Abstract

Recently, the classical Talbot effect (self-imaging of optical wave fields) has attracted a renewed interest, as the concept has been generalized to the domain of pulsed wave fields by several authors. In this paper we discuss the self-imaging of three-dimensional images. We construct pulsed wave fields that can be used as self-imaging “pixels” of a three-dimensional image and show that their superpositions reproduce the spatial separated copies of its initial three-dimensional intensity distribution at specific time intervals. The derived wave fields will be shown to be directly related to the fundamental localized wave solutions of the homogeneous scalar wave equation – focus wave modes. Our discussion is illustrated by some spectacular numerical simulations. We also propose a general idea for the optical generation of the derived wave fields. The results will be compared to the work, published so far on the subject.

© 2002 Optical Society of America

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References

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J. Mod. Opt. (1)

Z. Bouchal and M. Bertolotti, �Self-reconstruction of wave packets due to spatio-temporal couplings,� J. Mod. Opt. 47, 1455 - 1467 (2000).
[CrossRef]

J. Opt. A: Pure Appl. Opt. (1)

J. Salo and M. M. Salomaa, �Diffraction-free pulses at arbitrary speeds,� J. Opt. A: Pure Appl. Opt. 3, 366-373 (2001).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (3)

Laser Phys. (1)

P. Saari, H. S�najalg, �Pulsed Bessel beams,� Laser Phys. 7, 32-39 (1997).

Microwave and Opt. Technol. Lett. (1)

H. Wang, C. Zhou, L. Jianlang and L. Liu, �Talbot effect of a grating under ultrashort pulsed-laser illumination,� Microwave and Opt. Technol. Lett. 25, 184-187 (2000)
[CrossRef]

Opt. Commun. (3)

Zs. Bor and B. R�cz, �Group velocity dispersion in prisms and its application to pulse compression and traveling-wave excitation,� Opt. Commun. 54, 165-170 (1985).
[CrossRef]

Z. Bouchal and J. Wagner, �Self-reconstruction effect in free propagation of wavefield,� Opt. Commun. 176, 299-307 (2000).
[CrossRef]

P. Saari, J. Aaviksoo, A. Freiberg, K. Timpmann, �Elimination of excess pulse broadening at high spectral resolution of picosecond duration light emission,� Opt. Commun. 39, 94-98 (1981).
[CrossRef]

Opt. Lett. (3)

Opt.Commun. (1)

J. Wagner and Z. Bouchal, �Experimental realization of self-reconstruction of the 2D aperiodic objects,� Opt.Commun. 176, 309-311 (2000).
[CrossRef]

Phys. Rev. E (1)

K. Reivelt and P. Saari, �Optically realizable localized wave solutions of homogeneous scalar wave equation,� Phys. Rev. E (to be published) [accepted for publication].

Phys. Rev. Lett. (3)

P. Saari and K. Reivelt, �Evidence of X-shaped propagation-invariant localized light waves,� Phys. Rev. Lett. 79, 4135-4138 (1997).
[CrossRef]

J. Durnin, J. J. Miceli, Jr., and J. H. Eberly, �Diffraction-free beams,� Phys. Rev. Lett. 58, 1499�1501 (1987).
[CrossRef] [PubMed]

P. Di Trapani, D. Caironi, G. Valiulis, A. Dubietis, R. Danielius and A. Piskarskas, �Observation of Temporal Solitons in Second-Harmonic Generation with Tilted Pulses,� Phys. Rev. Lett. 81, 570-573 (1998).
[CrossRef]

Proc. SPIE (1)

Z. Bouchal, �Self-reconstruction ability of wave field,� Proc. SPIE, vol. 4356, 217-224, (2001).
[CrossRef]

Progr. In Electromagn. Research (1)

I. Besieris, M. Abdel-Rahman, A. Shaarawi, and A. Chatzipetros, �Two fundamental representations of localized pulse solutions of the scalar wave equation,� Progr. In Electromagn. Research 19, 1 (1998).
[CrossRef]

Pure Appl. Opt. (1)

J. Turunen and A. T. Friberg, �Self-imaging and propagation-invariance in electromagnetic fields,� Pure Appl. Opt. 2, 51-60 (1993).
[CrossRef]

Other (1)

O. Svelto, Principles of Lasers (3rd ed. Plenum Press 1989).

Supplementary Material (3)

» Media 1: MOV (853 KB)     
» Media 2: MOV (1223 KB)     
» Media 3: MOV (1387 KB)     

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Figures (7)

Fig. 1.
Fig. 1.

(0.85 MB) The video clip of the temporal evolution of the superposition of two interfering optical pulses (red and cyanide line) that have equal carrier wavelengths and group velocities but different phase velocities. The blue line and the dotted line denote the amplitude and the envelope of the sum of the two waves respectively.

Fig. 2.
Fig. 2.

(a) The temporal evolution of the spatial intensity distributions of the pulsed self-imaging wave field (the 1.2 MB movie); (b) The spatial amplitude of a monochromatic self-imaging wave field (see also text in Sec. 4).

Fig. 3.
Fig. 3.

