Abstract

We present and experimentally test a simple model for difference frequency generation (DFG) in periodically-poled crystals with gaussian pumping beams. Focusing of input beams originates several non-collinear quasi-phase-matching configurations of the interacting wavevectors, which contribute to the idler output field. In this picture, we accurately describe a number of effects, such as the occurrence of annular idler intensity profiles and the asymmetric trend of DFG power vs temperature. Finally, we quantitatively test the model by means of an indirect measurement of the crystal poling period.

© 2008 Optical Society of America

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  1. H. Huang and K. K. Lehmann, "Noise in cavity ring-down spectroscopy caused by transverse mode coupling," Opt. Express 15, 8745-8759 (2007).
    [CrossRef] [PubMed]
  2. J. L. Hall and C. Bordé, "Measurement of methane hyperfine structure using laser saturated absorption" Phys. Rev. Lett. 30, 1101-1104 (1973).
    [CrossRef]
  3. R. G. Hunsperger, Integrated Optics - Theory and Technology, (Springer 2002).
  4. S. Kulin, S. Aubin S. Christe, B. Peker, S. L. Rolston, and L. A. Orozco, " A single hollow-beam optical trap for cold atoms," J. Opt. B: Quantum semiclass. Opt. 3, 353-357 (2001).
    [CrossRef]
  5. A. Kaplan, N. Friedman, and N. Davidson "Optimized single beam dark optical trap," J. Opt. Soc. Am. B  19, 1233-1238 (2002).
    [CrossRef]
  6. G. Giusfredi, D. Mazzotti, P. Cancio, and P. De Natale, "Spatial mode control in radiation generated by frequency difference in periodically poled crystals," Phys Rev. Lett. 87, 113901 (2001).
    [CrossRef] [PubMed]
  7. D. Richter, P. Weibring, A. Fried, "High-power, tunable difference frequency generation source for absorption spectroscopy based on a ridge waveguide periodically poled lithium niobate crystal," Opt. Express 15, 564-571 (2007).
    [CrossRef] [PubMed]
  8. G. Imeshev, M. Proctor, and M. M. Fejer, "Lateral patterning of nonlinear frequency conversion with transversely varying quasi-phase-matching gratings," Opt. Lett. 23, 673-675 (1998).
    [CrossRef]
  9. G. D. Boyd and D. A. Kleinman, "Parametric interaction of focused Gaussian light beams," J. Appl. Phys. 39, 3597-3641 (1968).
    [CrossRef]
  10. Jean-Jacques Zondy, "The effects of focusing in type-I and type-II difference-frequency generations," Opt. Commun. 149, 181-206 (1998).
    [CrossRef]
  11. S. K. Wong, G. Fournier, P. Mathieu, and P. Pace. "Beam divergence effects on nonlinear frequency mixing," J. Appl. Phys. 71, 1091-1101 (1992).
    [CrossRef]
  12. D. Lu, L. Qian, Y. Li, H. Yang, H. Zhu, and D. Fan, "Phase velocity nonuniformity-resulted beam patterns in difference frequency generation," Opt. Express 15, 5050-5056 (2007).
    [CrossRef] [PubMed]
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    [CrossRef]
  14. H. H. Abu-Safe, "Difference frequency mixing of strongly focused Gaussian beams in periodically poled LiNbO3," Appl. Phys. Lett. 86, 231105 (2005).
    [CrossRef]
  15. G. M. Gibson, G. A. Turnbull, M. Ebrahimzadeh, M. H. Dunn, H. Karlsson, G. Arvidsson, and F. Laurell, "Temperature-tuned difference-frequency mixing in periodically poled KTiOPO4" Appl. Phys. B: Lasers.Opt 67, 675-677 (1998).
    [CrossRef]
  16. K. Fradkin, A. Arie, A. Skliar, and G. Rosenman, "Tunable midinfrared source by difference frequency generation in bulk periodically poled KTiOPO4," Appl. Phys. Lett. 74, 914-917 (1999).
    [CrossRef]
  17. F. Zernike and J. E. Midwinter, Applied Nonlinear Optics, (J. Wiley & Sons, 1973).
  18. P. Maddaloni, G. Gagliardi, P. Malara and P. De Natale, "A 3.5-mW continuous-wave difference-frequency source around 3 mm for sub-Doppler molecular spectroscopy," Appl. Phys. B 80, 141-145 (2005).
    [CrossRef]
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    [CrossRef]

