Abstract

We have analyzed optical parametric interaction in a 2D NPC. While in general the nonlinear coefficient is small compared to a 1D NPC, we show that at numerous orientations a multitude of reciprocal vectors contribute additively to enhance the gain in optical parametric amplification and oscillation in a 2D patterned crystal. In particular, we have derived the effective nonlinear coefficients for common-signal amplification and common-idler amplification for a tetragonal inverted domain pattern. We show that in the specific case of signal amplification with QPM by both G10 and G11, symmetry of the crystal results in coupled interaction with the corresponding signal amplification by G10 and G1,-1. As a consequence, this coupled utilization of all three reciprocal vectors leads to a substantial increase in parametric gain. Using PPLN we demonstrate numerically that a gain that comes close to that of a 1D QPM crystal could be realized in a 2D NPC with an inverted tetragonal domain pattern. This special mechanism produces two pairs of identical signal and idler beams propagating in mirror-imaged forward directions. In conjunction with this gain enhancement and multiple beams output we predict that there is a large pulling effect on the output wavelength due to dynamic signal build-up in the intrinsic noncollinear geometry of a 2D NPC OPO.

© 2008 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. V. Berger, "Nonlinear photonic crystals," Phys. Rev. Lett. 81, 4136-4139 (1998).
    [CrossRef]
  2. N. G. R. Broderick, G. W. Ross, H. L. Offerhaus, D. J. Richardson, and D. C. Hanna, "Hexagonally poled lithium niobate: a two-dimensional nonlinear photonic crystal," Phys. Rev. Lett. 84, 4345-4348 (2000).
    [CrossRef] [PubMed]
  3. L.-H. Peng, C.-C. Hsu, J. Ng, and A. H. Kung, "Wavelength tunability of second-harmonic generation from two-dimensional ?(2) nonlinear photonic crystals with a tetragonal lattice structure," Appl. Phys. Lett. 84, 3250-3252 (2004).
    [CrossRef]
  4. M. Seiter and M. W. Sigrist, "On-line multicomponent trace-gas analysis with a broadly tunable pulsed difference-frequency laser spectrometer," Appl. Opt. 38, 4691-4698 (1999).
    [CrossRef]
  5. M. H. Chou, I. Brener, G. Lenz, R. Scotti, E. E. Chaban, J. Shmulovich, D. Philen, S. Kosinski, K. R. Parameswaran, and M. M. Fejer, "Efficient wide-band and tunable midspan spectral inverter using cascaded nonlinearities in LiNbO3 waveguides," IEEE Photonics Technology Lett. 12, 82-84 (2000).
    [CrossRef]
  6. L. E. Myers, G. D. Miller, R. C. Eckardt, M. M. Fejer, and R. L. Byer, "Quasi-phase-matched 1.064 micron-pumped optical parametric oscillator in bulk periodically poled LiNbO3," Opt. Lett. 20, 52-54 (1995).
    [CrossRef] [PubMed]
  7. A. Arie, N. Habshoosh, and A. Bahabad, "Quasi phase matching in 2D nonlinear photonic crystals," Opt. Quantum Electron. 39, 361 (2007).
    [CrossRef]
  8. D. H. Jundt, "Temperature-dependent Sellmeier equation for the index of refraction, ne, in congruent lithium niobate," Opt. Lett. 22, 1553 (1997).
    [CrossRef]
  9. R. W. Boyd, Nonlinear Optics, 2nd Ed. (Academic Press, 2003) Chap. 2.
  10. L.-H. Peng, C.-C. Hsu, and Y.-C. Shih, "Second-harmonic green generation from two-dimensional ?(2) nonlinear photonic crystal with orthorhombic lattice structure," Appl. Phys. Lett. 83, 3447 (2003).
    [CrossRef]
  11. A. E. Siegman, Lasers, (University Science Books, 1986) Chap. 29, pp. 1169.

2007 (1)

A. Arie, N. Habshoosh, and A. Bahabad, "Quasi phase matching in 2D nonlinear photonic crystals," Opt. Quantum Electron. 39, 361 (2007).
[CrossRef]

2004 (1)

L.-H. Peng, C.-C. Hsu, J. Ng, and A. H. Kung, "Wavelength tunability of second-harmonic generation from two-dimensional ?(2) nonlinear photonic crystals with a tetragonal lattice structure," Appl. Phys. Lett. 84, 3250-3252 (2004).
[CrossRef]

2003 (1)

L.-H. Peng, C.-C. Hsu, and Y.-C. Shih, "Second-harmonic green generation from two-dimensional ?(2) nonlinear photonic crystal with orthorhombic lattice structure," Appl. Phys. Lett. 83, 3447 (2003).
[CrossRef]

