Abstract

We show that second-harmonic cross correlation of the optical field that propagates from an illuminated object with the illuminating field is a process that shares many features with holography. The wave fronts of the double-frequency field are able to reconstruct holographic images of the object whose dimensions and position can be calculated by use of the holographic formalism. Our theory fits the experimental results obtained by use of a Q-switched Nd:YAG laser as the illuminating source and a thin β-barium borate crystal in a noncollinear type I phase-matching geometry for cross correlation.

© 2000 Optical Society of America

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  1. D. Gabor, “Microscopy by reconstructed wavefronts,” Proc. R. Soc. London Ser. A 197, 454–468 (1949).
    [CrossRef]
  2. I. P. Woerdman, “Formation of transient free carrier hologram in Si,” Opt. Commun. 2, 212–217 (1970).
    [CrossRef]
  3. B. I. Stepanov, E. V. Ivakin, and A. S. Rubanov, “On recording flat and volume dynamical holograms in bleaching materials,” Dokl. Akad. Nauk SSSR 196, 567–570 (1971).
  4. S. N. Shostko, Ya. G. Podoba, Yu. A. Ananiev, B. D. Volosov, and A. M. Gorlanov, “On one possibility of the compensation of optical inhomogeneities in laser devices (in Russian),” Pis'ma Zh. Tekh. Fiz. 5, 29–31 (1979).
  5. A. M. Gorlanov, N. I. Grishmanova, N. A. Sventsitskaya, and V. D. Solov’yov, “Angular characteristics of radiation from neodymium laser with wavefront conjugation under three-wave parametric interaction (in Russian),” Kvant. Elektron. (Moscow) 5, 415–417 (1982).
  6. D. M. Pepper and A. Yariv, “Optical phase conjugation using three-wave and four-wave mixing via elastic photon scattering in transparent media,” in Optical Phase Conjugation, R. A. Fisher, ed. (Academic, New York, 1983), p. 23.
  7. B. Zel’dovich, “Wave front conjugation by three-wave mixing,” in Wave Front Conjugation (Science, Moscow, 1985), p. 194.
  8. Yu. N. Denisyuk, “On reflection of the optical properties of an object in the wave-field scattered by it,” Opt. Spectrosc. (USSR) 15, 522–532 (1963).
  9. A. Andreoni, M. Bondani, M. A. C. Potenza, and F. Villani, “Phase selection of image-bearing field components by frequency up-conversion in nonlinear crystals,” J. Nonlinear Opt. Phys. Mater. 8, 55–77 (1999).
    [CrossRef]
  10. A. Andreoni, M. Bondani, M. A. C. Potenza, and F. Villani, “Relevance of temporal coherence in the second-harmonic cross-correlation measurement of a multiply scattered laser pulse,” Eur. Phys. J. D 8, 111–116 (2000).
    [CrossRef]
  11. Yu. N. Denisyuk, A. Andreoni, and M. A. C. Potenza, “Holographic properties of the effect of second-order harmonic cross-correlation of optical wavefields,” Opt. Mem. Neural Netw. 8, 130–140 (1999).
  12. A. Andreoni, M. Bondani, and M. A. C. Potenza, “Ultra-broadband and chirp-free frequency doubling in β-barium borate,” Opt. Commun. 154, 376–382 (1998).
    [CrossRef]

2000 (1)

A. Andreoni, M. Bondani, M. A. C. Potenza, and F. Villani, “Relevance of temporal coherence in the second-harmonic cross-correlation measurement of a multiply scattered laser pulse,” Eur. Phys. J. D 8, 111–116 (2000).
[CrossRef]

1999 (2)

Yu. N. Denisyuk, A. Andreoni, and M. A. C. Potenza, “Holographic properties of the effect of second-order harmonic cross-correlation of optical wavefields,” Opt. Mem. Neural Netw. 8, 130–140 (1999).

A. Andreoni, M. Bondani, M. A. C. Potenza, and F. Villani, “Phase selection of image-bearing field components by frequency up-conversion in nonlinear crystals,” J. Nonlinear Opt. Phys. Mater. 8, 55–77 (1999).
[CrossRef]

1998 (1)

A. Andreoni, M. Bondani, and M. A. C. Potenza, “Ultra-broadband and chirp-free frequency doubling in β-barium borate,” Opt. Commun. 154, 376–382 (1998).
[CrossRef]

1982 (1)

A. M. Gorlanov, N. I. Grishmanova, N. A. Sventsitskaya, and V. D. Solov’yov, “Angular characteristics of radiation from neodymium laser with wavefront conjugation under three-wave parametric interaction (in Russian),” Kvant. Elektron. (Moscow) 5, 415–417 (1982).

