Exact analytical solutions and their applications for interacting waves in quadratic nonlinear medium
Optics Express, Vol. 10, Issue 1, pp. 83-97 (2002)
http://dx.doi.org/10.1364/OE.10.000083
Acrobat PDF (470 KB)
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.
© Optical Society of America
1. Introduction
1. G. I. Stegeman, D. J. Hagan, and L. Torner, “χ^{(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]
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]
7. J. A. Armstrong, N. Bloembergen, and 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]
2. S. J. B. Yoo, “Wavelength conversion technologies for WDM network applications,” J. Lightwave Technol. 14, 955–966 (1996). [CrossRef]
2. S. J. B. Yoo, “Wavelength conversion technologies for WDM network applications,” J. Lightwave Technol. 14, 955–966 (1996). [CrossRef]
9. M. Asghari, I. H. White, and R. V. Penty, “Wavelength conversion using semiconductor optical amplifiers,” J. Lighteave Technol. 15, 1181–1190 (1997). [CrossRef]
2. S. J. B. Yoo, “Wavelength conversion technologies for WDM network applications,” J. Lightwave Technol. 14, 955–966 (1996). [CrossRef]
1. G. I. Stegeman, D. J. Hagan, and L. Torner, “χ^{(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]
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]
7. J. A. Armstrong, N. Bloembergen, and N, J. Ducuing, et al. “Interaction between light waves in a nonlinear dielectric,” Phys. Rev. 127, 1918–1939 (1962). [CrossRef]
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]
13. A. R. C. Sibilia, E. Fazio, and 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, and J. M. Arnold, “An all-optical switch employing the cascaded second-order nonlinear effect,” IEEE. J. Quantum Elecron. 29, 2650–2654 (1993). [CrossRef]
8. A. Kobyakov and F. Lederer, “Cascading of quadratic nonlinearities: an analytical study,” Phys. Rev. A 54, 3455–3471 (1996). [CrossRef] [PubMed]
15. A. Kobyakov, U. Peschel, and 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, and M. Bertolotti, “Three-wave mixing in a quadratic material under perfect phase-matching,”Opt. Commun. 142, 75–78 (1997). [CrossRef]
2. Exact solutions of CME
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]
13. A. R. C. Sibilia, E. Fazio, and M. Bertolotti, “Field dependent effect in a quadratic nonlinear medium,” J. Mod. Opt. 42, 823–839 (1995). [CrossRef]
17. X. -M. Liu, H. - Y. Zhang, and 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 and H. Nishihara, “Theoretical analysis of waveguide second-harmonic generation phase matchedwith uniform and chirped gratings,“ IEEE. J. Quantum Elecron. 26, 1265–1270 (1990). [CrossRef]
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]
2.1. Exact solutions for SFG-CME
7. J. A. Armstrong, N. Bloembergen, and N, J. Ducuing, et al. “Interaction between light waves in a nonlinear dielectric,” Phys. Rev. 127, 1918–1939 (1962). [CrossRef]
- The phase evolution of the sum-frequency wave in the TWM process is linear to the interaction length l in the medium, and is independent of the intensities of the three interacting waves.
- Contrarily, the phase evolution of the two fundament-frequency waves relates not only to their intensities but also to that of the sum-frequency wave.
- The intensity evolutions of the three waves, which are proportional to the square of the electric field magnitude |E|^{2}, are mutually affected by their intensities while are unaffected by their phases.