2000 (2)

N. G. R. Broderick, G. W. Ross, H. L. Offerhaus, D. J. Richardson, and D. C. Hanna, "Hexagonally poled lithium niobate: a two-dimensional nonlinear photonic crystal," Phys. Rev. Lett. 84, 4345-4348 (2000).
[CrossRef] [PubMed]

M. H. Chou, I. Brener, G. Lenz, R. Scotti, E. E. Chaban, J. Shmulovich, D. Philen, S. Kosinski, K. R. Parameswaran, and M. M. Fejer, "Efficient wide-band and tunable midspan spectral inverter using cascaded nonlinearities in LiNbO3 waveguides," IEEE Photonics Technology Lett. 12, 82-84 (2000).
[CrossRef]

1999 (1)

1998 (1)

V. Berger, "Nonlinear photonic crystals," Phys. Rev. Lett. 81, 4136-4139 (1998).
[CrossRef]

1997 (1)

1995 (1)

Arie, A.

A. Arie, N. Habshoosh, and A. Bahabad, "Quasi phase matching in 2D nonlinear photonic crystals," Opt. Quantum Electron. 39, 361 (2007).
[CrossRef]

Bahabad, A.

A. Arie, N. Habshoosh, and A. Bahabad, "Quasi phase matching in 2D nonlinear photonic crystals," Opt. Quantum Electron. 39, 361 (2007).
[CrossRef]

Berger, V.

V. Berger, "Nonlinear photonic crystals," Phys. Rev. Lett. 81, 4136-4139 (1998).
[CrossRef]

Brener, I.

M. H. Chou, I. Brener, G. Lenz, R. Scotti, E. E. Chaban, J. Shmulovich, D. Philen, S. Kosinski, K. R. Parameswaran, and M. M. Fejer, "Efficient wide-band and tunable midspan spectral inverter using cascaded nonlinearities in LiNbO3 waveguides," IEEE Photonics Technology Lett. 12, 82-84 (2000).
[CrossRef]

Broderick, N. G. R.

N. G. R. Broderick, G. W. Ross, H. L. Offerhaus, D. J. Richardson, and D. C. Hanna, "Hexagonally poled lithium niobate: a two-dimensional nonlinear photonic crystal," Phys. Rev. Lett. 84, 4345-4348 (2000).
[CrossRef] [PubMed]

Byer, R. L.

Chaban, E. E.

M. H. Chou, I. Brener, G. Lenz, R. Scotti, E. E. Chaban, J. Shmulovich, D. Philen, S. Kosinski, K. R. Parameswaran, and M. M. Fejer, "Efficient wide-band and tunable midspan spectral inverter using cascaded nonlinearities in LiNbO3 waveguides," IEEE Photonics Technology Lett. 12, 82-84 (2000).
[CrossRef]

Chou, M. H.

M. H. Chou, I. Brener, G. Lenz, R. Scotti, E. E. Chaban, J. Shmulovich, D. Philen, S. Kosinski, K. R. Parameswaran, and M. M. Fejer, "Efficient wide-band and tunable midspan spectral inverter using cascaded nonlinearities in LiNbO3 waveguides," IEEE Photonics Technology Lett. 12, 82-84 (2000).
[CrossRef]

Eckardt, R. C.

Fejer, M. M.

M. H. Chou, I. Brener, G. Lenz, R. Scotti, E. E. Chaban, J. Shmulovich, D. Philen, S. Kosinski, K. R. Parameswaran, and M. M. Fejer, "Efficient wide-band and tunable midspan spectral inverter using cascaded nonlinearities in LiNbO3 waveguides," IEEE Photonics Technology Lett. 12, 82-84 (2000).
[CrossRef]

L. E. Myers, G. D. Miller, R. C. Eckardt, M. M. Fejer, and R. L. Byer, "Quasi-phase-matched 1.064 micron-pumped optical parametric oscillator in bulk periodically poled LiNbO3," Opt. Lett. 20, 52-54 (1995).
[CrossRef] [PubMed]

Habshoosh, N.

A. Arie, N. Habshoosh, and A. Bahabad, "Quasi phase matching in 2D nonlinear photonic crystals," Opt. Quantum Electron. 39, 361 (2007).
[CrossRef]

Hanna, D. C.

N. G. R. Broderick, G. W. Ross, H. L. Offerhaus, D. J. Richardson, and D. C. Hanna, "Hexagonally poled lithium niobate: a two-dimensional nonlinear photonic crystal," Phys. Rev. Lett. 84, 4345-4348 (2000).
[CrossRef] [PubMed]

Hsu, C.-C.

