dispersion,” Opt. Lett. 9, 156–158 (1984).
[Crossref] [PubMed]

1987 (1)

1982 (1)

C. Froehly, B. Colombeau, M. Vampouille, “Shaping and analysis of picosecond light pulses,” Prog. Opt. 20, 63–153 (1982).
[Crossref]

1969 (1)

E. B. Treacy, “Optical pulse compression with diffraction gratings,” IEEE J. Quantum Electron. QE-5454–458 (1969); “Measurement of picosecond pulse substructure using compression techniques,” Appl. Phys. Lett. 14, 112–114 (1969); “Compression of picosecond light pulses,” Phys. Lett. A 28, 34–35 (1968).
[Crossref]

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1980), pp. 375–378.

Caulfield, H. J.

H. J. Caulfield, J. E. Ludman, “Time-frequency interrelationships in optical information processing,” in Transformations in Optical Signal Processing, W. T. Rhodes, ed., Proc. Soc. Photo-Opt. Instrum. Eng.373, 63–67 (1981).

Colombeau, B.

C. Froehly, B. Colombeau, M. Vampouille, “Shaping and analysis of picosecond light pulses,” Prog. Opt. 20, 63–153 (1982).
[Crossref]

Davenport, W. B.

W. B. Davenport, W. L. Root, Random Signals and Noise (McGraw-Hill, New York, 1958), p. 129, Eqs. (8-22) and (8-23).

Feshback, H.

P. M. Morse, H. Feshback, Methods of Theoretical Physics (McGraw-Hill, New York, 1953), Part 1, Sec. 7.3, pp. 834–857.

Froehly, C.

C. Froehly, B. Colombeau, M. Vampouille, “Shaping and analysis of picosecond light pulses,” Prog. Opt. 20, 63–153 (1982).
[Crossref]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), pp. 48–54.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), Sec. 3.5, p. 47, Eq. (3-33).

J. W. Goodman, Statistical Optics (Wiley, New York, 1985), Sec. 2.7.2(c), p. 38.

Lohmann, A. W.

Ludman, J. E.

H. J. Caulfield, J. E. Ludman, “Time-frequency interrelationships in optical information processing,” in Transformations in Optical Signal Processing, W. T. Rhodes, ed., Proc. Soc. Photo-Opt. Instrum. Eng.373, 63–67 (1981).

Marathay, A. S.

A. W. Lohmann, A. S. Marathay, “Globality and speed of optical parallel processor,” Appl. Opt. 28, 3838–3842 (1989).
[Crossref] [PubMed]

A. S. Marathay, Elements of Optical Coherence Theory (Wiley, New York, 1982), pp. 140–156, 174–178.

Martinez, O. E.

O. E. Martinez, “Matrix formalism for pulse compressors,” IEEE J. Quantum Electron. 24, 2530–2536 (1988); “Grating and prism compressors in the case of finite beam size,” J. Opt. Soc. Am. B 3, 929–934 (1986); O. E. Martinez, J. P. Gordon, R. L. Fork, “Negative group-velocity dispersion using refraction,” J. Opt. Soc. Am. A1003–1006 (1984); R. L. Fork, O. E. Martinez, J. P. Gordon, “Negative dispersion using pairs of prisms,” Opt. Lett. 9, 150–152 (1984); J. P. Gordon, R. L. Fork, “Optical resonator with negative dispersion,” Opt. Lett. 9, 153–155 (1984); O. E. Martinez, R. L. Fork, J. P. Gordon, “Theory of passively mode-locked lasers including self-phase modulation and group-velocity dispersion,” Opt. Lett. 9, 156–158 (1984).
[Crossref] [PubMed]

Morse, P. M.

P. M. Morse, H. Feshback, Methods of Theoretical Physics (McGraw-Hill, New York, 1953), Part 1, Sec. 7.3, pp. 834–857.

Prise, M. E.

N. Streibl, M. E. Prise, “Digital optics,” Phys. Status Solidi B 150, 447–454 (1988).
[Crossref]

Root, W. L.

W. B. Davenport, W. L. Root, Random Signals and Noise (McGraw-Hill, New York, 1958), p. 129, Eqs. (8-22) and (8-23).

Shamir, J.

Sommerfeld, A.

A. Sommerfeld, Optics, Vol. 4 of Lectures on Theoretical Physics (Academic, New York, 1964), Vol. 4, pp. 195–207.

Streibl, N.

N. Streibl, M. E. Prise, “Digital optics,” Phys. Status Solidi B 150, 447–454 (1988).
[Crossref]

Treacy, E. B.

E. B. Treacy, “Optical pulse compression with diffraction gratings,” IEEE J. Quantum Electron. QE-5454–458 (1969); “Measurement of picosecond pulse substructure using compression techniques,” Appl. Phys. Lett. 14, 112–114 (1969); “Compression of picosecond light pulses,” Phys. Lett. A 28, 34–35 (1968).
[Crossref]

Vampouille, M.

