Abstract

We present a theoretical treatment of difference-frequency generation in conditions of phase mismatch for seed and generated fields that propagate at different angles to the pump. The analytical solution in the case of nondepleted plane-wave pump and experiments performed with seed fields strongly phase and amplitude modulated show that the pump-field wave fronts behave as efficient phase-conjugating mirrors.

© 2003 Optical Society of America

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  1. B. Y. Zeldovich, V. I. Popovichev, V. V. Ragulskii, and F. S. Faisullov, “Connection between the wavefronts of the reflected and exciting light in stimulated Mandel’shtam–Brillouin scattering,” Sov. Phys. JETP 15, 109–110 (1972).
  2. A. Yariv, “Phase conjugate optics and real-time holography,” IEEE J. Quantum Electron. QE-14, 650–660 (1978).
    [CrossRef]
  3. 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, San Diego, Calif., 1983), pp. 23–78.
  4. J. H. Marburger, “Optical pulse integration and chirp reversal in degenerate four-wave mixing,” Appl. Phys. Lett. 32, 372–374 (1978).
    [CrossRef]
  5. A. Yariv, D. Fekete, and D. M. Pepper, “Compensation for channel dispersion by nonlinear optical phase conjugation,” Opt. Lett. 4, 52–54 (1979).
    [CrossRef] [PubMed]
  6. A. Yariv, “Three-dimensional pictorial transmission in optical fibers,” Appl. Phys. Lett. 28, 88–89 (1976).
    [CrossRef]
  7. P. V. Avizonis, F. A. Hopf, W. D. Bomberger, S. F. Jacobs, A. Tomita, and K. H. Womack, “Optical phase conjugation in a lithium formate crystal,” Appl. Phys. Lett. 31, 435–437 (1977).
    [CrossRef]
  8. 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,” Kvant. Elektron. (Moscow) 5, 415–417 (1982), in Russian.
  9. L. Lefort and A. Barthelemy, “Revisiting optical phase conjugation by difference-frequency generation,” Opt. Lett. 21, 848–850 (1996).
    [CrossRef] [PubMed]
  10. M. Bondani and A. Andreoni, “Holographic nature of three-wave mixing,” Phys. Rev. A 66, 033805 (2002).
    [CrossRef]
  11. M. Bondani, A. Allevi, A. Brega, E. Puddu, and A. Andreoni, Dipartimento di Scienze Chimiche, Fisiche e Matematiche, Università degli Studi dell’Insubria and Istituto Nazionale di Fisica della Materia, Unità di Como, Via Valleggio 11, Como 22100, Italy, are preparing a manuscript to be called “Difference-frequency-generated holograms of 2D-objects.”
  12. G. S. Agarwal, A. T. Friberg, and E. Wolf, “Elimination of distortions by phase conjugation without losses or gains,” Opt. Commun. 43, 446–450 (1982).
    [CrossRef]
  13. V. G. Dmitriev, G. G. Gurzadyan, and D. N. Nikogosyan, Handbook of Nonlinear Optical Crystals (Springer-Verlag, Berlin, 1997).
  14. M. Bondani, A. Allevi, and A. Andreoni, “Holography bynondegenerate χ(2) interactions,” J. Opt. Soc. Am. B 20, 1–13 (2003).
    [CrossRef]
  15. M. R. Fewings and A. L. Gaeta, “Compensation of pulse distortions by phase conjugation via difference-frequency generation,” J. Opt. Soc. Am. B 17, 1522–1525 (2000).
    [CrossRef]
  16. R. Danielius, A. Piskarskas, P. Di Trapani, A. Andreoni, C. Solcia, and P. Foggi, “Matching of group velocities by spatial walkoff in collinear three-wave interaction with tilted pulses,” Opt. Lett. 21, 973–975 (1996).
    [CrossRef] [PubMed]
  17. A. Andreoni, M. Bondani, and M. A. C. Potenza, “Ultrabroadband and chirp-free frequency doubling by β-barium borate,” Opt. Commun. 154, 376–382 (1998).
    [CrossRef]

2003 (1)

2002 (1)

M. Bondani and A. Andreoni, “Holographic nature of three-wave mixing,” Phys. Rev. A 66, 033805 (2002).
[CrossRef]

2000 (1)

1998 (1)

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

1996 (2)

1982 (2)

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,” Kvant. Elektron. (Moscow) 5, 415–417 (1982), in Russian.

