Abstract

We report on the design and experimental characterization of aperiodically poled lithium niobate (APLN) crystals for use in monolithically integrated dual nonlinear-optical devices. A cascade and a single aperiodic-domain-structure designs based on simulated annealing method were constructed in LiNbO3 to simultaneously perform as 4-channel electro-optically active (EOA) filters and 4-channel frequency doublers in the telecom band. We found that we could obtain a 2.44-fold enhancement in second-harmonic-generation conversion efficiency and a 2.4-time reduction in filter transmission bandwidth with the single APLN device over the cascade one when the same device length of 2 cm and the EOA field of 1027 V/mm were used.

©2008 Optical Society of America

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  1. Y. H. Chen, Y. Y. Lin, C. H. Chen, and Y. C. Huang, “Monolithic quasi-phase-matched nonlinear crystal for simultaneous laser Q-switching and parametric oscillation in a Nd:YVO4 laser,” Opt. Lett. 30, 1045–1047 (2005).
    [Crossref] [PubMed]
  2. S. T. Lin, G. W. Chang, Y. Y. Lin, Y. C. Huang, A. C. Chiang, and Y. H. Chen, “Monolithically integrated laser Bragg Q-switch and wavelength converter in a PPLN crystal,” Opt. Exp. 15, 17093–17098 (2007).
    [Crossref]
  3. S. N. Zhu, Y. Y. Zhu, and N. B. Ming, “Quasi-phase-matched third-harmonic generation in a quasi-periodic optical superlattice,” Science 278, 843–846 (1997).
    [Crossref]
  4. M. H. Chou, K. R. Parameswaran, and M. M. Fejer, “Multiple-channel wavelength conversion by use of engineered quasi-phase-matching structures in LiNbO3 waveguides,” Opt. Lett. 24, 1157–1159 (1999).
    [Crossref]
  5. M. Asobe, O. Tadanaga, H. Miyazawa, Y. Nishida, and H. Suzuki, “Multiple quasi-phase-matched LiNbO3 wavelength converter with a continuously phase-modulated domain structure,” Opt. Lett. 28, 558–560 (2003).
    [Crossref] [PubMed]
  6. B. Y. Gu, B. Z. Dong, Y. Zhang, and G. Z. Yang, “Enhanced harmonic generation in aperiodic optical superlattices,” Appl. Phys. Lett. 75, 2175–2177 (1999).
    [Crossref]
  7. M. H. Chou, J. Hauden, M. A. Arbore, and M. M. Fejer, “1.5 µm band wavelength conversion based on difference frequency generation in LiNbO3 waveguides with integrated coupling structures,” Opt. Lett. 23, 1004–1006 (1998).
    [Crossref]
  8. C. H. Lin, Y. H. Chen, S. W. Lin, C. L. Chang, Y. C. Huang, and J. Y. Chang, “Electro-optic narrowband multi-wavelength filter in aperiodically poled lithium niobate,” Opt. Exp. 15, 9859–9866 (2007).
    [Crossref]
  9. Y. W. Lee, F. C. Fan, Y. C. Huang, B. Y. Gu, B. Z. Dong, and M. H. Chou, “Nonlinear multiwavelength conversion based on an aperiodic optical superlattice in lithium niobate,” Opt. Lett. 27, 2191–2193 (2002).
    [Crossref]
  10. S. Kirkpatrick, C. D. Gelatt, and M. P. Vecchi, “Optimization by simulated annealing,” Science 220, 671–680 (1983).
    [Crossref] [PubMed]
  11. B. Y. Gu, Y. Zhang, and B. Z. Dong, “Investigations of harmonic generations in aperiodic optical superlattices,” J. Appl. Phys. 87, 7629–7637 (2000).
    [Crossref]
  12. C. Y. Huang, C. H. Lin, Y. H. Chen, and Y. C. Huang, “Electro-optic Ti:PPLN waveguide as efficient optical wavelength filter and polarization mode converter,” Opt. Exp. 15, 2548–2554 (2007).
    [Crossref]
  13. M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, “Quasi-phase-matched second harmonic generation: tuning and tolerances,” IEEE J. Quantum Electron. 23, 2631–2654 (1992).
    [Crossref]
  14. L. E. Myers, R. C. Eckardt, M. M. Fejer, R. L. Byer, W. R. Bosenberg, and J. W. Pierce, “Quasi-phasematched optical parametric oscillators in bulk periodically poled LiNbO3,” J. Opt. Soc. Am. B 12, 2102–2116 (1995).
    [Crossref]
  15. G. Schreiber, H. Suche, Y. L. Lee, W. Grundkötter, V. Quiring, R. Ricken, and W. Sohler, “Efficient cascaded difference frequency conversion in periodically poled Ti:LiNbO3 waveguides using pulsed and cw pumping,” Appl. Phys. B 73, 501–504 (2001).
    [Crossref]