1979 (1)

S. N. Shostko, Ya. G. Podoba, Yu. A. Ananiev, B. D. Volosov, and A. M. Gorlanov, “On one possibility of the compensation of optical inhomogeneities in laser devices (in Russian),” Pis'ma Zh. Tekh. Fiz. 5, 29–31 (1979).

1971 (1)

B. I. Stepanov, E. V. Ivakin, and A. S. Rubanov, “On recording flat and volume dynamical holograms in bleaching materials,” Dokl. Akad. Nauk SSSR 196, 567–570 (1971).

1970 (1)

I. P. Woerdman, “Formation of transient free carrier hologram in Si,” Opt. Commun. 2, 212–217 (1970).
[CrossRef]

1963 (1)

Yu. N. Denisyuk, “On reflection of the optical properties of an object in the wave-field scattered by it,” Opt. Spectrosc. (USSR) 15, 522–532 (1963).

1949 (1)

D. Gabor, “Microscopy by reconstructed wavefronts,” Proc. R. Soc. London Ser. A 197, 454–468 (1949).
[CrossRef]

Ananiev, Yu. A.

S. N. Shostko, Ya. G. Podoba, Yu. A. Ananiev, B. D. Volosov, and A. M. Gorlanov, “On one possibility of the compensation of optical inhomogeneities in laser devices (in Russian),” Pis'ma Zh. Tekh. Fiz. 5, 29–31 (1979).

Andreoni, A.

A. Andreoni, M. Bondani, M. A. C. Potenza, and F. Villani, “Relevance of temporal coherence in the second-harmonic cross-correlation measurement of a multiply scattered laser pulse,” Eur. Phys. J. D 8, 111–116 (2000).
[CrossRef]

Yu. N. Denisyuk, A. Andreoni, and M. A. C. Potenza, “Holographic properties of the effect of second-order harmonic cross-correlation of optical wavefields,” Opt. Mem. Neural Netw. 8, 130–140 (1999).

A. Andreoni, M. Bondani, M. A. C. Potenza, and F. Villani, “Phase selection of image-bearing field components by frequency up-conversion in nonlinear crystals,” J. Nonlinear Opt. Phys. Mater. 8, 55–77 (1999).
[CrossRef]

A. Andreoni, M. Bondani, and M. A. C. Potenza, “Ultra-broadband and chirp-free frequency doubling in β-barium borate,” Opt. Commun. 154, 376–382 (1998).
[CrossRef]

Bondani, M.

A. Andreoni, M. Bondani, M. A. C. Potenza, and F. Villani, “Relevance of temporal coherence in the second-harmonic cross-correlation measurement of a multiply scattered laser pulse,” Eur. Phys. J. D 8, 111–116 (2000).
[CrossRef]

A. Andreoni, M. Bondani, M. A. C. Potenza, and F. Villani, “Phase selection of image-bearing field components by frequency up-conversion in nonlinear crystals,” J. Nonlinear Opt. Phys. Mater. 8, 55–77 (1999).
[CrossRef]

A. Andreoni, M. Bondani, and M. A. C. Potenza, “Ultra-broadband and chirp-free frequency doubling in β-barium borate,” Opt. Commun. 154, 376–382 (1998).
[CrossRef]

Denisyuk, Yu. N.

Yu. N. Denisyuk, A. Andreoni, and M. A. C. Potenza, “Holographic properties of the effect of second-order harmonic cross-correlation of optical wavefields,” Opt. Mem. Neural Netw. 8, 130–140 (1999).

Yu. N. Denisyuk, “On reflection of the optical properties of an object in the wave-field scattered by it,” Opt. Spectrosc. (USSR) 15, 522–532 (1963).

Gabor, D.

D. Gabor, “Microscopy by reconstructed wavefronts,” Proc. R. Soc. London Ser. A 197, 454–468 (1949).
[CrossRef]

Gorlanov, A. M.

