Abstract

A general study of phase-matching loci and associated angular acceptances is performed in the case of non-collinear parametric amplification. Numerical and analytical calculations, as well as measurements, are described for the uniaxial BBO crystal and the biaxial LBO crystal.

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References

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  1. D. Herrmann, R. Tautz, F. Tavella, F. Krausz, and L. Veisz, “Investigation of two-beam-pumped noncollinear optical parametric chirped-pulse amplification for the generation of few-cycle light pulses,” Opt. Express18(5), 4170–4183 (2010).
    [CrossRef] [PubMed]
  2. T. Kurita, K. Sueda, K. Tsubakimoto, and N. Miyanaga, “Experimental demonstration of spatially coherent beam combining using optical parametric amplification,” Opt. Express18(14), 14541–14546 (2010).
    [CrossRef] [PubMed]
  3. K. Kato, “Second-harmonic generation to 2048 Å in β-BaB2O4,” IEEE J. Quantum Electron.22(7), 1013–1014 (1986).
    [CrossRef]
  4. K. Kato, “Temperature-tuned 90° phase-matching properties of LiB3O5,” IEEE J. Quantum Electron.30(12), 2950–2952 (1994).
    [CrossRef]
  5. B. Boulanger and J. Zyss, Non-linear Optical properties, Chapter 1.8 in International Tables for Crystallography, Vol. D: Physical Properties of Crystals, (A. Authier Ed., 2006) International Union of Crystallography, Kluwer Academic Publisher, Dordrecht, Netherlands, 178–219.
  6. J. P. Fève, B. Boulanger, and G. Marnier, “Calculation and classification of the direction loci for collinear types I, II and III phase-matching of three-wave nonlinear optical parametric interactions in uniaxial and biaxial acentric crystals,” Opt. Commun.99(3-4), 284–302 (1993).
    [CrossRef]
  7. N. Boeuf, D. Branning, I. Chaperot, E. Dauler, S. Guérin, G. Jaeger, A. Muller, and A. Migdall, “Calculating Characteristics of Non-collinear Phase-matching in Uniaxial and Biaxial Crystals,” Opt. Eng.39(4), 1016–1039 (2000).
    [CrossRef]
  8. C. Chen, Y. Wu, A. Jiang, B. Wu, G. You, R. Li, and S. Lin, “New nonlinear-optical crystal: LiB3O5,” J. Opt. Soc. Am. B6(4), 616–621 (1989).
    [CrossRef]
  9. G. Aka, A. Kahn-Harari, F. Mougel, D. Vivien, F. Salin, P. Coquelin, P. Colin, D. Pelenc, and J. P. Damelet, “Linear and nonlinear optical properties of a new gadolinium calcium oxoborate crystal Ca4GdO(BO3)3,” J. Opt. Soc. Am. B14(9), 2238–2247 (1997).
    [CrossRef]
  10. SNLO nonlinear optics code available from A. V. Smith, AS-Photonics, Albuquerque, NM.

2010

2000

N. Boeuf, D. Branning, I. Chaperot, E. Dauler, S. Guérin, G. Jaeger, A. Muller, and A. Migdall, “Calculating Characteristics of Non-collinear Phase-matching in Uniaxial and Biaxial Crystals,” Opt. Eng.39(4), 1016–1039 (2000).
[CrossRef]

1997

1994

K. Kato, “Temperature-tuned 90° phase-matching properties of LiB3O5,” IEEE J. Quantum Electron.30(12), 2950–2952 (1994).
[CrossRef]

1993

J. P. Fève, B. Boulanger, and G. Marnier, “Calculation and classification of the direction loci for collinear types I, II and III phase-matching of three-wave nonlinear optical parametric interactions in uniaxial and biaxial acentric crystals,” Opt. Commun.99(3-4), 284–302 (1993).
[CrossRef]

1989

1986

K. Kato, “Second-harmonic generation to 2048 Å in β-BaB2O4,” IEEE J. Quantum Electron.22(7), 1013–1014 (1986).
[CrossRef]

Aka, G.

Boeuf, N.

N. Boeuf, D. Branning, I. Chaperot, E. Dauler, S. Guérin, G. Jaeger, A. Muller, and A. Migdall, “Calculating Characteristics of Non-collinear Phase-matching in Uniaxial and Biaxial Crystals,” Opt. Eng.39(4), 1016–1039 (2000).
[CrossRef]

Boulanger, B.

J. P. Fève, B. Boulanger, and G. Marnier, “Calculation and classification of the direction loci for collinear types I, II and III phase-matching of three-wave nonlinear optical parametric interactions in uniaxial and biaxial acentric crystals,” Opt. Commun.99(3-4), 284–302 (1993).
[CrossRef]

Branning, D.

N. Boeuf, D. Branning, I. Chaperot, E. Dauler, S. Guérin, G. Jaeger, A. Muller, and A. Migdall, “Calculating Characteristics of Non-collinear Phase-matching in Uniaxial and Biaxial Crystals,” Opt. Eng.39(4), 1016–1039 (2000).
[CrossRef]

Chaperot, I.