G. S. Agarwal, A. T. Friberg, and E. Wolf, “Elimination of distortions by phase conjugation without losses or gains,” Opt. Commun. 43, 446–450 (1982).
[CrossRef]

1979 (1)

1978 (2)

A. Yariv, “Phase conjugate optics and real-time holography,” IEEE J. Quantum Electron. QE-14, 650–660 (1978).
[CrossRef]

J. H. Marburger, “Optical pulse integration and chirp reversal in degenerate four-wave mixing,” Appl. Phys. Lett. 32, 372–374 (1978).
[CrossRef]

1977 (1)

P. V. Avizonis, F. A. Hopf, W. D. Bomberger, S. F. Jacobs, A. Tomita, and K. H. Womack, “Optical phase conjugation in a lithium formate crystal,” Appl. Phys. Lett. 31, 435–437 (1977).
[CrossRef]

1976 (1)

A. Yariv, “Three-dimensional pictorial transmission in optical fibers,” Appl. Phys. Lett. 28, 88–89 (1976).
[CrossRef]

1972 (1)

B. Y. Zeldovich, V. I. Popovichev, V. V. Ragulskii, and F. S. Faisullov, “Connection between the wavefronts of the reflected and exciting light in stimulated Mandel’shtam–Brillouin scattering,” Sov. Phys. JETP 15, 109–110 (1972).

Agarwal, G. S.

G. S. Agarwal, A. T. Friberg, and E. Wolf, “Elimination of distortions by phase conjugation without losses or gains,” Opt. Commun. 43, 446–450 (1982).
[CrossRef]

Allevi, A.

Andreoni, A.

Avizonis, P. V.

P. V. Avizonis, F. A. Hopf, W. D. Bomberger, S. F. Jacobs, A. Tomita, and K. H. Womack, “Optical phase conjugation in a lithium formate crystal,” Appl. Phys. Lett. 31, 435–437 (1977).
[CrossRef]

Barthelemy, A.

Bomberger, W. D.

P. V. Avizonis, F. A. Hopf, W. D. Bomberger, S. F. Jacobs, A. Tomita, and K. H. Womack, “Optical phase conjugation in a lithium formate crystal,” Appl. Phys. Lett. 31, 435–437 (1977).
[CrossRef]

Bondani, M.

M. Bondani, A. Allevi, and A. Andreoni, “Holography bynondegenerate χ(2) interactions,” J. Opt. Soc. Am. B 20, 1–13 (2003).
[CrossRef]

M. Bondani and A. Andreoni, “Holographic nature of three-wave mixing,” Phys. Rev. A 66, 033805 (2002).
[CrossRef]

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

Danielius, R.

Di Trapani, P.

Faisullov, F. S.

B. Y. Zeldovich, V. I. Popovichev, V. V. Ragulskii, and F. S. Faisullov, “Connection between the wavefronts of the reflected and exciting light in stimulated Mandel’shtam–Brillouin scattering,” Sov. Phys. JETP 15, 109–110 (1972).

Fekete, D.

Fewings, M. R.

Foggi, P.

Friberg, A. T.

G. S. Agarwal, A. T. Friberg, and E. Wolf, “Elimination of distortions by phase conjugation without losses or gains,” Opt. Commun. 43, 446–450 (1982).
[CrossRef]

Gaeta, A. L.

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,” Kvant. Elektron. (Moscow) 5, 415–417 (1982), in Russian.

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,” Kvant. Elektron. (Moscow) 5, 415–417 (1982), in Russian.

Hopf, F. A.

P. V. Avizonis, F. A. Hopf, W. D. Bomberger, S. F. Jacobs, A. Tomita, and K. H. Womack, “Optical phase conjugation in a lithium formate crystal,” Appl. Phys. Lett. 31, 435–437 (1977).
[CrossRef]

Jacobs, S. F.

P. V. Avizonis, F. A. Hopf, W. D. Bomberger, S. F. Jacobs, A. Tomita, and K. H. Womack, “Optical phase conjugation in a lithium formate crystal,” Appl. Phys. Lett. 31, 435–437 (1977).
[CrossRef]

Lefort, L.

Marburger, J. H.