2007 (3)

S. T. Lin, G. W. Chang, Y. Y. Lin, Y. C. Huang, A. C. Chiang, and Y. H. Chen, “Monolithically integrated laser Bragg Q-switch and wavelength converter in a PPLN crystal,” Opt. Exp. 15, 17093–17098 (2007).
[Crossref]

C. H. Lin, Y. H. Chen, S. W. Lin, C. L. Chang, Y. C. Huang, and J. Y. Chang, “Electro-optic narrowband multi-wavelength filter in aperiodically poled lithium niobate,” Opt. Exp. 15, 9859–9866 (2007).
[Crossref]

C. Y. Huang, C. H. Lin, Y. H. Chen, and Y. C. Huang, “Electro-optic Ti:PPLN waveguide as efficient optical wavelength filter and polarization mode converter,” Opt. Exp. 15, 2548–2554 (2007).
[Crossref]

2005 (1)

2003 (1)

2002 (1)

2001 (1)

G. Schreiber, H. Suche, Y. L. Lee, W. Grundkötter, V. Quiring, R. Ricken, and W. Sohler, “Efficient cascaded difference frequency conversion in periodically poled Ti:LiNbO3 waveguides using pulsed and cw pumping,” Appl. Phys. B 73, 501–504 (2001).
[Crossref]

2000 (1)

B. Y. Gu, Y. Zhang, and B. Z. Dong, “Investigations of harmonic generations in aperiodic optical superlattices,” J. Appl. Phys. 87, 7629–7637 (2000).
[Crossref]

1999 (2)

B. Y. Gu, B. Z. Dong, Y. Zhang, and G. Z. Yang, “Enhanced harmonic generation in aperiodic optical superlattices,” Appl. Phys. Lett. 75, 2175–2177 (1999).
[Crossref]

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

1998 (1)

1997 (1)

S. N. Zhu, Y. Y. Zhu, and N. B. Ming, “Quasi-phase-matched third-harmonic generation in a quasi-periodic optical superlattice,” Science 278, 843–846 (1997).
[Crossref]

1995 (1)

1992 (1)

M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, “Quasi-phase-matched second harmonic generation: tuning and tolerances,” IEEE J. Quantum Electron. 23, 2631–2654 (1992).
[Crossref]

1983 (1)

S. Kirkpatrick, C. D. Gelatt, and M. P. Vecchi, “Optimization by simulated annealing,” Science 220, 671–680 (1983).
[Crossref] [PubMed]

Arbore, M. A.

Asobe, M.

Bosenberg, W. R.

Byer, R. L.

L. E. Myers, R. C. Eckardt, M. M. Fejer, R. L. Byer, W. R. Bosenberg, and J. W. Pierce, “Quasi-phasematched optical parametric oscillators in bulk periodically poled LiNbO3,” J. Opt. Soc. Am. B 12, 2102–2116 (1995).
[Crossref]

M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, “Quasi-phase-matched second harmonic generation: tuning and tolerances,” IEEE J. Quantum Electron. 23, 2631–2654 (1992).
[Crossref]

Chang, C. L.