L.-H. Peng, C.-C. Hsu, J. Ng, and A. H. Kung, "Wavelength tunability of second-harmonic generation from two-dimensional ?(2) nonlinear photonic crystals with a tetragonal lattice structure," Appl. Phys. Lett. 84, 3250-3252 (2004).
[CrossRef]

L.-H. Peng, C.-C. Hsu, and Y.-C. Shih, "Second-harmonic green generation from two-dimensional ?(2) nonlinear photonic crystal with orthorhombic lattice structure," Appl. Phys. Lett. 83, 3447 (2003).
[CrossRef]

Jundt, D. H.

Kosinski, S.

M. H. Chou, I. Brener, G. Lenz, R. Scotti, E. E. Chaban, J. Shmulovich, D. Philen, S. Kosinski, K. R. Parameswaran, and M. M. Fejer, "Efficient wide-band and tunable midspan spectral inverter using cascaded nonlinearities in LiNbO3 waveguides," IEEE Photonics Technology Lett. 12, 82-84 (2000).
[CrossRef]

Kung, A. H.

L.-H. Peng, C.-C. Hsu, J. Ng, and A. H. Kung, "Wavelength tunability of second-harmonic generation from two-dimensional ?(2) nonlinear photonic crystals with a tetragonal lattice structure," Appl. Phys. Lett. 84, 3250-3252 (2004).
[CrossRef]

Lenz, G.

M. H. Chou, I. Brener, G. Lenz, R. Scotti, E. E. Chaban, J. Shmulovich, D. Philen, S. Kosinski, K. R. Parameswaran, and M. M. Fejer, "Efficient wide-band and tunable midspan spectral inverter using cascaded nonlinearities in LiNbO3 waveguides," IEEE Photonics Technology Lett. 12, 82-84 (2000).
[CrossRef]

Miller, G. D.

Myers, L. E.

Ng, J.

L.-H. Peng, C.-C. Hsu, J. Ng, and A. H. Kung, "Wavelength tunability of second-harmonic generation from two-dimensional ?(2) nonlinear photonic crystals with a tetragonal lattice structure," Appl. Phys. Lett. 84, 3250-3252 (2004).
[CrossRef]

Offerhaus, H. L.

N. G. R. Broderick, G. W. Ross, H. L. Offerhaus, D. J. Richardson, and D. C. Hanna, "Hexagonally poled lithium niobate: a two-dimensional nonlinear photonic crystal," Phys. Rev. Lett. 84, 4345-4348 (2000).
[CrossRef] [PubMed]

Parameswaran, K. R.

M. H. Chou, I. Brener, G. Lenz, R. Scotti, E. E. Chaban, J. Shmulovich, D. Philen, S. Kosinski, K. R. Parameswaran, and M. M. Fejer, "Efficient wide-band and tunable midspan spectral inverter using cascaded nonlinearities in LiNbO3 waveguides," IEEE Photonics Technology Lett. 12, 82-84 (2000).
[CrossRef]

Peng, L.-H.

L.-H. Peng, C.-C. Hsu, J. Ng, and A. H. Kung, "Wavelength tunability of second-harmonic generation from two-dimensional ?(2) nonlinear photonic crystals with a tetragonal lattice structure," Appl. Phys. Lett. 84, 3250-3252 (2004).
[CrossRef]

L.-H. Peng, C.-C. Hsu, and Y.-C. Shih, "Second-harmonic green generation from two-dimensional ?(2) nonlinear photonic crystal with orthorhombic lattice structure," Appl. Phys. Lett. 83, 3447 (2003).
[CrossRef]

Philen, D.

M. H. Chou, I. Brener, G. Lenz, R. Scotti, E. E. Chaban, J. Shmulovich, D. Philen, S. Kosinski, K. R. Parameswaran, and M. M. Fejer, "Efficient wide-band and tunable midspan spectral inverter using cascaded nonlinearities in LiNbO3 waveguides," IEEE Photonics Technology Lett. 12, 82-84 (2000).
[CrossRef]

Richardson, D. J.

N. G. R. Broderick, G. W. Ross, H. L. Offerhaus, D. J. Richardson, and D. C. Hanna, "Hexagonally poled lithium niobate: a two-dimensional nonlinear photonic crystal," Phys. Rev. Lett. 84, 4345-4348 (2000).
[CrossRef] [PubMed]

Ross, G. W.

