Abstract

The exact analytical solutions for the interacting waves in the quadratic nonlinear medium with the periodic structure are detailedly derived and obtained, and the properties of solutions are analyzed. Three applicable examples employing the exact solutions in the all-optical processing are given and analyzed. The optimized results show that the phase of signal can obviously be increased by proper choosing Δk, and that the intensity of pump can greatly be decreased in the all-optical switching by means of optimizing Δk, increasing medium length, and choosing sum-frequency generation.

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References

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  1. G. I. Stegeman, D. J. Hagan, L. Torner, "?(2) cascading phenomena and their applications to all-optical signal processing, mode-locking, pulse, compression and solitions," Opt. Quantum Electron. 28, 1691-1740 (1996).
    [CrossRef]
  2. S. J. B. Yoo, "Wavelength conversion technologies for WDM network applications," J. Lightwave Technol. 14, 955-966 (1996).
    [CrossRef]
  3. G. P. Banfi, P. K. Datta, V. Degiorgio, D. Fortusini, "Wavelength shifting and amplification of optical pulses through cascaded second-order processes in periodically poled lithium niobate," Appl. Phys. Lett. 73, 136-138 (1998).
    [CrossRef]
  4. G. Assanto and I. Torelli, "Cascading effects in type II second-harmonic generation: application to all-optical processing," Opt. Commun. 119, 143-148 (1995).
    [CrossRef]
  5. J. Leuthold, P. A. Besse, E. Gamper, M. Dulk, S. Fischer, G. Guekos, H. Melchior,, "All-optical Mach-Zehnder interferometer wavelength converters and switches with integrated data- and control-signal separation scheme," J. lightwave Technol. 17, 1056-1065 (1999).
    [CrossRef]
  6. G. S. Kanter and P. Kumar, "Optical devices based on internally seeded cascaded nonlinearities," IEEE. J. Quantum Elecron. 35, 891-896 (1999).
    [CrossRef]
  7. J. A. Armstrong, N. Bloembergen N, J. Ducuing, et al. "Interaction between light waves in a nonlinear dielectric," Phys. Rev. 127, 1918-1939 (1962).
    [CrossRef]
  8. A. Kobyakov and F. Lederer, "Cascading of quadratic nonlinearities: an analytical study," Phys. Rev. A 54, 3455-3471 (1996).
    [CrossRef] [PubMed]
  9. M. Asghari, I. H. White, R. V. Penty, "Wavelength conversion using semiconductor optical amplifiers," J. Lighteave Technol. 15, 1181-1190 (1997).
    [CrossRef]
  10. R. W. Boyd, Nonlinear Optics (Academic Press, San Diego, 1992), Chap.2.
  11. M. J. T. Milton, "General expressions for the efficiency of phase-matched and nonphase-matched second-order nonlinear interactions between plane waves," IEEE. J. Quantum Elecron. 28, 739-749 (1992).
    [CrossRef]
  12. Y. R. Shen, The Principles of Nonlinear Optics (Wiley, New York, 1984), Chap.6-7.
  13. A. R. C. Sibilia, E. Fazio, M. Bertolotti, "Field dependent effect in a quadratic nonlinear medium," J. Mod. Opt. 42, 823-839 (1995).
    [CrossRef]
  14. C. N. Ironside, J. S. Aitchison, J. M. Arnold, "An all-optical switch employing the cascaded second-order nonlinear effect," IEEE. J. Quantum Elecron. 29, 2650-2654 (1993).
    [CrossRef]
  15. A. Kobyakov, U. Peschel, F. Lederer, "Vectorial type-II interaction in cascaded quadratic nonlinearities-an analytical approach," Opt. Commun. 124, 184-194 (1996).
    [CrossRef]
  16. G. D'Aguanno, C. Sibilia, E. Fazio, M. Bertolotti, "Three-wave mixing in a quadratic material under perfect phase-matching,"Opt. Commun. 142, 75-78 (1997).
    [CrossRef]
  17. X. -M. Liu, H. -Y. Zhang, Y. -L. Guo, "Theoretical analyses and optimizations for wavelength conversion by quasi-phase-matching difference-frequency generation," J. Lightwave Technol. 19, 1785-1792 (2001).
    [CrossRef]
  18. T. Suhara, H. Nishihara, "Theoretical analysis of waveguide second-harmonic generation phase matched with uniform and chirped gratings," IEEE. J. Quantum Elecron. 26, 1265-1270 (1990).
    [CrossRef]
  19. X. -M. Liu and M. -D. Zhang, "Theoretical studies for the special states of the cascaded quadratic nonlinear effects", J. Opt. Soc. Am. B 18, 1659-1666 (2001).
    [CrossRef]
  20. [USA] G. A. Kehen, T. M. Kehen, Handbook of Mathematics (Worker Press, Beijing, 1987), (in Chinese).
  21. M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover Publications, Dover, 1965), Chap. 16-17.
  22. H. Kanbara, H. Itoh, M. Asobe, K. Noguchi, H. Miyazawa, T. Yanagawa, I. Yokohama, "All-optical switching based on cascading of second-order nonlinearities in a periodically poled titanium-diffused lithium niobate waveguide," IEEE. Photon. Technol. Lett. 11, 328-330 (1999).
    [CrossRef]
  23. C. Q. Xu, H. Okayama, M. Kawahara, "Optical frequency conversions in nonlinear medium with periodically modulated linear and nonlinear optical parameters," IEEE. J. Quantum Elecron. 31, 981-987 (1995).
    [CrossRef]
  24. X. -M. Liu, H. -Y. Zhang, Y. -H Li, "Optimal design for the quasi-phase-matching three-wave mixing," Opt. Express 9, 631-636 (2001), <a href="http://www.opticsexpress.org/oearchive/source/37804.htm">http://www.opticsexpress.org/oearchive/source/37804.htm</a>
    [CrossRef] [PubMed]