C. H. Lin, Y. H. Chen, S. W. Lin, C. L. Chang, Y. C. Huang, and J. Y. Chang, “Electro-optic narrowband multi-wavelength filter in aperiodically poled lithium niobate,” Opt. Exp. 15, 9859–9866 (2007).
[Crossref]

Chang, G. W.

S. T. Lin, G. W. Chang, Y. Y. Lin, Y. C. Huang, A. C. Chiang, and Y. H. Chen, “Monolithically integrated laser Bragg Q-switch and wavelength converter in a PPLN crystal,” Opt. Exp. 15, 17093–17098 (2007).
[Crossref]

Chang, J. Y.

C. H. Lin, Y. H. Chen, S. W. Lin, C. L. Chang, Y. C. Huang, and J. Y. Chang, “Electro-optic narrowband multi-wavelength filter in aperiodically poled lithium niobate,” Opt. Exp. 15, 9859–9866 (2007).
[Crossref]

Chen, C. H.

Chen, Y. H.

C. Y. Huang, C. H. Lin, Y. H. Chen, and Y. C. Huang, “Electro-optic Ti:PPLN waveguide as efficient optical wavelength filter and polarization mode converter,” Opt. Exp. 15, 2548–2554 (2007).
[Crossref]

S. T. Lin, G. W. Chang, Y. Y. Lin, Y. C. Huang, A. C. Chiang, and Y. H. Chen, “Monolithically integrated laser Bragg Q-switch and wavelength converter in a PPLN crystal,” Opt. Exp. 15, 17093–17098 (2007).
[Crossref]

C. H. Lin, Y. H. Chen, S. W. Lin, C. L. Chang, Y. C. Huang, and J. Y. Chang, “Electro-optic narrowband multi-wavelength filter in aperiodically poled lithium niobate,” Opt. Exp. 15, 9859–9866 (2007).
[Crossref]

Y. H. Chen, Y. Y. Lin, C. H. Chen, and Y. C. Huang, “Monolithic quasi-phase-matched nonlinear crystal for simultaneous laser Q-switching and parametric oscillation in a Nd:YVO4 laser,” Opt. Lett. 30, 1045–1047 (2005).
[Crossref] [PubMed]

Chiang, A. C.

S. T. Lin, G. W. Chang, Y. Y. Lin, Y. C. Huang, A. C. Chiang, and Y. H. Chen, “Monolithically integrated laser Bragg Q-switch and wavelength converter in a PPLN crystal,” Opt. Exp. 15, 17093–17098 (2007).
[Crossref]

Chou, M. H.

Dong, B. Z.

Y. W. Lee, F. C. Fan, Y. C. Huang, B. Y. Gu, B. Z. Dong, and M. H. Chou, “Nonlinear multiwavelength conversion based on an aperiodic optical superlattice in lithium niobate,” Opt. Lett. 27, 2191–2193 (2002).
[Crossref]

B. Y. Gu, Y. Zhang, and B. Z. Dong, “Investigations of harmonic generations in aperiodic optical superlattices,” J. Appl. Phys. 87, 7629–7637 (2000).
[Crossref]

B. Y. Gu, B. Z. Dong, Y. Zhang, and G. Z. Yang, “Enhanced harmonic generation in aperiodic optical superlattices,” Appl. Phys. Lett. 75, 2175–2177 (1999).
[Crossref]

Eckardt, R. C.

Fan, F. C.

Fejer, M. M.

Gelatt, C. D.

S. Kirkpatrick, C. D. Gelatt, and M. P. Vecchi, “Optimization by simulated annealing,” Science 220, 671–680 (1983).
[Crossref] [PubMed]

Grundkötter, W.