2007 (3)

2005 (2)

H. H. Abu-Safe, "Difference frequency mixing of strongly focused Gaussian beams in periodically poled LiNbO3," Appl. Phys. Lett. 86, 231105 (2005).
[CrossRef]

P. Maddaloni, G. Gagliardi, P. Malara and P. De Natale, "A 3.5-mW continuous-wave difference-frequency source around 3 mm for sub-Doppler molecular spectroscopy," Appl. Phys. B 80, 141-145 (2005).
[CrossRef]

2002 (1)

2001 (2)

G. Giusfredi, D. Mazzotti, P. Cancio, and P. De Natale, "Spatial mode control in radiation generated by frequency difference in periodically poled crystals," Phys Rev. Lett. 87, 113901 (2001).
[CrossRef] [PubMed]

S. Kulin, S. Aubin S. Christe, B. Peker, S. L. Rolston, and L. A. Orozco, " A single hollow-beam optical trap for cold atoms," J. Opt. B: Quantum semiclass. Opt. 3, 353-357 (2001).
[CrossRef]

1999 (1)

K. Fradkin, A. Arie, A. Skliar, and G. Rosenman, "Tunable midinfrared source by difference frequency generation in bulk periodically poled KTiOPO4," Appl. Phys. Lett. 74, 914-917 (1999).
[CrossRef]

1998 (3)

G. M. Gibson, G. A. Turnbull, M. Ebrahimzadeh, M. H. Dunn, H. Karlsson, G. Arvidsson, and F. Laurell, "Temperature-tuned difference-frequency mixing in periodically poled KTiOPO4" Appl. Phys. B: Lasers.Opt 67, 675-677 (1998).
[CrossRef]

G. Imeshev, M. Proctor, and M. M. Fejer, "Lateral patterning of nonlinear frequency conversion with transversely varying quasi-phase-matching gratings," Opt. Lett. 23, 673-675 (1998).
[CrossRef]

Jean-Jacques Zondy, "The effects of focusing in type-I and type-II difference-frequency generations," Opt. Commun. 149, 181-206 (1998).
[CrossRef]

1997 (1)

1992 (1)

S. K. Wong, G. Fournier, P. Mathieu, and P. Pace. "Beam divergence effects on nonlinear frequency mixing," J. Appl. Phys. 71, 1091-1101 (1992).
[CrossRef]

1977 (1)

J. R. Morris and Y. R. Shen, "Theory of far-infrared generation by optical mixing," Phys. Rev. A 15, 1143-1156 (1977).
[CrossRef]

1973 (1)

J. L. Hall and C. Bordé, "Measurement of methane hyperfine structure using laser saturated absorption" Phys. Rev. Lett. 30, 1101-1104 (1973).
[CrossRef]

1968 (1)

G. D. Boyd and D. A. Kleinman, "Parametric interaction of focused Gaussian light beams," J. Appl. Phys. 39, 3597-3641 (1968).
[CrossRef]

Abu-Safe, H. H.

H. H. Abu-Safe, "Difference frequency mixing of strongly focused Gaussian beams in periodically poled LiNbO3," Appl. Phys. Lett. 86, 231105 (2005).
[CrossRef]

Arie, A.

K. Fradkin, A. Arie, A. Skliar, and G. Rosenman, "Tunable midinfrared source by difference frequency generation in bulk periodically poled KTiOPO4," Appl. Phys. Lett. 74, 914-917 (1999).
[CrossRef]

Arvidsson, G.