N. Boeuf, D. Branning, I. Chaperot, E. Dauler, S. Guérin, G. Jaeger, A. Muller, and A. Migdall, “Calculating Characteristics of Non-collinear Phase-matching in Uniaxial and Biaxial Crystals,” Opt. Eng.39(4), 1016–1039 (2000).
[CrossRef]

Chen, C.

Colin, P.

Coquelin, P.

Damelet, J. P.

Dauler, E.

N. Boeuf, D. Branning, I. Chaperot, E. Dauler, S. Guérin, G. Jaeger, A. Muller, and A. Migdall, “Calculating Characteristics of Non-collinear Phase-matching in Uniaxial and Biaxial Crystals,” Opt. Eng.39(4), 1016–1039 (2000).
[CrossRef]

Fève, J. P.

J. P. Fève, B. Boulanger, and G. Marnier, “Calculation and classification of the direction loci for collinear types I, II and III phase-matching of three-wave nonlinear optical parametric interactions in uniaxial and biaxial acentric crystals,” Opt. Commun.99(3-4), 284–302 (1993).
[CrossRef]

Guérin, S.

N. Boeuf, D. Branning, I. Chaperot, E. Dauler, S. Guérin, G. Jaeger, A. Muller, and A. Migdall, “Calculating Characteristics of Non-collinear Phase-matching in Uniaxial and Biaxial Crystals,” Opt. Eng.39(4), 1016–1039 (2000).
[CrossRef]

Herrmann, D.

Jaeger, G.

N. Boeuf, D. Branning, I. Chaperot, E. Dauler, S. Guérin, G. Jaeger, A. Muller, and A. Migdall, “Calculating Characteristics of Non-collinear Phase-matching in Uniaxial and Biaxial Crystals,” Opt. Eng.39(4), 1016–1039 (2000).
[CrossRef]

Jiang, A.

Kahn-Harari, A.

Kato, K.

K. Kato, “Temperature-tuned 90° phase-matching properties of LiB3O5,” IEEE J. Quantum Electron.30(12), 2950–2952 (1994).
[CrossRef]

K. Kato, “Second-harmonic generation to 2048 Å in β-BaB2O4,” IEEE J. Quantum Electron.22(7), 1013–1014 (1986).
[CrossRef]

Krausz, F.

Kurita, T.

Li, R.

Lin, S.

Marnier, G.

J. P. Fève, B. Boulanger, and G. Marnier, “Calculation and classification of the direction loci for collinear types I, II and III phase-matching of three-wave nonlinear optical parametric interactions in uniaxial and biaxial acentric crystals,” Opt. Commun.99(3-4), 284–302 (1993).
[CrossRef]

Migdall, A.

N. Boeuf, D. Branning, I. Chaperot, E. Dauler, S. Guérin, G. Jaeger, A. Muller, and A. Migdall, “Calculating Characteristics of Non-collinear Phase-matching in Uniaxial and Biaxial Crystals,” Opt. Eng.39(4), 1016–1039 (2000).
[CrossRef]

Miyanaga, N.

Mougel, F.

Muller, A.

N. Boeuf, D. Branning, I. Chaperot, E. Dauler, S. Guérin, G. Jaeger, A. Muller, and A. Migdall, “Calculating Characteristics of Non-collinear Phase-matching in Uniaxial and Biaxial Crystals,” Opt. Eng.39(4), 1016–1039 (2000).
[CrossRef]

Pelenc, D.

Salin, F.

Sueda, K.

Tautz, R.

Tavella, F.

Tsubakimoto, K.

Veisz, L.

Vivien, D.

Wu, B.

Wu, Y.

You, G.

IEEE J. Quantum Electron.

K. Kato, “Second-harmonic generation to 2048 Å in β-BaB2O4,” IEEE J. Quantum Electron.22(7), 1013–1014 (1986).
[CrossRef]

K. Kato, “Temperature-tuned 90° phase-matching properties of LiB3O5,” IEEE J. Quantum Electron.30(12), 2950–2952 (1994).
[CrossRef]

J. Opt. Soc. Am. B

Opt. Commun.

J. P. Fève, B. Boulanger, and G. Marnier, “Calculation and classification of the direction loci for collinear types I, II and III phase-matching of three-wave nonlinear optical parametric interactions in uniaxial and biaxial acentric crystals,” Opt. Commun.99(3-4), 284–302 (1993).
[CrossRef]

Opt. Eng.

N. Boeuf, D. Branning, I. Chaperot, E. Dauler, S. Guérin, G. Jaeger, A. Muller, and A. Migdall, “Calculating Characteristics of Non-collinear Phase-matching in Uniaxial and Biaxial Crystals,” Opt. Eng.39(4), 1016–1039 (2000).
[CrossRef]

Opt. Express

Other

B. Boulanger and J. Zyss, Non-linear Optical properties, Chapter 1.8 in International Tables for Crystallography, Vol. D: Physical Properties of Crystals, (A. Authier Ed., 2006) International Union of Crystallography, Kluwer Academic Publisher, Dordrecht, Netherlands, 178–219.