J. H. Marburger, “Optical pulse integration and chirp reversal in degenerate four-wave mixing,” Appl. Phys. Lett. 32, 372–374 (1978).
[CrossRef]

Pepper, D. M.

Piskarskas, A.

Popovichev, V. I.

B. Y. Zeldovich, V. I. Popovichev, V. V. Ragulskii, and F. S. Faisullov, “Connection between the wavefronts of the reflected and exciting light in stimulated Mandel’shtam–Brillouin scattering,” Sov. Phys. JETP 15, 109–110 (1972).

Potenza, M. A. C.

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

Ragulskii, V. V.

B. Y. Zeldovich, V. I. Popovichev, V. V. Ragulskii, and F. S. Faisullov, “Connection between the wavefronts of the reflected and exciting light in stimulated Mandel’shtam–Brillouin scattering,” Sov. Phys. JETP 15, 109–110 (1972).

Solcia, C.

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,” Kvant. Elektron. (Moscow) 5, 415–417 (1982), in Russian.

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,” Kvant. Elektron. (Moscow) 5, 415–417 (1982), in Russian.

Tomita, A.

P. V. Avizonis, F. A. Hopf, W. D. Bomberger, S. F. Jacobs, A. Tomita, and K. H. Womack, “Optical phase conjugation in a lithium formate crystal,” Appl. Phys. Lett. 31, 435–437 (1977).
[CrossRef]

Wolf, E.

G. S. Agarwal, A. T. Friberg, and E. Wolf, “Elimination of distortions by phase conjugation without losses or gains,” Opt. Commun. 43, 446–450 (1982).
[CrossRef]

Womack, K. H.

P. V. Avizonis, F. A. Hopf, W. D. Bomberger, S. F. Jacobs, A. Tomita, and K. H. Womack, “Optical phase conjugation in a lithium formate crystal,” Appl. Phys. Lett. 31, 435–437 (1977).
[CrossRef]

Yariv, A.

A. Yariv, D. Fekete, and D. M. Pepper, “Compensation for channel dispersion by nonlinear optical phase conjugation,” Opt. Lett. 4, 52–54 (1979).
[CrossRef] [PubMed]

A. Yariv, “Phase conjugate optics and real-time holography,” IEEE J. Quantum Electron. QE-14, 650–660 (1978).
[CrossRef]

A. Yariv, “Three-dimensional pictorial transmission in optical fibers,” Appl. Phys. Lett. 28, 88–89 (1976).
[CrossRef]

Zeldovich, B. Y.

B. Y. Zeldovich, V. I. Popovichev, V. V. Ragulskii, and F. S. Faisullov, “Connection between the wavefronts of the reflected and exciting light in stimulated Mandel’shtam–Brillouin scattering,” Sov. Phys. JETP 15, 109–110 (1972).

Appl. Phys. Lett. (3)

J. H. Marburger, “Optical pulse integration and chirp reversal in degenerate four-wave mixing,” Appl. Phys. Lett. 32, 372–374 (1978).
[CrossRef]

A. Yariv, “Three-dimensional pictorial transmission in optical fibers,” Appl. Phys. Lett. 28, 88–89 (1976).
[CrossRef]

P. V. Avizonis, F. A. Hopf, W. D. Bomberger, S. F. Jacobs, A. Tomita, and K. H. Womack, “Optical phase conjugation in a lithium formate crystal,” Appl. Phys. Lett. 31, 435–437 (1977).
[CrossRef]

IEEE J. Quantum Electron. (1)

A. Yariv, “Phase conjugate optics and real-time holography,” IEEE J. Quantum Electron. QE-14, 650–660 (1978).
[CrossRef]

J. Opt. Soc. Am. B (2)

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,” Kvant. Elektron. (Moscow) 5, 415–417 (1982), in Russian.

Opt. Commun. (2)

G. S. Agarwal, A. T. Friberg, and E. Wolf, “Elimination of distortions by phase conjugation without losses or gains,” Opt. Commun. 43, 446–450 (1982).
[CrossRef]

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

Opt. Lett. (3)

Phys. Rev. A (1)

M. Bondani and A. Andreoni, “Holographic nature of three-wave mixing,” Phys. Rev. A 66, 033805 (2002).
[CrossRef]

Sov. Phys. JETP (1)

B. Y. Zeldovich, V. I. Popovichev, V. V. Ragulskii, and F. S. Faisullov, “Connection between the wavefronts of the reflected and exciting light in stimulated Mandel’shtam–Brillouin scattering,” Sov. Phys. JETP 15, 109–110 (1972).