G. M. Gibson, G. A. Turnbull, M. Ebrahimzadeh, M. H. Dunn, H. Karlsson, G. Arvidsson, and F. Laurell, "Temperature-tuned difference-frequency mixing in periodically poled KTiOPO4" Appl. Phys. B: Lasers.Opt 67, 675-677 (1998).
[CrossRef]

Bordé, C.

J. L. Hall and C. Bordé, "Measurement of methane hyperfine structure using laser saturated absorption" Phys. Rev. Lett. 30, 1101-1104 (1973).
[CrossRef]

Boyd, G. D.

G. D. Boyd and D. A. Kleinman, "Parametric interaction of focused Gaussian light beams," J. Appl. Phys. 39, 3597-3641 (1968).
[CrossRef]

Cancio, P.

G. Giusfredi, D. Mazzotti, P. Cancio, and P. De Natale, "Spatial mode control in radiation generated by frequency difference in periodically poled crystals," Phys Rev. Lett. 87, 113901 (2001).
[CrossRef] [PubMed]

Davidson, N.

De Natale, P.

P. Maddaloni, G. Gagliardi, P. Malara and P. De Natale, "A 3.5-mW continuous-wave difference-frequency source around 3 mm for sub-Doppler molecular spectroscopy," Appl. Phys. B 80, 141-145 (2005).
[CrossRef]

G. Giusfredi, D. Mazzotti, P. Cancio, and P. De Natale, "Spatial mode control in radiation generated by frequency difference in periodically poled crystals," Phys Rev. Lett. 87, 113901 (2001).
[CrossRef] [PubMed]

Dunn, M. H.

G. M. Gibson, G. A. Turnbull, M. Ebrahimzadeh, M. H. Dunn, H. Karlsson, G. Arvidsson, and F. Laurell, "Temperature-tuned difference-frequency mixing in periodically poled KTiOPO4" Appl. Phys. B: Lasers.Opt 67, 675-677 (1998).
[CrossRef]

Ebrahimzadeh, M.

G. M. Gibson, G. A. Turnbull, M. Ebrahimzadeh, M. H. Dunn, H. Karlsson, G. Arvidsson, and F. Laurell, "Temperature-tuned difference-frequency mixing in periodically poled KTiOPO4" Appl. Phys. B: Lasers.Opt 67, 675-677 (1998).
[CrossRef]

Fan, D.

Fejer, M. M.

Fournier, G.

S. K. Wong, G. Fournier, P. Mathieu, and P. Pace. "Beam divergence effects on nonlinear frequency mixing," J. Appl. Phys. 71, 1091-1101 (1992).
[CrossRef]

Fradkin, K.

K. Fradkin, A. Arie, A. Skliar, and G. Rosenman, "Tunable midinfrared source by difference frequency generation in bulk periodically poled KTiOPO4," Appl. Phys. Lett. 74, 914-917 (1999).
[CrossRef]

Fried, A.

Friedman, N.

Gagliardi, G.

P. Maddaloni, G. Gagliardi, P. Malara and P. De Natale, "A 3.5-mW continuous-wave difference-frequency source around 3 mm for sub-Doppler molecular spectroscopy," Appl. Phys. B 80, 141-145 (2005).
[CrossRef]

Gibson, G. M.

G. M. Gibson, G. A. Turnbull, M. Ebrahimzadeh, M. H. Dunn, H. Karlsson, G. Arvidsson, and F. Laurell, "Temperature-tuned difference-frequency mixing in periodically poled KTiOPO4" Appl. Phys. B: Lasers.Opt 67, 675-677 (1998).
[CrossRef]

Giusfredi, G.

G. Giusfredi, D. Mazzotti, P. Cancio, and P. De Natale, "Spatial mode control in radiation generated by frequency difference in periodically poled crystals," Phys Rev. Lett. 87, 113901 (2001).
[CrossRef] [PubMed]

Hall, J. L.