N. G. R. Broderick, G. W. Ross, H. L. Offerhaus, D. J. Richardson, and D. C. Hanna, "Hexagonally poled lithium niobate: a two-dimensional nonlinear photonic crystal," Phys. Rev. Lett. 84, 4345-4348 (2000).
[CrossRef] [PubMed]

Scotti, R.

M. H. Chou, I. Brener, G. Lenz, R. Scotti, E. E. Chaban, J. Shmulovich, D. Philen, S. Kosinski, K. R. Parameswaran, and M. M. Fejer, "Efficient wide-band and tunable midspan spectral inverter using cascaded nonlinearities in LiNbO3 waveguides," IEEE Photonics Technology Lett. 12, 82-84 (2000).
[CrossRef]

Seiter, M.

Shih, Y.-C.

L.-H. Peng, C.-C. Hsu, and Y.-C. Shih, "Second-harmonic green generation from two-dimensional ?(2) nonlinear photonic crystal with orthorhombic lattice structure," Appl. Phys. Lett. 83, 3447 (2003).
[CrossRef]

Shmulovich, J.

M. H. Chou, I. Brener, G. Lenz, R. Scotti, E. E. Chaban, J. Shmulovich, D. Philen, S. Kosinski, K. R. Parameswaran, and M. M. Fejer, "Efficient wide-band and tunable midspan spectral inverter using cascaded nonlinearities in LiNbO3 waveguides," IEEE Photonics Technology Lett. 12, 82-84 (2000).
[CrossRef]

Sigrist, M. W.

Appl. Opt. (1)

Appl. Phys. Lett. (2)

L.-H. Peng, C.-C. Hsu, and Y.-C. Shih, "Second-harmonic green generation from two-dimensional ?(2) nonlinear photonic crystal with orthorhombic lattice structure," Appl. Phys. Lett. 83, 3447 (2003).
[CrossRef]

L.-H. Peng, C.-C. Hsu, J. Ng, and A. H. Kung, "Wavelength tunability of second-harmonic generation from two-dimensional ?(2) nonlinear photonic crystals with a tetragonal lattice structure," Appl. Phys. Lett. 84, 3250-3252 (2004).
[CrossRef]

IEEE Photonics Technology Lett. (1)

M. H. Chou, I. Brener, G. Lenz, R. Scotti, E. E. Chaban, J. Shmulovich, D. Philen, S. Kosinski, K. R. Parameswaran, and M. M. Fejer, "Efficient wide-band and tunable midspan spectral inverter using cascaded nonlinearities in LiNbO3 waveguides," IEEE Photonics Technology Lett. 12, 82-84 (2000).
[CrossRef]

Opt. Lett. (2)

Opt. Quantum Electron. (1)

A. Arie, N. Habshoosh, and A. Bahabad, "Quasi phase matching in 2D nonlinear photonic crystals," Opt. Quantum Electron. 39, 361 (2007).
[CrossRef]

Phys. Rev. Lett. (2)

V. Berger, "Nonlinear photonic crystals," Phys. Rev. Lett. 81, 4136-4139 (1998).
[CrossRef]

N. G. R. Broderick, G. W. Ross, H. L. Offerhaus, D. J. Richardson, and D. C. Hanna, "Hexagonally poled lithium niobate: a two-dimensional nonlinear photonic crystal," Phys. Rev. Lett. 84, 4345-4348 (2000).
[CrossRef] [PubMed]

Other (2)

R. W. Boyd, Nonlinear Optics, 2nd Ed. (Academic Press, 2003) Chap. 2.

A. E. Siegman, Lasers, (University Science Books, 1986) Chap. 29, pp. 1169.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (8)

Fig. 1.
Fig. 1.

Reciprocal lattice of a tetragonal structure

Fig. 2.
Fig. 2.

the wavelength- θs (θi ) diagram of G 10 , G 11 , and G 1,-1 . Black lines are the signal, while blue lines are the idler.

Fig. 3.
Fig. 3.

(Left) Phase-matching condition in point E. (Right) 2D OPG diagram in point E.

Fig. 4.
Fig. 4.

(a) Hexagonal inverted domain in the simulation. (b) Square inverted domain. The small dashed square is half the size of the period.

Fig. 5.
Fig. 5.

The gain-length product gL as a function of signal wavelength and θ s ¯ . w0 =160 µm. The solid lines are the QPM solution of G10 , G11 , and G1,-1 .

Fig. 6.
Fig. 6.

The gain-length product gL (a) and the output signal wavelength (b) versus θ s ¯ . w0 =160 µm. The solid lines in the right figure are the QPM solutions for G10 and G11 , respectively from the Sellmeier equations of PPLN.

