Abstract

We describe a method for designing 1-dimensional aperiodic poled grating structures of finite length that quasi-phase match multiple χ (2) processes. The poling functions for such gratings are best aligned, in terms of the dot product in Fourier space, with a design target. No restrictions are placed on the quasi-phase matching wave numbers. A grating designed for third harmonic generation is simulated.

© 2004 Optical Society of America

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References

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  1. K. Fradkin-Kashi and A. Arie, �??Multiple-wavelength quasi-phase-matched nonlinear interactions,�?? IEEE J. Quantum Electron. 35, 1649�??1656 (1999).
    [CrossRef]
  2. O. Bang, C. B. Clausen, P. L. Christiansen, and L. Torner, �??Engineering competing nonlinearities,�?? Opt. Lett. 24, 1413�??1415 (1999).
    [CrossRef]
  3. M. Asobe, O. Tadanaga, H. Miyazawa, Y. Nishida, and H. Suzuki, �??Multiple quasi-phase-matched LiNbO3 wave length converter with a continuously phase-modulated domain structure,�?? Opt. Lett. 28, 558�??560 (2003).
    [CrossRef] [PubMed]
  4. M. L. Bortz, �??Quasi-Phasematched Optical Frequency Conversion in Lithium Niobate Waveguides,�?? Ph.D. thesis, Stanford University (1994).
  5. M. H. Chou, K. R. Parameswaran, M. M. Fejer, and I. Brener, �??Multiple-channel wavelength conversion by use of engineered quasi-phase-matching structures in LiNbO3 waveguides,�?? Opt. Lett. 24, 1157�??1159 (1999).
    [CrossRef]
  6. Z.-W. Liu, Y. Du, J. Liao, S.-N. Zhu, Y.-Y. Zhu, Y.-Q. Qin, H.-T. Wang, J.-L. He, C. Zhang, and N.-B. Ming, �??Engineering of a dual-periodic optical superlattice used in a coupled optical parametric interaction,�?? J. Opt. Soc. Am. B 19, 1676�??1684 (2002).
    [CrossRef]
  7. S. M. Saltiel, A. A. Sukhorukov, and Y. S. Kivshar, �??Multistep parametric processes in nonlinear optics,�?? <a href="http://arxiv.org/abs/nlin.PS/0311013"> (2003).http://arxiv.org/abs/nlin.PS/0311013</a>
  8. D. H. Jundt, �??Temperature-dependent Sellmeier equation for the index of refraction, ne, in congruent lithium niobate,�?? Opt. Lett. 22, 1553�??1555 (1997).
    [CrossRef]
  9. C. Zhang, Y. Zhu, S. Yang, Y. Qin, S. Zhu, Y. Chen, H. Liu, and N. Ming, �??Crucial effects of coupling coefficients on quasi-phase-matched harmonic generation in an optical superlattice,�?? Opt. Lett. 25, 436�??438 (2000).
    [CrossRef]
  10. A. H. Norton and C. M. de Sterke, �??Two-dimensional poling patterns for 3rd and 4th harmonic generation,�?? Opt. Express 11, 1008�??1014 (2003), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-9-1008">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-9-1008</a>.
    [CrossRef] [PubMed]

IEEE J. Quantum Electron. (1)

K. Fradkin-Kashi and A. Arie, �??Multiple-wavelength quasi-phase-matched nonlinear interactions,�?? IEEE J. Quantum Electron. 35, 1649�??1656 (1999).
[CrossRef]

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

Z.-W. Liu, Y. Du, J. Liao, S.-N. Zhu, Y.-Y. Zhu, Y.-Q. Qin, H.-T. Wang, J.-L. He, C. Zhang, and N.-B. Ming, �??Engineering of a dual-periodic optical superlattice used in a coupled optical parametric interaction,�?? J. Opt. Soc. Am. B 19, 1676�??1684 (2002).
[CrossRef]

Opt. Express (1)

A. H. Norton and C. M. de Sterke, �??Two-dimensional poling patterns for 3rd and 4th harmonic generation,�?? Opt. Express 11, 1008�??1014 (2003), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-9-1008">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-9-1008</a>.
[CrossRef] [PubMed]

Opt. Lett. (5)

M. H. Chou, K. R. Parameswaran, M. M. Fejer, and I. Brener, �??Multiple-channel wavelength conversion by use of engineered quasi-phase-matching structures in LiNbO3 waveguides,�?? Opt. Lett. 24, 1157�??1159 (1999).
[CrossRef]

D. H. Jundt, �??Temperature-dependent Sellmeier equation for the index of refraction, ne, in congruent lithium niobate,�?? Opt. Lett. 22, 1553�??1555 (1997).
[CrossRef]

C. Zhang, Y. Zhu, S. Yang, Y. Qin, S. Zhu, Y. Chen, H. Liu, and N. Ming, �??Crucial effects of coupling coefficients on quasi-phase-matched harmonic generation in an optical superlattice,�?? Opt. Lett. 25, 436�??438 (2000).
[CrossRef]

O. Bang, C. B. Clausen, P. L. Christiansen, and L. Torner, �??Engineering competing nonlinearities,�?? Opt. Lett. 24, 1413�??1415 (1999).
[CrossRef]