J. L. Hall and C. Bordé, "Measurement of methane hyperfine structure using laser saturated absorption" Phys. Rev. Lett. 30, 1101-1104 (1973).
[CrossRef]

Huang, H.

Imeshev, G.

Jundt, D. H.

Kaplan, A.

Karlsson, H.

G. M. Gibson, G. A. Turnbull, M. Ebrahimzadeh, M. H. Dunn, H. Karlsson, G. Arvidsson, and F. Laurell, "Temperature-tuned difference-frequency mixing in periodically poled KTiOPO4" Appl. Phys. B: Lasers.Opt 67, 675-677 (1998).
[CrossRef]

Kleinman, D. A.

G. D. Boyd and D. A. Kleinman, "Parametric interaction of focused Gaussian light beams," J. Appl. Phys. 39, 3597-3641 (1968).
[CrossRef]

Kulin, S.

S. Kulin, S. Aubin S. Christe, B. Peker, S. L. Rolston, and L. A. Orozco, " A single hollow-beam optical trap for cold atoms," J. Opt. B: Quantum semiclass. Opt. 3, 353-357 (2001).
[CrossRef]

Laurell, F.

G. M. Gibson, G. A. Turnbull, M. Ebrahimzadeh, M. H. Dunn, H. Karlsson, G. Arvidsson, and F. Laurell, "Temperature-tuned difference-frequency mixing in periodically poled KTiOPO4" Appl. Phys. B: Lasers.Opt 67, 675-677 (1998).
[CrossRef]

Lehmann, K.K.

Li, Y.

Lu, D.

Maddaloni, P.

P. Maddaloni, G. Gagliardi, P. Malara and P. De Natale, "A 3.5-mW continuous-wave difference-frequency source around 3 mm for sub-Doppler molecular spectroscopy," Appl. Phys. B 80, 141-145 (2005).
[CrossRef]

Malara, P.

P. Maddaloni, G. Gagliardi, P. Malara and P. De Natale, "A 3.5-mW continuous-wave difference-frequency source around 3 mm for sub-Doppler molecular spectroscopy," Appl. Phys. B 80, 141-145 (2005).
[CrossRef]

Mathieu, P.

S. K. Wong, G. Fournier, P. Mathieu, and P. Pace. "Beam divergence effects on nonlinear frequency mixing," J. Appl. Phys. 71, 1091-1101 (1992).
[CrossRef]

Mazzotti, D.

G. Giusfredi, D. Mazzotti, P. Cancio, and P. De Natale, "Spatial mode control in radiation generated by frequency difference in periodically poled crystals," Phys Rev. Lett. 87, 113901 (2001).
[CrossRef] [PubMed]

Morris, J. R.

J. R. Morris and Y. R. Shen, "Theory of far-infrared generation by optical mixing," Phys. Rev. A 15, 1143-1156 (1977).
[CrossRef]

Pace, P.

S. K. Wong, G. Fournier, P. Mathieu, and P. Pace. "Beam divergence effects on nonlinear frequency mixing," J. Appl. Phys. 71, 1091-1101 (1992).
[CrossRef]

Proctor, M.

Qian, L.

Richter, D.

Rosenman, G.

K. Fradkin, A. Arie, A. Skliar, and G. Rosenman, "Tunable midinfrared source by difference frequency generation in bulk periodically poled KTiOPO4," Appl. Phys. Lett. 74, 914-917 (1999).
[CrossRef]

Shen, Y. R.

J. R. Morris and Y. R. Shen, "Theory of far-infrared generation by optical mixing," Phys. Rev. A 15, 1143-1156 (1977).
[CrossRef]

Skliar, A.

K. Fradkin, A. Arie, A. Skliar, and G. Rosenman, "Tunable midinfrared source by difference frequency generation in bulk periodically poled KTiOPO4," Appl. Phys. Lett. 74, 914-917 (1999).
[CrossRef]

Turnbull, G. A.