2.2. Exact solutions for SHG-CME
2.3. Exact solutions for DFG-CME
2.4. Small-signal approximation
3. Calculation and analysis
3.1. Comparing the results from Eq. (1) with those from Eqs. (18) and (24)
6. G. S. Kanter and P. Kumar, “Optical devices based on internally seeded cascaded nonlinearities,” IEEE. J. Quantum Elecron. 35, 891–896 (1999). [CrossRef]
22. H. Kanbara, H. Itoh, M. Asobe, K. Noguchi, H. Miyazawa, T. Yanagawa, and 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, and 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]
22. H. Kanbara, H. Itoh, M. Asobe, K. Noguchi, H. Miyazawa, T. Yanagawa, and 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]
3.2. Phase evolutions of three interacting waves
4. Optimizations and applications
17. X. -M. Liu, H. - Y. Zhang, and 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]
24. X. -M. Liu, H. - Y. Zhang, and Y. -H Li, “Optimal design for the quasi-phase-matching three-wave mixing,” Opt. Express 9, 631–636 (2001), http://www.opticsexpress.org/oearchive/source/37804.htm. [CrossRef] [PubMed]
4.1. Applications in the wavelength conversion
17. X. -M. Liu, H. - Y. Zhang, and 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]
2. S. J. B. Yoo, “Wavelength conversion technologies for WDM network applications,” J. Lightwave Technol. 14, 955–966 (1996). [CrossRef]
4.2. Optimization of the nonlinear phase shift
1. G. I. Stegeman, D. J. Hagan, and L. Torner, “χ^{(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]
22. H. Kanbara, H. Itoh, M. Asobe, K. Noguchi, H. Miyazawa, T. Yanagawa, and 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]
4.3. Applications in all-optical switching
14. C. N. Ironside, J. S. Aitchison, and J. M. Arnold, “An all-optical switch employing the cascaded second-order nonlinear effect,” IEEE. J. Quantum Elecron. 29, 2650–2654 (1993). [CrossRef]
5. Conclusion
Acknowledgement:
Footnotes
1) | Eqs.(4) and (5) have the generality for the TWM processing. Provided ʌ→∞, Δk=k
_{t3}-k
_{t2}-k
_{t1}, as belongs to TWM for the homogeneous medium without the spatial periodical structure. In this situation, the most common procedure for achieving phase-matching is to make use of the birefringence displayed by many crystals [7 7. J. A. Armstrong, N. Bloembergen, and N, J. Ducuing, et al. “Interaction between light waves in a nonlinear dielectric,” Phys. Rev. 127, 1918–1939 (1962). [CrossRef] |
2) | sn and dn in Eqs. (18) and (24) can be directly utilized by function of “ellipj” in the MATLAB, and the integral operation for the calculations of phase can also be obtained from the function of “quad” in MATLAB after only a few changes. |
3) | This periodical process is also named as the “cascading” of SFG-DFG in [1 1. G. I. Stegeman, D. J. Hagan, and L. Torner, “χ^{(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] 22. H. Kanbara, H. Itoh, M. Asobe, K. Noguchi, H. Miyazawa, T. Yanagawa, and 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] |
4) | The periodical change of the DFG and its reverse process is called as the “cascading” of DFG-SFG. |
5) | This SFG process is consistent with the “cascading” of SFG-DFG in [1 1. G. I. Stegeman, D. J. Hagan, and L. Torner, “χ^{(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] 22. H. Kanbara, H. Itoh, M. Asobe, K. Noguchi, H. Miyazawa, T. Yanagawa, and 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] |
References and links
1. | G. I. Stegeman, D. J. Hagan, and L. Torner, “χ^{(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, and 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, and 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, and 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, and 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, and 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, and 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, and 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, and 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, and 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 and H. Nishihara, “Theoretical analysis of waveguide second-harmonic generation phase matchedwith 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. | G. A. Kehen [USA] and T. M. Kehen, Handbook of Mathematics (Worker Press, Beijing, 1987), (in Chinese). |
21. | M. Abramowitz and 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, and 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, and 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, and Y. -H Li, “Optimal design for the quasi-phase-matching three-wave mixing,” Opt. Express 9, 631–636 (2001), http://www.opticsexpress.org/oearchive/source/37804.htm. [CrossRef] [PubMed] |
OCIS Codes
(060.2630) Fiber optics and optical communications : Frequency modulation
(190.0190) Nonlinear optics : Nonlinear optics
(190.2620) Nonlinear optics : Harmonic generation and mixing
(230.1150) Optical devices : All-optical devices
ToC Category:
Research Papers
History
Original Manuscript: December 21, 2001
Published: January 14, 2002
Citation
Xueming Liu, Hanyi Zhang, and Mingde Zhang, "Exact analytical solutions and their applications for interacting waves in quadratic nonlinear medium," Opt. Express 10, 83-97 (2002)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-10-1-83
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References
- 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]
- 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]
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