A. M. Gorlanov, N. I. Grishmanova, N. A. Sventsitskaya, and V. D. Solov’yov, “Angular characteristics of radiation from neodymium laser with wavefront conjugation under three-wave parametric interaction (in Russian),” Kvant. Elektron. (Moscow) 5, 415–417 (1982).

S. N. Shostko, Ya. G. Podoba, Yu. A. Ananiev, B. D. Volosov, and A. M. Gorlanov, “On one possibility of the compensation of optical inhomogeneities in laser devices (in Russian),” Pis'ma Zh. Tekh. Fiz. 5, 29–31 (1979).

Grishmanova, N. I.

A. M. Gorlanov, N. I. Grishmanova, N. A. Sventsitskaya, and V. D. Solov’yov, “Angular characteristics of radiation from neodymium laser with wavefront conjugation under three-wave parametric interaction (in Russian),” Kvant. Elektron. (Moscow) 5, 415–417 (1982).

Ivakin, E. V.

B. I. Stepanov, E. V. Ivakin, and A. S. Rubanov, “On recording flat and volume dynamical holograms in bleaching materials,” Dokl. Akad. Nauk SSSR 196, 567–570 (1971).

Podoba, Ya. G.

S. N. Shostko, Ya. G. Podoba, Yu. A. Ananiev, B. D. Volosov, and A. M. Gorlanov, “On one possibility of the compensation of optical inhomogeneities in laser devices (in Russian),” Pis'ma Zh. Tekh. Fiz. 5, 29–31 (1979).

Potenza, M. A. C.

A. Andreoni, M. Bondani, M. A. C. Potenza, and F. Villani, “Relevance of temporal coherence in the second-harmonic cross-correlation measurement of a multiply scattered laser pulse,” Eur. Phys. J. D 8, 111–116 (2000).
[CrossRef]

Yu. N. Denisyuk, A. Andreoni, and M. A. C. Potenza, “Holographic properties of the effect of second-order harmonic cross-correlation of optical wavefields,” Opt. Mem. Neural Netw. 8, 130–140 (1999).

A. Andreoni, M. Bondani, M. A. C. Potenza, and F. Villani, “Phase selection of image-bearing field components by frequency up-conversion in nonlinear crystals,” J. Nonlinear Opt. Phys. Mater. 8, 55–77 (1999).
[CrossRef]

A. Andreoni, M. Bondani, and M. A. C. Potenza, “Ultra-broadband and chirp-free frequency doubling in β-barium borate,” Opt. Commun. 154, 376–382 (1998).
[CrossRef]

Rubanov, A. S.

B. I. Stepanov, E. V. Ivakin, and A. S. Rubanov, “On recording flat and volume dynamical holograms in bleaching materials,” Dokl. Akad. Nauk SSSR 196, 567–570 (1971).

Shostko, S. N.

S. N. Shostko, Ya. G. Podoba, Yu. A. Ananiev, B. D. Volosov, and A. M. Gorlanov, “On one possibility of the compensation of optical inhomogeneities in laser devices (in Russian),” Pis'ma Zh. Tekh. Fiz. 5, 29–31 (1979).

Solov’yov, V. D.

A. M. Gorlanov, N. I. Grishmanova, N. A. Sventsitskaya, and V. D. Solov’yov, “Angular characteristics of radiation from neodymium laser with wavefront conjugation under three-wave parametric interaction (in Russian),” Kvant. Elektron. (Moscow) 5, 415–417 (1982).

Stepanov, B. I.

B. I. Stepanov, E. V. Ivakin, and A. S. Rubanov, “On recording flat and volume dynamical holograms in bleaching materials,” Dokl. Akad. Nauk SSSR 196, 567–570 (1971).

Sventsitskaya, N. A.

A. M. Gorlanov, N. I. Grishmanova, N. A. Sventsitskaya, and V. D. Solov’yov, “Angular characteristics of radiation from neodymium laser with wavefront conjugation under three-wave parametric interaction (in Russian),” Kvant. Elektron. (Moscow) 5, 415–417 (1982).

Villani, F.