G. M. Gibson, G. A. Turnbull, M. Ebrahimzadeh, M. H. Dunn, H. Karlsson, G. Arvidsson, and F. Laurell, "Temperature-tuned difference-frequency mixing in periodically poled KTiOPO4" Appl. Phys. B: Lasers.Opt 67, 675-677 (1998).
[CrossRef]

Weibring, P.

Wong, S. K.

S. K. Wong, G. Fournier, P. Mathieu, and P. Pace. "Beam divergence effects on nonlinear frequency mixing," J. Appl. Phys. 71, 1091-1101 (1992).
[CrossRef]

Yang, H.

Zhu, H.

Appl. Phys. B (1)

P. Maddaloni, G. Gagliardi, P. Malara and P. De Natale, "A 3.5-mW continuous-wave difference-frequency source around 3 mm for sub-Doppler molecular spectroscopy," Appl. Phys. B 80, 141-145 (2005).
[CrossRef]

Appl. Phys. Lett. (2)

K. Fradkin, A. Arie, A. Skliar, and G. Rosenman, "Tunable midinfrared source by difference frequency generation in bulk periodically poled KTiOPO4," Appl. Phys. Lett. 74, 914-917 (1999).
[CrossRef]

H. H. Abu-Safe, "Difference frequency mixing of strongly focused Gaussian beams in periodically poled LiNbO3," Appl. Phys. Lett. 86, 231105 (2005).
[CrossRef]

J. Appl. Phys. (2)

S. K. Wong, G. Fournier, P. Mathieu, and P. Pace. "Beam divergence effects on nonlinear frequency mixing," J. Appl. Phys. 71, 1091-1101 (1992).
[CrossRef]

G. D. Boyd and D. A. Kleinman, "Parametric interaction of focused Gaussian light beams," J. Appl. Phys. 39, 3597-3641 (1968).
[CrossRef]

J. Opt. Soc. Am. B (1)

Opt (1)

G. M. Gibson, G. A. Turnbull, M. Ebrahimzadeh, M. H. Dunn, H. Karlsson, G. Arvidsson, and F. Laurell, "Temperature-tuned difference-frequency mixing in periodically poled KTiOPO4" Appl. Phys. B: Lasers.Opt 67, 675-677 (1998).
[CrossRef]

Opt. (1)

S. Kulin, S. Aubin S. Christe, B. Peker, S. L. Rolston, and L. A. Orozco, " A single hollow-beam optical trap for cold atoms," J. Opt. B: Quantum semiclass. Opt. 3, 353-357 (2001).
[CrossRef]

Opt. Commun. (1)

Jean-Jacques Zondy, "The effects of focusing in type-I and type-II difference-frequency generations," Opt. Commun. 149, 181-206 (1998).
[CrossRef]

Opt. Express (3)

Opt. Lett. (2)

Phys Rev. Lett. (1)

G. Giusfredi, D. Mazzotti, P. Cancio, and P. De Natale, "Spatial mode control in radiation generated by frequency difference in periodically poled crystals," Phys Rev. Lett. 87, 113901 (2001).
[CrossRef] [PubMed]

Phys. Rev. A (1)

J. R. Morris and Y. R. Shen, "Theory of far-infrared generation by optical mixing," Phys. Rev. A 15, 1143-1156 (1977).
[CrossRef]

Phys. Rev. Lett. (1)

J. L. Hall and C. Bordé, "Measurement of methane hyperfine structure using laser saturated absorption" Phys. Rev. Lett. 30, 1101-1104 (1973).
[CrossRef]

Other (2)

R. G. Hunsperger, Integrated Optics - Theory and Technology, (Springer 2002).

F. Zernike and J. E. Midwinter, Applied Nonlinear Optics, (J. Wiley & Sons, 1973).

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Figures (7)

Fig. 1.
Fig. 1.