C. Froehly, B. Colombeau, M. Vampouille, “Shaping and analysis of picosecond light pulses,” Prog. Opt. 20, 63–153 (1982).
[Crossref]

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1980), pp. 375–378.

Appl. Opt. (2)

IEEE J. Quantum Electron. (2)

O. E. Martinez, “Matrix formalism for pulse compressors,” IEEE J. Quantum Electron. 24, 2530–2536 (1988); “Grating and prism compressors in the case of finite beam size,” J. Opt. Soc. Am. B 3, 929–934 (1986); O. E. Martinez, J. P. Gordon, R. L. Fork, “Negative group-velocity dispersion using refraction,” J. Opt. Soc. Am. A1003–1006 (1984); R. L. Fork, O. E. Martinez, J. P. Gordon, “Negative dispersion using pairs of prisms,” Opt. Lett. 9, 150–152 (1984); J. P. Gordon, R. L. Fork, “Optical resonator with negative dispersion,” Opt. Lett. 9, 153–155 (1984); O. E. Martinez, R. L. Fork, J. P. Gordon, “Theory of passively mode-locked lasers including self-phase modulation and group-velocity dispersion,” Opt. Lett. 9, 156–158 (1984).
[Crossref] [PubMed]

E. B. Treacy, “Optical pulse compression with diffraction gratings,” IEEE J. Quantum Electron. QE-5454–458 (1969); “Measurement of picosecond pulse substructure using compression techniques,” Appl. Phys. Lett. 14, 112–114 (1969); “Compression of picosecond light pulses,” Phys. Lett. A 28, 34–35 (1968).
[Crossref]

Phys. Status Solidi B (1)

N. Streibl, M. E. Prise, “Digital optics,” Phys. Status Solidi B 150, 447–454 (1988).
[Crossref]

Prog. Opt. (1)

C. Froehly, B. Colombeau, M. Vampouille, “Shaping and analysis of picosecond light pulses,” Prog. Opt. 20, 63–153 (1982).
[Crossref]

Other (9)

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), pp. 48–54.

A. S. Marathay, Elements of Optical Coherence Theory (Wiley, New York, 1982), pp. 140–156, 174–178.

H. J. Caulfield, J. E. Ludman, “Time-frequency interrelationships in optical information processing,” in Transformations in Optical Signal Processing, W. T. Rhodes, ed., Proc. Soc. Photo-Opt. Instrum. Eng.373, 63–67 (1981).

P. M. Morse, H. Feshback, Methods of Theoretical Physics (McGraw-Hill, New York, 1953), Part 1, Sec. 7.3, pp. 834–857.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1980), pp. 375–378.

A. Sommerfeld, Optics, Vol. 4 of Lectures on Theoretical Physics (Academic, New York, 1964), Vol. 4, pp. 195–207.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), Sec. 3.5, p. 47, Eq. (3-33).

W. B. Davenport, W. L. Root, Random Signals and Noise (McGraw-Hill, New York, 1958), p. 129, Eqs. (8-22) and (8-23).

J. W. Goodman, Statistical Optics (Wiley, New York, 1985), Sec. 2.7.2(c), p. 38.

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Figures (2)

Fig. 1
Fig. 1

Coordinate system and notation used for light propagation in the right-half space, z > 0.

Fig. 2
Fig. 2

Geometry for temporal distortion or stretching of the pulse by the grating in the x′, y′ plane.

Equations (33)

