Abstract

An analytical expression for the time evolution of the diffraction pattern of an ultrashort laser pulse passing through a circular aperture is obtained in the Fresnel regime. The diffraction is not constant in time as the pulse travels through the aperture. This may have implications in experiments involving fast dynamics. Examples of the evolution of the diffraction pattern are given.

© 2003 Optical Society of America

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References

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Acta Phys. Sin.

H. Quan-Sheng and Zhu-Zhen-he,�??Diffraction of ultrashort pulses,�?? Acta Phys. Sin. 37, 1432-1437 (1988).

Appl. Opt.

J. Mod. Opt.

C. F. R. Caron and R. M. Potvliege, "Free-space propagation of ultrashort pulses: space-time couplings in Gaussian pulse beams," J. Mod. Opt. 46, 1881-1891 (1999).
[CrossRef]

J. Opt Soc. Am. A

M. Kempe and W. Rudolph, ,�??Analysis of confocal microscopy under ultrashort light pulse illumination,�??J. Opt Soc. Am. A 10, 240-245 (1993).
[CrossRef]

J. Opt. Soc. Am. A

J. Opt. Soc. Am. B

Opt. Lett.

Other

G. N. Watson, A treatise on the theory of Bessel functions (Cambridge University Press, Cambridge, 1966).

W. Goodman, Introduction to Fourier optics (Mc Graw-Hill, NY, 1968).

M. Born and E. Wolf, Principles of Optics (Pergamon, NY, 1980)(p.437-439, chap. 8.8.1).

Supplementary Material (4)

» Media 1: AVI (234 KB)     
» Media 2: AVI (140 KB)     
» Media 3: AVI (118 KB)     
» Media 4: AVI (281 KB)     

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Figures (2)

Fig. 1.
Fig. 1.

Time evolution diffraction of a short pulse. Parameters are: spectrum width is 6.1012 rad/s (about THz), which corresponds to a pulse duration of about 1 ps, the frequency shift is 1014 rad/s (about 16 THz or 20 µm). The Fresnel number is equal to 0.04 and the parameter s=1. (a) to (d) corresponds to the following times t=-5, 0, 0.1, 0.5 ps, respectively. (e) [239 KB] film showing the unnormalised 2D pattern and (f) [120 KB] film showing the profiles.

Fig. 2.
Fig. 2.

Time evolution of the diffraction pattern with the same parameters as in Fig. 1 except for the Fresnel number, which here is 0.17. (a) to (f) correspond to the following times t=0, 0.2, 0.7, 0.8, 1, and 1.5 ps. (g) [143 KB] Film of the unnormalised 2D pattern, and (h) [287 KB] film showing the profiles.

Equations (21)