(a) An excerpt of the spatial amplitude distribution of a tilted pulse. vg and vp denote its group and phase velocities respectively, k0 and λ0 are the wave vector and the wavelength of the plane wave of the carrier wave number, t is an arbitrary time and ϕ is the tilt angle of the pulse; (b) On the phase and group velocities of the tilted pulses (see text).

Fig. 4.
Fig. 4.

The examples of the supports of the angular spectrums of plane waves of (a) the specified tilted pulse for β= 40 rad m-1 and γ= 1 (k0 is the wave vector of the carrier wavelength), (b) the self-imaging superposition of the tilted pulses (see text). The “rainbow” on the pictures denotes the visible spectral region.

Fig. 5.
Fig. 5.

(a) The Fourier spectrum and (b) the spatial amplitude of a train of sinusoidal waves.

Fig. 6.
Fig. 6.

The instantaneous intensity distribution of a FWM if β= 40 rad m-1, θ (T)(k0 ,β) = 0.22 deg, γ= 1 and the frequency spectrum A(k) has a rectangular shape and extends over the wavelengths 400–800nm.

Fig. 7.
Fig. 7.

(1.4 MB movie) A numerical example of the evolution of a self-imaging spatial image (smiling human face) consisting of eight self-imaging pixels, each consisting of five FWM’s of different constant β(see text).

Equations (38)

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2 Ψ 1 c 2 2 Ψ t 2 = 0 ,
Ψ x y z t = exp [ iωt ] 0 π sin θ 0 2 π dϕA k θ ϕ
× exp [ ik ( x cos ϕ si n θ + y sin ϕ sin θ + z cos θ ) ]
Ψ x y z t = Ψ ( x , y , z + d , t ) ,
kd cos θ = ψ + 2 πq ,
k z = k cos θ = ψ + 2 πq d ,
Ψ ρ z t = exp [ iωt ] n c n exp [ inϕ ] 0 π sin θ A k θ ,
× J n ( sin θ ) exp [ ikz cos θ ]
Ψ ρ z t = exp [ iωt ] q a q J 0 [ 1 ( 2 πq d ) 2 ] exp [ i 2 πq d z ] .
Ψ 0 z t = exp [ iωt ] q a q exp [ i 2 πq d z ] ,
Ψ ( T ) x z t = 0 dkA ( k ) exp [ ik ( x sin θ ( k ) + z cos θ ( k ) ct ) ] .
θ ( T ) ( k , β ) = arccos [ γ ( k 2 β ) k ] ,
v g ( k ) = ( dk z ) 1 ,
k z = k cos θ ( T ) k β = γk 2 βγ
v g ( k ) = c γ ,
v p ( β ) = c cos θ ( T ) ( k 0 , β ) = ck 0 γ ( k 0 2 β ) ,
Ψ ( SI ) x z t = q a q 0 dkA ( k ) exp [ ik ( x sin θ ( T ) k β q + z cos θ ( T ) k β q ct ) ] ,
k x k β = k x k 0 β + dk x k β dk k 0 ( k k 0 )
k z k β = k z k 0 β + dk z k β dk k 0 ( k k 0 )
= k z k 0 β + γ ( k k 0 )
Ψ ( SI ) x z t q a q C q x ct exp ik 0 ( x sin θ ( T ) k 0 β q + z cos θ ( T ) k 0 β q ct ) ,
C q ( x , ct ) = k 0 dkA ( k + k 0 ) exp [ ik ( x dk x k β q dk k 0 + ct ) ]
Ψ ( SI ) x z t C x ct q a q exp ik 0 ( x sin θ ( T ) k 0 β q + z cos θ ( T ) ( k 0 , β 0 ) ct ) .
kd cos θ ( T ) ( k 0 , β q ) = ψ + 2 πq .
2 πq d = γk 0 2 γβ ,
β q = k 0 2 πq γd
θ ( T ) k β q = arccos ( γ ( k k 0 ) k + 2 π kd q ) ,
θ ( T ) k 0 β q = arccos ( 2 π k 0 d q ) .
0 < q < k 0 d 2 π
Ψ ( z ) = A 0 sin [ ½ ( 2 n + 1 ) Δ k z z ] sin ( ½ Δ k z z ) ,
Ψ ρ z t = 0 dkA ( k ) J 0 [ sin θ ( k ) ] exp [ ik ( z cos θ ( k ) ct ) ] .
Ψ ( SI ) ρ z t = q a q 0 dkA ( k ) ,
× J 0 [ sin θ ( T ) k β q ] exp [ ik ( z cos θ ( T ) k β q ct ) ]
Ψ ( SI ) ρ z t = q a q exp [ i 2 γ β q z ] 0 dkA ( k ) .
× J 0 [ 1 ( γ ( k 2 β q ) k ) 2 ] exp [ ik ( ct ) ]
k z , q = γ ( k k 0 ) + 2 π d q ,
k z , q = γk + 2 π d q ,
U ' ν ψ ω = U ν ψ ω m = M M n = 0 N δ ( ω ω m ) δ ( ν ν mn ) ,

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