SNLO nonlinear optics code available from A. V. Smith, AS-Photonics, Albuquerque, NM.

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Figures (7)

Fig. 1
Fig. 1

Wave vector configurations and corresponding relevant angles of phase-matched (a) and non-phase-matched (b) non-collinear three-wave parametric interactions.

Fig. 2
Fig. 2

(a) Relative orientation of the dielectric frame (x, y, z) and of the laboratory frame (x”, y”, z”) linked to the signal wave vector k s ; (θs, φs) are the angles of spherical coordinates of k s in the dielectric frame; the parallelepiped stands for the nonlinear crystal ; (θ, φ) are the spherical coordinates angles of an arbitrary direction u . (b) Orientation of phase-matched signal, pump and idler wave vectors k s , k p and k i resp. in the laboratory frame.

Fig. 3
Fig. 3

Setup for the measurements of the non-collinear OPA phase-matching loci and angular acceptances.

Fig. 4
Fig. 4

(a) Configuration of orientation of the BBO crystal with respect to the signal wave vector , the dielectric frame (x, y, z), and the laboratory frame (x”, y”, z”). (b) Non-collinear phase-matching angles (αp, ψp) of the pump wave vector in the laboratory frame; the circular dots correspond to experimental data and the continuous line to numerical calculations; the encircled dot at (αp = 0°, ψp = 0°) corresponds to z” and.

Fig. 5
Fig. 5

Angular acceptance properties of the non-collinear OPA of a signal at λs = 720 nm in a 15-mm-long BBO crystal cut at (θ = 23°, φ = 0°) and pumped at λp = 532 nm. (a) Parametric gain at ψp = 180° as a function of the phase-matching angle αp; (b) angular acceptance L.Δαp as a function of the phase-matching angle ψp.

Fig. 6
Fig. 6

(a) Orientation of the LBO crystal with respect to the signal wave vector, the dielectric frame (x, y, z) and the laboratory frame (x”, y”, z”). (b) Non-collinear phase-matching angles (αp, ψp) of the pump wave vector in the laboratory frame ; the circular dots correspond to experimental data and the continuous line to numerical calculations; the encircled dot at (αp = 0°, ψp = 0°) corresponds to z” and .

Fig. 7
Fig. 7

Angular acceptance properties of the non-collinear OPA of a signal at λs = 720 nm in a 22-mm-long LBO crystal cut at (θ = 90°, φ = 0°) and pumped at λp = 532 nm. (a) Parametric gain at ψp = 0° as a function of the phase-matching angle αp; (b) angular acceptance L.Δαp as a function of the phase-matching angle ψp.

Equations (11)

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1 λ p = 1 λ s + 1 λ i
Δk = k p k s k i = 0
n p λ p cos( α p )= n s λ s + n i λ i cos( α i ) n p λ p sin( α p )= n i λ i sin( α i )
n ± = [ 2 B ( B 2 4C ) 1/2 ] 1/2 B= u x 2 (b+c) u y 2 (a+c) u z 2 (a+b) C= u x 2 bc+ u y 2 ac+ u z 2 ab a= n x 2 , b= n y 2 , c= n z 2
u x =cosφsinθ u y =sinφsinθ u z =cosθ
Δk =2π( n p λ p sin( α p ).cos( ψ p ) n 1 λ 1 sin( α 1 ).cos( ψ 1 ) n p λ p sin( α p ).sin( ψ p ) n 1 λ 1 sin( α 1 ).sin( ψ 1 ) n p λ p cos( α p ) n s λ s n 1 λ 1 cos( α 1 ) )
cos α p,i =sin θ s cos φ s .sin θ p,i cos φ p,i +sin θ s sin φ s .sin θ p,i sin φ p,i +cos θ s .cos θ p,i sin ψ p,i = cos φ s .sin θ p,i sin φ p,i sin φ s .sin θ p,i cos φ p,i sin α p,i
Δk . L LΔ α p α p ( Δk k s k s )=4π
LΔ α p = 2 λ p n p [ 1 2 sin(2 θ p ) n p 2 ( 1 n o 2 ( λ p ) 1 n e 2 ( λ p ) ) ( cos α p +sin( θ i θ s )sin( θ p θ i ) ) sin( θ p θ s )+sin( θ i θ s )cos( θ p θ i ) ] 1
LΔ α p = 2 λ p n p [ 1 2 sin(2 φ p ) n p 2 ( 1 n x 2 ( λ p ) 1 n y 2 ( λ p ) ) ( cos α p +sin( φ i φ s )sin( φ p φ i ) ) sin( φ p φ s )+sin( φ i φ s )cos( φ p φ i ) ] 1
L.Δ α p = 2λ n p [ ( n i cos( θ i θ p )+ n i 'sin( θ i θ p ) )( n i sin( θ i θ s ) n i 'cos( θ i θ s ) ) n i ²+ n i '² sin( θ p θ s ) ] 1 n i = 1 2 sin(2 θ i ) n i 3 ( 1 n x 2 ( λ i ) 1 n y 2 ( λ i ) ).

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