A. Andreoni, M. Bondani, M. A. C. Potenza, and F. Villani, “Relevance of temporal coherence in the second-harmonic cross-correlation measurement of a multiply scattered laser pulse,” Eur. Phys. J. D 8, 111–116 (2000).
[CrossRef]

A. Andreoni, M. Bondani, M. A. C. Potenza, and F. Villani, “Phase selection of image-bearing field components by frequency up-conversion in nonlinear crystals,” J. Nonlinear Opt. Phys. Mater. 8, 55–77 (1999).
[CrossRef]

Volosov, B. D.

S. N. Shostko, Ya. G. Podoba, Yu. A. Ananiev, B. D. Volosov, and A. M. Gorlanov, “On one possibility of the compensation of optical inhomogeneities in laser devices (in Russian),” Pis'ma Zh. Tekh. Fiz. 5, 29–31 (1979).

Woerdman, I. P.

I. P. Woerdman, “Formation of transient free carrier hologram in Si,” Opt. Commun. 2, 212–217 (1970).
[CrossRef]

Dokl. Akad. Nauk SSSR (1)

B. I. Stepanov, E. V. Ivakin, and A. S. Rubanov, “On recording flat and volume dynamical holograms in bleaching materials,” Dokl. Akad. Nauk SSSR 196, 567–570 (1971).

Eur. Phys. J. D (1)

A. Andreoni, M. Bondani, M. A. C. Potenza, and F. Villani, “Relevance of temporal coherence in the second-harmonic cross-correlation measurement of a multiply scattered laser pulse,” Eur. Phys. J. D 8, 111–116 (2000).
[CrossRef]

J. Nonlinear Opt. Phys. Mater. (1)

A. Andreoni, M. Bondani, M. A. C. Potenza, and F. Villani, “Phase selection of image-bearing field components by frequency up-conversion in nonlinear crystals,” J. Nonlinear Opt. Phys. Mater. 8, 55–77 (1999).
[CrossRef]

Kvant. Elektron. (Moscow) (1)

A. M. Gorlanov, N. I. Grishmanova, N. A. Sventsitskaya, and V. D. Solov’yov, “Angular characteristics of radiation from neodymium laser with wavefront conjugation under three-wave parametric interaction (in Russian),” Kvant. Elektron. (Moscow) 5, 415–417 (1982).

Opt. Commun. (2)

I. P. Woerdman, “Formation of transient free carrier hologram in Si,” Opt. Commun. 2, 212–217 (1970).
[CrossRef]

A. Andreoni, M. Bondani, and M. A. C. Potenza, “Ultra-broadband and chirp-free frequency doubling in β-barium borate,” Opt. Commun. 154, 376–382 (1998).
[CrossRef]

Opt. Mem. Neural Netw. (1)

Yu. N. Denisyuk, A. Andreoni, and M. A. C. Potenza, “Holographic properties of the effect of second-order harmonic cross-correlation of optical wavefields,” Opt. Mem. Neural Netw. 8, 130–140 (1999).

Opt. Spectrosc. (USSR) (1)

Yu. N. Denisyuk, “On reflection of the optical properties of an object in the wave-field scattered by it,” Opt. Spectrosc. (USSR) 15, 522–532 (1963).

Pis'ma Zh. Tekh. Fiz. (1)

S. N. Shostko, Ya. G. Podoba, Yu. A. Ananiev, B. D. Volosov, and A. M. Gorlanov, “On one possibility of the compensation of optical inhomogeneities in laser devices (in Russian),” Pis'ma Zh. Tekh. Fiz. 5, 29–31 (1979).

Proc. R. Soc. London Ser. A (1)

D. Gabor, “Microscopy by reconstructed wavefronts,” Proc. R. Soc. London Ser. A 197, 454–468 (1949).
[CrossRef]

Other (2)

D. M. Pepper and A. Yariv, “Optical phase conjugation using three-wave and four-wave mixing via elastic photon scattering in transparent media,” in Optical Phase Conjugation, R. A. Fisher, ed. (Academic, New York, 1983), p. 23.

B. Zel’dovich, “Wave front conjugation by three-wave mixing,” in Wave Front Conjugation (Science, Moscow, 1985), p. 194.

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Figures (7)

Fig. 1
Fig. 1

Optical scheme of a SHG hologram: O is the object, R is the reference source, and S is an arbitrary flat section inside the volume of a SHG hologram, H. The wave fronts EO and ER of the object and the reference fields at frequency ω are sketched. The fields E2ω, E2ω, E2ω at frequency 2ω are those generated by the SHG hologram, among which E2ω is the reconstructed object wave.