Frame adopted to describe the nonlinear interaction in a plane orthogonal to the crystal axis

Fig. 2.
Fig. 2.

Map, parametric in T, of plane wave interactions (ϑ s , ϑ p ) that cancel the vector mismatch of Eq. (3). Schemes B, A and A’ are examples of NCQPM configurations. In particular A and A’ (±ϑ s ,0) represent the configurations allowed in the case of interaction between a focused signal beam and a plane pump wave propagating along kg .

Fig. 3.
Fig. 3.

Normalized idler intensity I(T, ϑ) in paraxial direction, in proximity of the collinear quasi phase matching temperature T 0=340.4 K Calculated for λs =1053.05 nm, λp =1551.60 nm and Λ0=29.89 µm. Below T 0, due to NCQPM, the idler intensity is peaked around angles that depart from the axial direction as the temperature decrease.

Fig. 4.
Fig. 4.

Numerical evaluation of Eq. (12): overall output power for a difference frequency generation with focused signal beam (λs =1053.05 nm, λp =1551.60 nm and Λ0=29.89 µm). Different curves correspond to different values of the signal waist ws . The vertical line indicates the collinear QPM temperature T0 .

Fig. 5.
Fig. 5.

Recorded intensity distributions of the far field DFG beam for increasing values of temperature. On the right side, the corresponding theoretical I(T, ϑ) predicted by Eq. (11) are also reported.

Fig. 6.
Fig. 6.

Asymmetry in the output DFG power vs temperature, attributed to NCQPM at T<T0 in presence of focused beams (ws =20 µm, wp =50 µm). Continuous line: theoretical prediction by Eq. (12) for a focused-signal beam DFG (ws =15 µm) and a plane pump wave.

Fig. 7.
Fig. 7.

Half aperture angle of NCQPM idler emission for T<T0 . The continuous line is the fitting curve given by Eq. (7), used to extract the poling period at room temperature.

Equations (12)

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E ( ϑ ) = E 0 w 2 4 π e w 2 k 2 ϑ 2 4
Δ k 0 ( T , λ s , λ p ) = 2 π ( n ( T , λ s ) λ s n ( T , λ p ) λ p n ( T , λ i ) λ i 1 Λ ( T ) ) .
{ k s cos ϑ s k p cos ϑ p k i cos ϑ k g = Δ k z k s sin ϑ s k p sin ϑ p k i sin ϑ = Δ k y
k s ( k p + k g ) ϑ s 2 + k p ( k s k g ) ϑ p 2 2 k s k p ϑ s ϑ p + Δ k 0 ( 2 k i + Δ k 0 ) = 0
E idl ( ϑ s , ϑ p ) E 0 , s E 0 , p e w s 2 k s 2 ϑ s 2 + k p 2 k p 2 ϑ p 2 4 .
{ k s ( 1 ϑ s 2 2 ) k p k i ( 1 ϑ 2 2 ) k g = 0 k s ϑ s k i ϑ = 0 .
ϑ = 2 Δ k 0 k i ( 1 k i k s ) .
Δ k eff ( T , ϑ ) = Δ k 0 + k i 2 ( 1 k i k s ) ϑ 2
E i ( T ) E 0 , s E 0 , p L 2 L 2 e i Δ k 0 z d z
E i ( T , ϑ ) E 0 , s E 0 , p e ( w s k i ϑ 2 ) 2 L 2 ϑ 2 L 2 ϑ 2 e i Δ k eff z d z ,
I ( T , ϑ ) E 0 . s 2 E 0 , p 2 e ( w s k i ϑ ) 2 4 Sinc 2 [ Δ k eff · L 2 ϑ 2 ]
P ( T ) E 0 , s 2 E 0 , p 2 d L d L e ( w s k i ϑ ) 2 4 Sinc 2 ( Δ k eff · L 2 ϑ 2 ) d ϑ .

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