Opt. Express

Other

G. I. Stegeman, D. J. Hagan, L. Torner, "?(2) cascading phenomena and their applications to all-optical signal processing, mode-locking, pulse, compression and solitions," Opt. Quantum Electron. 28, 1691-1740 (1996).
[CrossRef]

S. J. B. Yoo, "Wavelength conversion technologies for WDM network applications," J. Lightwave Technol. 14, 955-966 (1996).
[CrossRef]

G. P. Banfi, P. K. Datta, V. Degiorgio, D. Fortusini, "Wavelength shifting and amplification of optical pulses through cascaded second-order processes in periodically poled lithium niobate," Appl. Phys. Lett. 73, 136-138 (1998).
[CrossRef]

G. Assanto and I. Torelli, "Cascading effects in type II second-harmonic generation: application to all-optical processing," Opt. Commun. 119, 143-148 (1995).
[CrossRef]

J. Leuthold, P. A. Besse, E. Gamper, M. Dulk, S. Fischer, G. Guekos, H. Melchior,, "All-optical Mach-Zehnder interferometer wavelength converters and switches with integrated data- and control-signal separation scheme," J. lightwave Technol. 17, 1056-1065 (1999).
[CrossRef]

G. S. Kanter and P. Kumar, "Optical devices based on internally seeded cascaded nonlinearities," IEEE. J. Quantum Elecron. 35, 891-896 (1999).
[CrossRef]

J. A. Armstrong, N. Bloembergen N, J. Ducuing, et al. "Interaction between light waves in a nonlinear dielectric," Phys. Rev. 127, 1918-1939 (1962).
[CrossRef]

A. Kobyakov and F. Lederer, "Cascading of quadratic nonlinearities: an analytical study," Phys. Rev. A 54, 3455-3471 (1996).
[CrossRef] [PubMed]

M. Asghari, I. H. White, R. V. Penty, "Wavelength conversion using semiconductor optical amplifiers," J. Lighteave Technol. 15, 1181-1190 (1997).
[CrossRef]

R. W. Boyd, Nonlinear Optics (Academic Press, San Diego, 1992), Chap.2.

M. J. T. Milton, "General expressions for the efficiency of phase-matched and nonphase-matched second-order nonlinear interactions between plane waves," IEEE. J. Quantum Elecron. 28, 739-749 (1992).
[CrossRef]