M. Asobe, O. Tadanaga, H. Miyazawa, Y. Nishida, and H. Suzuki, �??Multiple quasi-phase-matched LiNbO3 wave length converter with a continuously phase-modulated domain structure,�?? Opt. Lett. 28, 558�??560 (2003).
[CrossRef] [PubMed]

Progress in Optics (1)

S. M. Saltiel, A. A. Sukhorukov, and Y. S. Kivshar, �??Multistep parametric processes in nonlinear optics,�?? <a href="http://arxiv.org/abs/nlin.PS/0311013"> (2003).http://arxiv.org/abs/nlin.PS/0311013</a>

Other (1)

M. L. Bortz, �??Quasi-Phasematched Optical Frequency Conversion in Lithium Niobate Waveguides,�?? Ph.D. thesis, Stanford University (1994).

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Figures (3)

Fig. 1.
Fig. 1.

(a) The target Fourier transform (k) (red) and the Fourier transform of the best aligned poled grating (k) (blue) of length L=1cm. The two QPM peaks were chosen for THG (see Section 5). (b) Close-up of the QPM peak at k=G 1. (c) The peak at k=G 2. The target peaks have half-widths ΔG=2π/L.

Fig. 2.
Fig. 2.

Part of a THG poled grating defined by p(x)=sign(n(x)) where n(x)=w 1 cos(G 1 x)+w 2 cos(G 2 x). Values for wj and Gj are given in the text (Section 5). The grating length is L=1cm, of which 0.5mm is shown. Domains below fabrication resolution (≈1 µm for LiNbO3) can be deleted by inversion before the grating is simulated.

Fig. 3.
Fig. 3.

THG simulation. Relative energy of the fundamental, 2nd and 3rd harmonic waves (red, green and blue curves respectively) as a function of distance through the poled grating.

Equations (26)

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p ( x ) = { ± 1 for x [ L 2 , L 2 ] , 0 otherwise ,
k 1 + k 1 + G 1 = k 2 , k 1 + k 2 + G 2 = k 3 .
u k = 1 2 π e ikx .
f ( x ) = f ̂ ( k ) u k d k = 1 2 π f ̂ ( k ) e ikx d k ,
f 1 , f 2 = 2 π f 1 ( x ) f 2 ( x ) * d x .
f ̂ ( k ) = f , u k = f ( x ) e ikx d x .
f = Re { f ̂ ( k ) } u k d k + Im { f ̂ ( k ) } i u k d k .
f 1 · f 2 = 1 2 ( f 1 , f 2 + f 2 , f 1 )
= 2 π ( Re { f 1 ( x ) } Re { f 2 ( x ) } + Im { f 1 ( x ) } Im { f 2 ( x ) } ) d x .
f 1 · f 2 = ( Re { f ̂ 1 ( k ) } Re { f ̂ 2 ( k ) } + Im { f ̂ 1 ( k ) } Im { f ̂ 2 ( k ) } ) d k .
p · n = 2 π L 2 L 2 p ( x ) Re { n ( x ) } d x ,
p ( x ) = { sign ( Re { n ( x ) } ) for x [ L 2 , L 2 ] , 0 otherwise .
p ̂ ( k ) = j = 1 r 1 p j h ̂ j ( k ) ,
h j ( x ) = { 1 for x ( x j , x j + 1 ) , 0 otherwise ,
h ̂ j ( k ) = i k ( exp ( i k x j + 1 ) exp ( i k x j ) ) .
H j ( k ) = { 1 for G j Δ G j < k < G j + Δ G j , 0 otherwise ,
n ̂ ( k ) = j = 1 N ( a j ( H j ( k ) + H j ( k ) ) + i b j ( H j ( k ) H j ( k ) ) ) ,
n ( x ) = 1 2 π j = 1 N ( a j ( H j ( k ) + H j ( k ) ) + i b j ( H j ( k ) H j ( k ) ) ) e ikx d k
= 1 π j = 1 N 0 ( a j cos ( k x ) b j sin ( k x ) ) H j ( k ) d k
= j = 1 N 2 sin ( Δ G j x ) π x ( a j cos ( G j x ) b j sin ( G j x ) )
= j = 1 N 2 sin ( Δ G j x ) π x w j cos ( G j x + ϕ j )
p ( x ) = { sign ( Σ j = 1 N sin ( Δ G j x ) x w j cos ( G j x + ϕ j ) ) for x [ L 2 , L 2 ] , 0 otherwise .
p ( x ) = { sign ( Σ j = 1 N w j cos ( G j x + ϕ j ) ) for x [ L 2 , L 2 ] , 0 otherwise .
d a 1 dx = i ω n 1 c χ p ( x ) ( a 2 a 1 * e i G 1 x + a 3 a 2 * e i G 2 x ) ,
d a 2 dx = i ω n 2 c χ p ( x ) ( 2 a 3 a 1 * e i G 2 x + a 1 2 e i G 1 x ) ,
d a 3 dx = 3 i ω n 3 c χ p ( x ) a 1 a 2 e i G 2 x ,

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