Fig. 7.
Fig. 7.

The gain-length product gL as a function of signal wavelength and signal angle. w0 =1000 µm. The solid lines are the QPM solution of G10 , G11 , and G1,-1 .

Fig. 8.
Fig. 8.

The gain-length product gL (a) and the output signal wavelength (b) versus θ s ¯ . w0 =1000 µm. The solid lines in the right figure are the QPM solutions for G10 and G11 , respectively from the Sellmeier equations of PPLN.

Tables (3)

Tables Icon

Table 1. The Fourier coefficients amn of the first 3 terms of equation 6 for Fig. 4 (a) and (b)

Tables Icon

Table 2. The signal wavelength and QPM direction relative to G 10 for the points A–F in Fig. 2 for the 2D PPLN crystal used in the numerical example.

Tables Icon

Table 3. The normalized effective nonlinear coefficient d eff ¯ = d eff d 33 for various beam radiuses.

Equations (25)

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

G mn = m 2 π Λ x x ̂ + n 2 π Λ y y ̂ ,
k p = k s + k i + G mn ,
cos ( θ s θ ) = k 2 + k s 2 k i 2 2 k k s
cos ( θ i θ ) = k 2 + k i 2 k s 2 2 k k i ,
d E p ( x ) d x = i 2 μ 0 w p 2 k p d 33 ( r ) E s ( x ) E i ( x ) exp [ i Δ k · r ]
d E s ( x ) d x = i 2 μ 0 w s 2 k s d 33 ( r ) E p ( x ) E i * ( x ) exp [ i Δ k · r ] ,
d E i ( x ) d x = i 2 μ 0 w i 2 k i d 33 ( r ) E p ( x ) E s * ( x ) exp [ i Δ k · r ]
d 33 ( r ) = d 33 m = n = a mn exp [ i G mn · r ] ,
d 2 E s ( x ) d x 2 = 4 μ 0 2 w s 2 w i 2 d 33 2 E p 2 k s k i · a mn 2 E s ( x ) .
E s ( x ) = E s ( 0 ) cosh ( gx ) ,
g = 2 μ 0 w s w i d 33 E p k s k i · a mn
d E s ( x ) d x = i 2 μ 0 w s 2 d 33 E p k s ( a 1 E i 1 * ( x ) + a 2 E i 2 * ( x ) )
d E i 1 ( x ) d x = i 2 μ 0 w i 2 d 33 E p k i a 1 E s * ( x )
d E i 2 ( x ) d x = i 2 μ 0 w i 2 d 33 E p k i a 2 E s * ( x ) .
g = 2 μ 0 w s w i d 33 E p k s k i · a 1 2 + a 2 2 .
d E s 1 ( x ) d x = i 2 μ 0 w s 2 d 33 E p k s a 1 E i * ( x )
d E s 2 ( x ) d x = i 2 μ 0 w s 2 d 33 E p k s a 2 E i * ( x )
d E i ( x ) d x = i 2 μ 0 w i 2 d 33 E p k i ( a 1 E s 1 * ( x ) + a 2 E s 2 * ( x ) )
E s 1 ( x ) = E s 1 ( 0 ) [ a 2 2 a 1 2 + a 2 2 + a 1 2 a 1 2 + a 2 2 cosh ( gx ) ] ,
d E s 1 ( x ) d x = i 2 μ 0 w s 2 d 33 E p k s ( a 11 E i 1 * ( x ) + a 10 E i 2 * ( x ) )
d E s 2 ( x ) d x = i 2 μ 0 w s 2 d 33 E p k s ( a 10 E i 1 * ( x ) + a 1 , 1 E i 2 * ( x ) )
d E i 1 ( x ) d x = i 2 μ 0 w i 2 d 33 E p k i ( a 11 E s 1 * ( x ) + a 10 E s 2 * ( x ) )
d E i 2 ( x ) d x = i 2 μ 0 w i 2 d 33 E p k i ( a 10 E s 1 * ( x ) + a 1 , 1 E s 2 * ( x ) ) .
d 2 d x 2 [ E s 1 ( x ) E s 2 ( x ) ] = 4 μ 0 2 w s 2 w i 2 d 33 2 E p 2 k s k i [ a 10 2 + a 11 2 a 11 a 10 * + a 10 a 1 , 1 * a 10 a 11 * + a 1 , 1 a 10 * a 10 2 + a 1 , 1 2 ] [ E s 1 ( x ) E s 2 ( x ) ] .
d eff = ( a 10 + a 11 ) d 33 .

Metrics