Other (3)

M. Bondani, A. Allevi, A. Brega, E. Puddu, and A. Andreoni, Dipartimento di Scienze Chimiche, Fisiche e Matematiche, Università degli Studi dell’Insubria and Istituto Nazionale di Fisica della Materia, Unità di Como, Via Valleggio 11, Como 22100, Italy, are preparing a manuscript to be called “Difference-frequency-generated holograms of 2D-objects.”

V. G. Dmitriev, G. G. Gurzadyan, and D. N. Nikogosyan, Handbook of Nonlinear Optical Crystals (Springer-Verlag, Berlin, 1997).

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, San Diego, Calif., 1983), pp. 23–78.

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Figures (9)

Fig. 1
Fig. 1

Geometry of the type 1 interaction among three noncollinearly propagating fields, E1, E2, and E3, that propagate in different directions (wave vectors k1,2,3) in the (y, z) plane; χ(2) nonlinear medium at z>0.

Fig. 2
Fig. 2

Wave vectors k1,2,3 at angles ϑ1,2,3 to the z axis, traces of the k1 and k2 surfaces (both spherical, i.e., type I interaction) in the (y, z) plane, and Δk=k3-k1-k2. The dashed line bisects the k1-to-k2 angle.

Fig. 3
Fig. 3

As in Fig. 2 with Δk parallel to the bisector (dashed line) of the k1-to-k2 angle; see Eq. (18). The dotted circle (center C) is tangent to the k1 and k2 surfaces.

Fig. 4
Fig. 4

(a) Layout of the experimental setup (general) and details of (b) exp-I and (c) exp-II. See text for the nonlinear crystals BBO I (exp-I) and BBO II (exp-II). Lenses L1–L7, pinholes PH1 and PH2, mirrors M1–M4, neutral and bandpass filters F1–F3, photodiodes PD1 and PD2; BS1, high-energy harmonic separator (wedged substrate); DM, high energy dichroic mirror; DIFF, diffusing glass plate; P, film polarizer.

Fig. 5
Fig. 5

Photon flux in the DFG field at BBO I output Φ2(r), normalized to seeded photon flux Φ1(0), as a function of pump photon flux density |A3|2 (lower scale). Upper scale, pump intensity; straight line: least-squares fit of experimental points; inset, typical fluence map of a DFG spot obtained with the highest pump intensity.

Fig. 6
Fig. 6

Phase relations according to Eq. (24). The pump-field phase on plane Π is constant, (e.g., φ3(P)=-π/2). See text for the values of φ1 and φ2 at P and φ1 and ψ2 at origin O.

Fig. 7
Fig. 7

Fluence maps of the seed field after removal of the diffusing glass plate, DIFF in Fig. 4(c), and in the absence of the pump as detected by the CCD camera at distances of (a) 51 and (b) 291 cm from BBO II.

Fig. 8
Fig. 8

Speckle pattern of the seed field measured at a distance of 6 cm from the diffusing glass plate.

Fig. 9
Fig. 9

Fluence maps of the DFG field after reflection by mirror M4 in Fig. 4(c) and backpropagation through DIFF. The CCD camera is at distances of (a) 51 and (b) 291 cm from BBO II.

Equations (49)

Equations on this page are rendered with MathJax. Learn more.