Y. R. Shen, The Principles of Nonlinear Optics (Wiley, New York, 1984), Chap.6-7.

A. R. C. Sibilia, E. Fazio, M. Bertolotti, "Field dependent effect in a quadratic nonlinear medium," J. Mod. Opt. 42, 823-839 (1995).
[CrossRef]

C. N. Ironside, J. S. Aitchison, J. M. Arnold, "An all-optical switch employing the cascaded second-order nonlinear effect," IEEE. J. Quantum Elecron. 29, 2650-2654 (1993).
[CrossRef]

A. Kobyakov, U. Peschel, F. Lederer, "Vectorial type-II interaction in cascaded quadratic nonlinearities-an analytical approach," Opt. Commun. 124, 184-194 (1996).
[CrossRef]

G. D'Aguanno, C. Sibilia, E. Fazio, M. Bertolotti, "Three-wave mixing in a quadratic material under perfect phase-matching,"Opt. Commun. 142, 75-78 (1997).
[CrossRef]

X. -M. Liu, H. -Y. Zhang, Y. -L. Guo, "Theoretical analyses and optimizations for wavelength conversion by quasi-phase-matching difference-frequency generation," J. Lightwave Technol. 19, 1785-1792 (2001).
[CrossRef]

T. Suhara, H. Nishihara, "Theoretical analysis of waveguide second-harmonic generation phase matched with uniform and chirped gratings," IEEE. J. Quantum Elecron. 26, 1265-1270 (1990).
[CrossRef]

X. -M. Liu and M. -D. Zhang, "Theoretical studies for the special states of the cascaded quadratic nonlinear effects", J. Opt. Soc. Am. B 18, 1659-1666 (2001).
[CrossRef]

[USA] G. A. Kehen, T. M. Kehen, Handbook of Mathematics (Worker Press, Beijing, 1987), (in Chinese).

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover Publications, Dover, 1965), Chap. 16-17.

H. Kanbara, H. Itoh, M. Asobe, K. Noguchi, H. Miyazawa, T. Yanagawa, I. Yokohama, "All-optical switching based on cascading of second-order nonlinearities in a periodically poled titanium-diffused lithium niobate waveguide," IEEE. Photon. Technol. Lett. 11, 328-330 (1999).
[CrossRef]

C. Q. Xu, H. Okayama, M. Kawahara, "Optical frequency conversions in nonlinear medium with periodically modulated linear and nonlinear optical parameters," IEEE. J. Quantum Elecron. 31, 981-987 (1995).
[CrossRef]

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Figures (8)

Fig.1.
Fig.1.

Evolutions of the intensities and phases in QPM-SFG with the interacting length z, where Δk=0.0419 mm-1.

Fig.2.
Fig.2.

Evolutions of the intensities and phases in QPM-DFG with the interacting length z, where Δk=-0.0507 mm-1.

Fig.3.
Fig.3.

The phase evolutions of three interacting waves with z under the difference phase mismatching Δk. (a) for SFG, and (b) for DFG

Fig.4.
Fig.4.

The influence of phase mismatching Δk in the QPM-SFG process, (a) for the normalized electrical field |E 3N| of the sum-frequency wave, (b) for one-forth period (i.e., T/4) which corresponds to the interacting length of |E 3N|, and (c) for the phase evolution of the signal wave.

Fig.5.
Fig.5.

The influence of phase mismatching Δk in the QPM-DFG process, (a) for the normalized electrical field |E 3N| of the difference-frequency wave, (b) for one-forth period (i.e., T/4) which corresponds to the interacting length of |E 3N|, and (c) for the phase evolution of the signal wave.

Fig.6.
Fig.6.

Relationships of the conversion efficiency η and the phase φ 3 of the converted wave with the signal wavelength λ 2.

Fig.7.
Fig.7.

The optimized maximum phase φ 2_max of signal, and the corresponded Δk and I 2_out /I 2 (I 2_out and I 2 denote the output and input intensity of signal, respectively) with the different medium length l. (a) for QPM-SFG, and (b) for QPM-DFG

Fig.8.
Fig.8.

the minimum intensity I 1 of pump and its corresponding Δk for the “push-pull” all-optical switching (i.e., Δφ=π/2) under I 2_out /I 2>0.8 with the different medium length l. (a) for QPM-DFG, and (b) QPM-SFG.