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2 Ψ - 1 c 2 2 Ψ t 2 = 0 ,
Ψ ( r , t ) = 1 2 π d x d y × [ z R 3 Ψ ( r , t ) + z c R 2 Ψ ( r , t ) t ] t = t - ( R / c ) ,
r = i ^ x + j ^ y , r = i ^ x + j ^ y + k ^ z .
R = r - r = [ z 2 + ( x - x ) 2 + ( y - y ) 2 ] 1 / 2 .
Ψ ( r , t ) = exp [ - i 2 π ( ν 0 t - ξ 0 x - η 0 y ) ] × ϕ ( r - r 0 , t - t 0 ) ,
Ψ ( r , t ) = z 2 π d x d y { 1 R 2 exp [ - i 2 π ( ν 0 t - ξ 0 x - η 0 y ) ] × [ ( 1 R - i 2 π λ 0 ) ϕ + 1 c ϕ t ] } .
Ψ ( r , t ) - i z λ 0 d x d y { 1 R 2 exp [ - i 2 π ( ν 0 t - ξ 0 x - η 0 y ) ] × ( ϕ + i λ 0 2 π c ϕ t ) } .
λ 0 2 π c | ϕ t | ϕ .
Ψ ( r , t ) - i z λ 0 d x d y ( 1 R 2 exp { - i 2 π [ ν 0 ( t - R / c ) - ξ 0 x - η 0 y ] } ϕ ( r - r 0 , t - t 0 - R / c ) ) .
r 1 = r - r 0 or x 1 = x - x 0 , y 1 = y - y 0 .
R R 0 + Δ R ,
Δ R R = - [ ( x - x 0 ) x 1 + ( y - y 0 ) y 1 R 0 2 ] + ( x 1 2 2 R 0 2 ) [ 1 + ( x - x 0 ) 2 R 0 2 ] + ( y 1 2 R 0 2 ) [ 1 + ( y - y 0 ) 2 R 0 2 ] + [ ( x - x 0 ) ( y - y 0 ) x 1 y 1 R 0 4 ] .
ϕ ( r - r 0 , t - t 0 ) = ϕ 0 ( r - r 0 , t - t 0 ) a ( x , y ) .
Ψ ( r , t ) - i z λ 0 R 0 2 exp [ - i 2 π ( ν 0 t - ν 0 R 0 / c - ξ 0 x 0 - η 0 y 0 ) ] × d x 1 d y 1 exp { + i 2 π [ ν 0 ( Δ R / c ) + ξ 0 x 1 + η 0 y 1 ] } × a ( x 1 + x 0 , y 1 + y 0 ) × ϕ 0 ( r 1 , t - t 0 - R 0 / c - Δ R / c ) .
ϕ 0 ( r - r 0 , t - t 0 ) = P ( r - r 0 ) S ( t - t 0 ) .
Ψ ( r , t ) = - i z λ 0 R 0 2 exp [ - i 2 π ν 0 ( t - R 0 / c ) ] × exp [ + i 2 π ( ξ 0 x 0 + η 0 y 0 ) ] × exp [ + i 2 π ( ν 0 Δ R / c + ξ 0 x 1 + η 0 y 1 ) ] × P ( x 1 , y 1 ) a ( x 1 + x 0 , y 1 + y 0 ) × S ( t - t 0 - R 0 / c - Δ R / c ) d x 1 d y 1 .
S ( τ 0 - Δ R / c ) S ( τ 0 ) - Δ R c S τ 0 ,
| Δ R c S t | S .
Ψ ( r , t ) = - i z λ 0 R 0 2 exp [ - i 2 π ν 0 ( t - R 0 / c ) ] × exp [ i 2 π ( ξ 0 x 0 + η 0 y 0 ) ] S ( t - t 0 - R 0 / c ) × d x 1 d y 1 { P ( x 1 , y 1 ) a ( x 1 + x 0 , y 1 + y 0 ) × exp [ i 2 π ( ν 0 Δ R / c + ξ 0 x 1 + η 0 y 1 ) ] } .
P ( x 1 , y 1 ) = exp ( - x 1 2 2 σ x 2 - y 1 2 2 σ y 2 )
S ( t - t 0 - R / c ) = exp ( - t - t 0 - R / c 2 σ t 2 ) .
a ( x 1 + x 0 , y 1 + y 0 ) = exp [ - ( x 1 + x 0 ) 2 2 σ a 2 - ( y 1 + y 0 ) 2 2 σ b 2 ] .
Ψ ( r , t ) = ( - i z λ 0 R 0 2 ) exp [ - i 2 π ( ν 0 t - ν 0 R 0 / c - ξ 0 x 0 - η 0 y 0 ) ] × a ( x 0 , y 0 ) S ( t - t 0 - R 0 / c ) [ π ( C A - B 2 ) 1 / 2 ] × exp [ - π 2 ( C ξ 2 - 2 B ξ η + A η 2 ) ( C A - B 2 ) ] .
d 0 c ( t - t 0 ) - R 0 σ d c σ t k 0 2 π / λ 0 ξ ξ 0 + i x 0 2 π σ a 2 - K ( x - x 0 ) 2 π R 0 η η 0 + i y 0 2 π σ b 2 - K ( y - y 0 ) 2 π R 0 A 1 2 σ x 2 + 1 2 σ a 2 - ( x - x 0 ) 2 2 σ d 2 R 0 2 + i K 2 R 0 [ 1 + ( x - x 0 ) 2 R 0 2 ] C 1 2 σ y 2 + 1 2 σ b 2 - ( y - y 0 ) 2 2 σ a 2 R 0 2 + i K 2 R 0 [ 1 + ( y - y 0 ) 2 R 0 2 ] B ( 1 σ d 2 - i K R 0 ) ( x - x 0 ) ( y - y 0 ) 2 R 0 2 , K k 0 ( 1 - i d 0 k 0 σ d 2 ) + 1 k 0 σ d 2 .
d 0 2 π σ d 2 λ 0 = k 0 σ d 2 .
K k 0 + 1 k 0 σ d 2 ,
d 0 4 π ( ν 0 Δ ν ) 2 λ 0 = 2 π σ d 2 λ 0 .
a ( x , y ) = n = - N n = N m = - m m = M rect [ ( x - n p x ) Δ ] rect [ ( y - m p y ) Δ ] ,
Ψ ( r , t ) = - i z Δ a λ 0 r 2 P 0 exp ( - i 2 π ν 0 t ) × n = - N n = N m = - M m = M S ( t - t 0 - R n m c ) exp ( i 2 π ν 0 R n m c ) ,
T D = ( R + N - R - N ) c .
T D = ( 2 N p x x ) c r .
sin α = λ 0 p x = x r ,
T D = 2 N ν 0 .

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