E1(r, t)=xˆ22η0ω1n11/2×{a1(r)exp[-i(k1  r-ω1t)]+c.c.},
E2(r, t)=xˆ22η0ω2n21/2×{a2(r)exp[-i(k2  r-ω2t)]+c.c.},
k^1a1(r)=igeff a3(0)a2*(r)exp(-iΔk  r),
k^2a2(r)=igeff a3(0)a1*(r)exp(-iΔk  r),
geff=(d22cos α+d31sin α)2ω1ω2ω3η03n1n2n31/2,
|a1(r)|2k^1=|a2(r)|2k^2,
a1(r)=a1(0)coshQΔkˆ2r+iΔkQsinhQΔkˆ2r×exp-iΔk2r,
a2(r)=a1*(0) 2iA3(Δkˆk^2)QsinhQΔkˆ2r×exp-iΔk2r,
Q=4|A3|2(Δkˆk^1)(Δkˆk^2)-Δk21/2,
E1(r, t)=xˆ2η0ω1n11/2|a1(r)|×cos[φ1(r)+ω1t],
E2(r, t)=xˆ2η0ω2n21/2|a2(r)|×cos[φ2(r)+ω2t]
φ1(r)=φ1(0)-k1r-Δk2r+tan-1ΔkQtanhQΔkˆ2r,
φ2(r)=-φ1(0)+φ3(0)+π2-k2r-Δk2r
φ1(r)+φ2(r)
=φ3(r)+π2+tan-1ΔkQtanhQΔkˆ2r,
|a1(r)|=|a1(0)|QQ2+4|A3|2(Δkˆk^1)(Δkˆk^2)×sinh2QΔkˆ2r1/2,
|a2(r)|=|a1(0)| 2|A3|(Δkˆk^2)QsinhQΔkˆ2r.
|a1(r)|2=|a1(0)|2+Δkˆk^2Δkˆk^1 |a2(r)|2
|a1(r)|2=|a1(0)|2Q4|A3|2(Δkˆk^1)(Δkˆk^2)×sinhQΔkˆ2rcoshQΔkˆ2rΔkˆ,
|a2(r)|2=|a1(0)|2Q4|A3|2(Δkˆk^2)2sinhQΔkˆ2r×coshQΔkˆ2rΔkˆ.
φ1(r)=-k1-Δk2+Δk2×|a1(0)|2|a1(r)|2,
φ2(r)=-k2-Δk2
S1(r)=ω1k1 |a1(r)|2k1+Δk2-Δk2×|a1(0)|2|a1(r)|2,
S2(r)=ω2k2 |a2(r)|2k2+Δk2,
S1(r)=ω1|a1(0)|2k^1+ω1k1 |a2(r)|2Δkk^2Δkk^1×k1+Δk2,
Δk/2k1, k2,
|a2PM(r)|=|a1(0)|sinh|A3|cosϑ1-ϑ22  ×sinϑ1+ϑ22 y+cosϑ1+ϑ22 z,
|a2(r)|=|a1(0)|Δkˆk^1Δkˆk^21/2×sinh|A3|(Δkˆk^1)(Δkˆk^2) (Δkˆr),
Δkˆk^1=Δkˆk^2,
|S2(r)|ω2|S1(r)|ω1-|a1(0)|2,
(Δkk^1)(Δkk^2)4|A3|2.
|a2(r)|2=|a1(0)|2|A3|2(zˆr)2cos2 ϑ2sinhQ2 (zˆr)Q2 (zˆr)2
Φ2(r)=|a1(0)|2|A3|2d2cos2 ϑ1sinhQ2 (zˆr)Q2 (zˆr)2×k^2+Δk2kω zˆzˆ|a1(0)|2|A3|2d2cos2 ϑ1sinhQ2 (zˆr)Q2 (zˆr)2k^1zˆ,
Φ2(r)Φ1(0)
=|A3|2d2sinh24|A3|2cos2 ϑ1-4kω2(1-cos ϑ1)21/2d2cos2 ϑ14|A3|2cos2 ϑ1-4kω2(1-cos ϑ1)2d22,
φ1(r)+φ2(r)=φ3(r)+π/2,
S1,2(r)=Re12E1,2(r)×H1,2*(r),
×E1,2(r)=-iω1,2μ0H1,2(r).
E1,2(r)=xˆ2η0ω1,2n1,21/2|a1,2(r)|exp[iφ1,2(r)],
Φ1,2(r)=S1,2(r)Aˆ/ω1,2,
12ik1 2a1(r)+k^1a1(r)
=igeffa3(0)a2*(r)exp(-iΔkr),
12ik2 2a2(r)+k^2a2(r)
=igeffa3(0)a1*(r)exp(-iΔkr).
12ik1,2 2a1,2(r)|k^1,2a1,2(r)|,
(Q2+Δk2)2sinh2QΔkˆ2r+16Q2Δk2Δkˆ2
16(Δkˆk1,2)2(Q2+Δk2)sinh2QΔkˆ2r+16Q2(Δkˆk1,2)2.
(Q2+Δk2)16(Δkˆk1,2)2,
Δk2Δkˆ2(Δkˆk1,2)2.

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