Equations (48)

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E t 1 z = i ω t 1 d eff n t 1 c E t 2 * E t 3 exp ( i Δ kz ) ,
E t 2 z = i ω t 2 d eff n t 2 c E t 1 * E t 3 exp ( i Δ kz ) ,
E t 3 z = i ω t 3 d eff n t 3 c E t 1 E t 2 exp ( i Δ kz ) ,
I = ε 0 cn E 2 / 2 ,
E j = e j f j exp ( i φ j ) ,
e j = 2 ħ ω j / ( n j ε 0 c ) ( j = 1,2,3 ) ,
ζ ( f 1 f 2 f 3 ) = ( f 2 f 3 f 1 f 3 f 1 f 2 ) sin θ ,
ζ ( φ 1 φ 2 φ 3 ) = ( f 2 f 3 / f 1 f 1 f 3 / f 2 f 1 f 2 / f 3 ) cos θ
ζ ( f 1 f 2 f 3 ) = ( f 2 f 3 f 1 f 3 f 1 f 2 ) sin θ′ ,
ζ ( φ 1 φ 2 φ 3 ) = ( f 2 f 3 / f 1 f 1 f 3 / f 2 f 1 f 2 / f 3 ) cos θ′
ζ ( ln ( f 1 ) ln ( f 2 ) ln ( f 3 ) ) = ( f 2 f 3 / f 1 f 1 f 3 / f 2 f 1 f 2 / f 3 ) sin θ .
= Δ k C + c tan ( θ ) d ln ( f 1 f 2 f 3 ) .
f 1 f 2 f 3 cos θ + Δ k f 3 2 / ( 2 C ) = Γ 1 ,
f 1 d f 1 = f 2 d f 2 = f 3 d f 3 = f 1 f 2 f 3 sin θ .
( f 1 2 ( 0 ) + f 3 2 ( 0 ) f 2 2 ( 0 ) + f 3 2 ( 0 ) f 1 2 ( 0 ) f 2 2 ( 0 ) ) = ( f 1 2 ( ζ ) + f 3 2 ( ζ ) f 2 2 ( ζ ) + f 3 2 ( ζ ) f 1 2 ( ζ ) f 2 2 ( ζ ) ) .
1 2 d ( f 3 2 ) = f 1 2 f 2 2 f 3 2 ( Γ 1 Δ k f 3 2 / ( 2 C ) ) 2 .
0 ξ = 1 2 0 F 3 d ( f 3 2 ) / ( F 1 f 3 2 ) ( F 2 f 3 2 ) f 3 2 [ Δ k f 3 2 / ( 2 C ) ] 2 ,
ξ F 1 = 0 X f [ ( 1 x 2 ) ( σ x 2 ) κ x 2 ] 0.5 dx ,
f 3 ( ξ ) = F 1 B sn ( A ξ F 1 , u ) ,
f j ( ξ ) = F j F 1 B 2 sn 2 ( A ξ F 1 , u ) ( j = 1,2 )
φ 1 ζ = ( Γ 1 Δ k f 3 2 2 C ) / f 1 2 .
φ 1 ( ξ ) = 0 ξ Δ k f 3 2 ( ζ ) 2 C f 1 2 ( ζ ) + φ 1 ( 0 ) .
φ j ( ξ ) = 0 ξ Δ k f 3 2 ( ζ ) 2 C f j 2 ( ζ ) + φ j ( 0 ) ( j = 2,3 ) ,
cos θ ( z 0 ) = cos [ φ 3 ( z 0 ) φ 2 ( z 0 ) φ 1 ( z 0 ) ] 0 .
sin [ φ 3 ( z 0 ) φ 2 ( z 0 ) φ 1 ( z 0 ) ] > 0 .
E j ( ξ ) = e j F j F 1 B 2 sn 2 ( F 1 , u ) · e i ( φ j ( 0 ) 0 ξ Δ k f 3 2 ( ζ ) 2 C f j 2 ( ζ ) ) ( j = 1,2 ) ,