G. Schreiber, H. Suche, Y. L. Lee, W. Grundkötter, V. Quiring, R. Ricken, and W. Sohler, “Efficient cascaded difference frequency conversion in periodically poled Ti:LiNbO3 waveguides using pulsed and cw pumping,” Appl. Phys. B 73, 501–504 (2001).
[Crossref]

Gu, B. Y.

Y. W. Lee, F. C. Fan, Y. C. Huang, B. Y. Gu, B. Z. Dong, and M. H. Chou, “Nonlinear multiwavelength conversion based on an aperiodic optical superlattice in lithium niobate,” Opt. Lett. 27, 2191–2193 (2002).
[Crossref]

B. Y. Gu, Y. Zhang, and B. Z. Dong, “Investigations of harmonic generations in aperiodic optical superlattices,” J. Appl. Phys. 87, 7629–7637 (2000).
[Crossref]

B. Y. Gu, B. Z. Dong, Y. Zhang, and G. Z. Yang, “Enhanced harmonic generation in aperiodic optical superlattices,” Appl. Phys. Lett. 75, 2175–2177 (1999).
[Crossref]

Hauden, J.

Huang, C. Y.

C. Y. Huang, C. H. Lin, Y. H. Chen, and Y. C. Huang, “Electro-optic Ti:PPLN waveguide as efficient optical wavelength filter and polarization mode converter,” Opt. Exp. 15, 2548–2554 (2007).
[Crossref]

Huang, Y. C.

C. Y. Huang, C. H. Lin, Y. H. Chen, and Y. C. Huang, “Electro-optic Ti:PPLN waveguide as efficient optical wavelength filter and polarization mode converter,” Opt. Exp. 15, 2548–2554 (2007).
[Crossref]

C. H. Lin, Y. H. Chen, S. W. Lin, C. L. Chang, Y. C. Huang, and J. Y. Chang, “Electro-optic narrowband multi-wavelength filter in aperiodically poled lithium niobate,” Opt. Exp. 15, 9859–9866 (2007).
[Crossref]

S. T. Lin, G. W. Chang, Y. Y. Lin, Y. C. Huang, A. C. Chiang, and Y. H. Chen, “Monolithically integrated laser Bragg Q-switch and wavelength converter in a PPLN crystal,” Opt. Exp. 15, 17093–17098 (2007).
[Crossref]

Y. H. Chen, Y. Y. Lin, C. H. Chen, and Y. C. Huang, “Monolithic quasi-phase-matched nonlinear crystal for simultaneous laser Q-switching and parametric oscillation in a Nd:YVO4 laser,” Opt. Lett. 30, 1045–1047 (2005).
[Crossref] [PubMed]

Y. W. Lee, F. C. Fan, Y. C. Huang, B. Y. Gu, B. Z. Dong, and M. H. Chou, “Nonlinear multiwavelength conversion based on an aperiodic optical superlattice in lithium niobate,” Opt. Lett. 27, 2191–2193 (2002).
[Crossref]

Jundt, D. H.

M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, “Quasi-phase-matched second harmonic generation: tuning and tolerances,” IEEE J. Quantum Electron. 23, 2631–2654 (1992).
[Crossref]

Kirkpatrick, S.

S. Kirkpatrick, C. D. Gelatt, and M. P. Vecchi, “Optimization by simulated annealing,” Science 220, 671–680 (1983).
[Crossref] [PubMed]

Lee, Y. L.

G. Schreiber, H. Suche, Y. L. Lee, W. Grundkötter, V. Quiring, R. Ricken, and W. Sohler, “Efficient cascaded difference frequency conversion in periodically poled Ti:LiNbO3 waveguides using pulsed and cw pumping,” Appl. Phys. B 73, 501–504 (2001).
[Crossref]

Lee, Y. W.

Lin, C. H.