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E ˜ ( ρ , z , λ ) = 2 i π a 2 λ z e i 2 π z λ e i N ρ 2 × 0 1 E ˜ 1 ( ρ 1 , λ ) . J 0 ( 2 N ρ 1 ρ ) ρ 1 e i N ρ 1 2 d ρ 1
E ˜ 1 ( ρ 1 , ω ) = f s ( ω ) = ( s + 1 ω 1 ) s + 1 ( ω ω 0 ) s Γ ( s + 1 ) e ( s + 1 ) ω ω 0 ω 1 if ω ω 0 , 0 else ( plane wave )
0 1 J 0 ( 2 N ρ 1 ρ ) ρ 1 e i N ρ 1 2 d ρ 1 = e i N 2 N ( U 1 ( 2 N , 2 N ρ ) + i U 2 ( 2 N , 2 N ρ ) ) .
U n ( u , v ) = p = 0 ( 1 ) p ( u v ) n + 2 p J n + 2 p ( v )
E ˜ ( ρ , z , ω ) = i a 2 ω 2 c z e i ω c z e i a 2 ω 2 c z ρ 2 ( s + 1 ω 1 ) s + 1 ( ω ω 0 ) s Γ ( s + 1 ) ×
e ( s + 1 ) ω ω 0 ω 1 e i a 2 ω 2 c z 2 c z a 2 ω ( U 1 ( 2 N , 2 N ρ ) + i U 2 ( 2 N , 2 N ρ ) )
= i α s + 1 ( ω ω 0 ) s Γ ( s + 1 ) e α ( ω ω 0 ) e i ω β [ U 1 ( a 2 ω cz , a 2 ωρ cz ) + iU 2 ( a 2 ω cz , a 2 ω ρ cz ) ] ,
E j ( ρ , z , t ) = i Γ ( s + 1 ) α s + 1 e i ω 0 ( β t ) ω 0 e ( ω ω 0 ) ( α + i ( β t ) ) ( ω ω 0 ) s × U j ( a 2 ω c z , a 2 ω ρ c z ) d ω .
E j ( ρ , z , t ) = K e i ω 0 t 0 e ω ( α + i ( β t ) ) ω s × U j ( a 2 c z ( ω + ω 0 ) , a 2 ρ c z ( ω + ω 0 ) ) d ω ,
where K = i Γ ( s + 1 ) α s + 1 e i ω 0 β .
E j ( ρ , z , t ) = K p = 0 ( 1 ) p ρ j 2 p 0 J j + 2 p ( a 2 ρ c z ( ω + ω 0 ) ) ω s e ω ( α + i ( β t ) ) d ω .
J v ( a + b ) = m = ( 1 ) m J v + m ( a ) × J m ( b )
J v ( a + b ) = m = 0 ( 1 ) m J v + m ( a ) × J m ( b ) + m = 1 v J v m ( a ) × J m ( b ) + m = v + 1 ( 1 ) m v J m v ( a ) × J m ( b )
E j ( ρ , z , t ) = K p = 0 m = ( 1 ) p + m ρ j 2 p × J m + j + 2 p ( ρ N 0 ) × 0 J m ( ρ a 2 ω c z ) ω s e ω ( α + i ( β t ) ) d ω
0 J v ( b ω ) ω s e a ω d ω = k = 0 ( 1 ) k ( b 2 ) v + 2 k k ! Γ ( v + k + 1 ) × Γ ( s + 1 + v + 2 k ) a s + 1 + v + 2 k
E j ( ρ , z , t ) = K e i ω 0 t p = 0 m = k = 0 ( 1 ) p + m + k k ! ρ j 2 p × J m + j + 2 p ( ρ N 0 ) ×
( ρ a 2 2 c z ) m + 2 k Γ ( s + 1 + m + 2 k ) Γ ( k + m + 1 ) ( α + i ( β t ) ) ( s + 1 + m + 2 k )
E j ( ρ , z , t ) =
K e i ω 0 t p = 0 k = 0 m = 0 ( 1 ) p + m + k k ! ρ j 2 p × J m + j + 2 p ( ρ N 0 ) × ( ρ a 2 2 c z ) m + 2 k Γ ( s + 1 + m + 2 k ) Γ ( k + m + 1 ) ( α + i ( β t ) ) ( s + 1 + m + 2 k ) +
K e i ω 0 t p = 0 k = 0 m = 1 j + 2 p ( 1 ) p + k k ! ρ j 2 p × J j + 2 p m ( ρ N 0 ) × ( ρ a 2 2 c z ) m + 2 k Γ ( s + 1 + m + 2 k ) Γ ( k + m + 1 ) ( α + i ( β t ) ) ( s + 1 + m + 2 k ) +
K e i ω 0 t p = 0 k = 0 m = j + 2 p + 1 ( 1 ) p + m + k j k ! ρ j 2 p × J m j 2 p ( ρ N 0 ) × ( ρ a 2 2 c z ) m + 2 k Γ ( s + 1 + m + 2 k ) Γ ( k + m + 1 ) ( α + i ( β t ) ) ( s + 1 + m + 2 k )

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