E 3 ( ξ ) = e 3 F 1 B sn ( A ξ F 1 , u ) · e i ( π 2 + φ 1 ( 0 ) + φ 2 ( 0 ) l Δ k 2 ) .
E 1 ( ξ ) = E 2 ( ξ ) = e 1 F 1 F 1 tanh 2 ( ξ F 1 ) exp ( i φ 1 ( 0 ) ) ,
E 3 ( ξ ) = e 2 F 1 tanh ( ξ F 1 ) exp ( i φ 3 ( 0 ) ) .
tan ( θ′ ) dθ′ = ( Δ k / C ) tan ( θ′ ) + d [ ln ( f 1 f 2 f 3 ) ] .
f 1 f 2 f 3 cos θ′ Δ k f 3 2 / ( 2 C ) = Γ 1 ,
f j ( ξ ) = F j + ( 1 ) j F 2 B′ 2 u′ 2 sn 2 ( τ , u′ ) / dn 2 ( τ , u′ ) ( j = 1,2 ) ,
f 3 ( ξ ) = F 2 B u sn ( τ , u′ ) / dn ( τ , u′ ) ,
φ j ( ξ ) = 0 ξ Δ k f 3 2 ( ζ ) 2 C f j 2 ( ζ ) + φ j ( 0 ) ( j = 2,3 ) ,
E j ( ξ ) = e j F j + ( 1 ) j F 2 B′ 2 u′ 2 sn 2 ( τ , u′ ) dn 2 ( τ , u′ ) · e i ( φ j ( 0 ) + 0 ξ Δ k f 3 2 ( ζ ) 2 C f j 2 ( ζ ) ) ( j = 1,2 ) ,
E 3 ( ξ ) = e 3 F 2 B′ u′ [ sn ( τ , u′ ) / dn ( τ , u′ ) ] · e i ( π 2 + φ 1 ( 0 ) φ 2 ( 0 ) + l Δ k 2 ) .
E 3 ( ξ ) = e 3 F 2 / ( κ + 1 ) sin ( κ + 1 ξ F 1 ) · e i ( π 2 + φ 1 ( 0 ) + φ 2 ( 0 ) l Δ k 2 ) .
E 3 ( ξ ) = [ M E 1 ( 0 ) E 2 ( 0 ) / g ] sin ( g l ) · e i l Δ k 2 ,
E 3 ( ξ ) = i [ M 2 E 2 * ( 0 ) / Q ] sinh ( Q l ) · e i l Δ k 2 ,
φ 3 ( ξ ) φ 2 ( ξ ) φ 1 ( ξ ) = π / 2 .
( Δ k ) 0 ξ Δ k 2 C f 3 2 ( ζ , Δ k ) 2 C f 2 2 ( ζ , Δ k ) = 0 .
0 ξ f 3 2 ( ζ , Δ k ) f 2 2 ( ζ , Δ k ) + Δ k ( Δ k ) 0 ξ f 3 2 ( ζ , Δ k ) f 2 2 ( ζ , Δ k ) = 0 .
f 3 = ( C F 1 F 2 / g ) sin ( gl ) ,
f 2 = F 2 1 ( C 2 F 1 / g 2 ) sin 2 ( gl ) ,
f 3 = ( F 2 N / Q ) sinh ( Ql ) ,
f 2 = F 2 1 + ( N 2 / Q 2 ) sin h 2 ( Ql ) ,
( Δ k ) { Δ k 2 g [ ( 1 N 0 N 0 2 N 0 ) 1 N 1 atan ( tan ( gl / 2 ) N 1 ) + ( 1 + N 0 N 0 2 N 0 ) 1 N 2 atan ( tan ( gl / 2 ) N 2 ) 2 atan ( tan ( gl 2 ) ) ] } = 0
( Δ k ) { Δ k 2 Q [ ( 1 + N 0 N′ 0 2 N 0 ) 1 N 1 a tan ( tanh ( Ql / 2 ) N 1 ) + ( 1 N 0 N′ 0 2 N 0 ) 1 N 2 a tan ( tanh ( Ql / 2 ) N 2 ) ln ( tanh ( Ql / 2 ) 1 tanh ( Ql / 2 ) + 1 ) ] } = 0

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