Fig. 2
Fig. 2

Geometrical construction of the wave front WH generated by a SHG hologram in the case in which both object and reference wave fronts, WO and WR, are planar. See the text for the relations among iconals (LH, LO, and LR) and among wave vectors (kH, kO, and kR).

Fig. 3
Fig. 3

As in Fig. 2, the case in which WO is arbitrary and WR is spherical. The object wave front WO (iconal LO) is emitted by the object, O1; O2 is the intermediate image (iconal LO/2; see text); O3 is the final image reconstructed by a SHG hologram [iconal (LO+LR)/2; see text, Eq. (7)].

Fig. 4
Fig. 4

Scheme of the experimental setup for recording a SHG hologram: H, BBO I crystal; p¯, laser output beam; Z, 50% beam splitter; Z, high-reflectivity mirror; o¯ and r¯, beams that propagate along the object and the reference arms; WO, object wave that forms the object (focal spot of lens L3, which is either spherical or cylindrical) to be recorded on the SHG hologram; CCD, camera used for recording the reconstructed image of the object. Lenses L1 to L5 have focal distances f1=-5 mm, f2=+100 mm, f3=+30 mm (spherical lens) or f3=+210 mm (cylindrical lens), f4=-20 mm, and f5=+150 mm. All distances between optical elements are in millimeters. The values of distances d3 and d23 are given in the text.

Fig. 5
Fig. 5

Optical scheme of the formation of the image reconstructed by an experimental SHG hologram. R is the reference source emitting the reference wave WR; WO is the object wave that converges to the pointlike or 1D object O; BS is the bisector of the angle α between beams o¯ and r¯; Oexp is the position of the reconstructed image observed in the experiment, its distance from BBO I being equal to δexp=205 mm; Oth is the calculated position of the reconstructed image, its distance from BBO I being equal to δth=188 mm; other relevant distances from BBO I are those of O (δ=138.8 mm), O (δ=2δ=277.6 mm), and R(δR=290 mm); kO, kR, and kH are the wave vectors of object, reference, and reconstructed waves, respectively.

Fig. 6
Fig. 6

Intensity maps of the reconstructed pointlike image as they were displayed by the CCD camera: (a) the image focused on the sensor of a CCD camera and (b) a defocused image.

Fig. 7
Fig. 7

Intensity maps of the reconstructed quasi-1D images as they were displayed by the CCD camera in focus. The object image is the focal spot of a cylindrical lens with (a) vertical, (b) horizontal, or (c) 45°-inclined axes. (d) Reconstructed image for an object field as in (b), but 0-to-100% periodically modulated in amplitude along the horizontal direction. The white spots in the upper-right corners of the panels display single pixels (32 µm×32 µm).

Equations (10)

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EO=AO(x, y)exp{i[k1LO(x, y)+ωt]},
ER=AR exp{i[k1LR(x, y)+ωt]},
Eω=AO(x, y)exp{i[k1LO(x, y)+ωt]}+AR exp{i[k1LR(x, y)+ωt]}.
E2ωEωEω=AO2(x, y)exp{i[2k1LO(x, y)+2ωt]}+2AO(x, y)AR exp{i[k1(LO(x, y)+LR(x, y))+2ωt]}+AR2 exp{i[2k1LR(x, y)+2ωt]}
E2ω=AO(x, y)AR×expik2 LO(x, y)+LR(x, y)2+2ωt.
LH(x, y)=LO(x, y)2+LR(x, y)2.
E2ω=AO(x, y)exp[ik2LO(x, y)/2]AR×exp[ik2LR(x, y)/2]exp(i2ωt)
Iω=AO2(r)+AO(r)AR exp{ik1[LO(r)-LR(r)]}+AO(r)AR exp{ik1[LR(r)-LO(r)]}+AR2.
ER2ω=AR exp{i[k2LR(r)+2ωt]}.
EH=ARAO2(r)exp{i[k2LR(r)+2ωt]}+AR2AO(r)expik2 LO(r)+LR(r)2+2ωt+AR2AO(r)expik2 3LR(r)-LO(r)2+2ωt+AR3 exp{i[k2LR(r)+2ωt]}.

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