C. Y. Huang, C. H. Lin, Y. H. Chen, and Y. C. Huang, “Electro-optic Ti:PPLN waveguide as efficient optical wavelength filter and polarization mode converter,” Opt. Exp. 15, 2548–2554 (2007).
[Crossref]

C. H. Lin, Y. H. Chen, S. W. Lin, C. L. Chang, Y. C. Huang, and J. Y. Chang, “Electro-optic narrowband multi-wavelength filter in aperiodically poled lithium niobate,” Opt. Exp. 15, 9859–9866 (2007).
[Crossref]

Lin, S. T.

S. T. Lin, G. W. Chang, Y. Y. Lin, Y. C. Huang, A. C. Chiang, and Y. H. Chen, “Monolithically integrated laser Bragg Q-switch and wavelength converter in a PPLN crystal,” Opt. Exp. 15, 17093–17098 (2007).
[Crossref]

Lin, S. W.

C. H. Lin, Y. H. Chen, S. W. Lin, C. L. Chang, Y. C. Huang, and J. Y. Chang, “Electro-optic narrowband multi-wavelength filter in aperiodically poled lithium niobate,” Opt. Exp. 15, 9859–9866 (2007).
[Crossref]

Lin, Y. Y.

S. T. Lin, G. W. Chang, Y. Y. Lin, Y. C. Huang, A. C. Chiang, and Y. H. Chen, “Monolithically integrated laser Bragg Q-switch and wavelength converter in a PPLN crystal,” Opt. Exp. 15, 17093–17098 (2007).
[Crossref]

Y. H. Chen, Y. Y. Lin, C. H. Chen, and Y. C. Huang, “Monolithic quasi-phase-matched nonlinear crystal for simultaneous laser Q-switching and parametric oscillation in a Nd:YVO4 laser,” Opt. Lett. 30, 1045–1047 (2005).
[Crossref] [PubMed]

Magel, G. A.

M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, “Quasi-phase-matched second harmonic generation: tuning and tolerances,” IEEE J. Quantum Electron. 23, 2631–2654 (1992).
[Crossref]

Ming, N. B.

S. N. Zhu, Y. Y. Zhu, and N. B. Ming, “Quasi-phase-matched third-harmonic generation in a quasi-periodic optical superlattice,” Science 278, 843–846 (1997).
[Crossref]

Miyazawa, H.

Myers, L. E.

Nishida, Y.

Parameswaran, K. R.

Pierce, J. W.

Quiring, V.

G. Schreiber, H. Suche, Y. L. Lee, W. Grundkötter, V. Quiring, R. Ricken, and W. Sohler, “Efficient cascaded difference frequency conversion in periodically poled Ti:LiNbO3 waveguides using pulsed and cw pumping,” Appl. Phys. B 73, 501–504 (2001).
[Crossref]

Ricken, R.

G. Schreiber, H. Suche, Y. L. Lee, W. Grundkötter, V. Quiring, R. Ricken, and W. Sohler, “Efficient cascaded difference frequency conversion in periodically poled Ti:LiNbO3 waveguides using pulsed and cw pumping,” Appl. Phys. B 73, 501–504 (2001).
[Crossref]

Schreiber, G.

G. Schreiber, H. Suche, Y. L. Lee, W. Grundkötter, V. Quiring, R. Ricken, and W. Sohler, “Efficient cascaded difference frequency conversion in periodically poled Ti:LiNbO3 waveguides using pulsed and cw pumping,” Appl. Phys. B 73, 501–504 (2001).
[Crossref]

Sohler, W.

G. Schreiber, H. Suche, Y. L. Lee, W. Grundkötter, V. Quiring, R. Ricken, and W. Sohler, “Efficient cascaded difference frequency conversion in periodically poled Ti:LiNbO3 waveguides using pulsed and cw pumping,” Appl. Phys. B 73, 501–504 (2001).
[Crossref]

Suche, H.

G. Schreiber, H. Suche, Y. L. Lee, W. Grundkötter, V. Quiring, R. Ricken, and W. Sohler, “Efficient cascaded difference frequency conversion in periodically poled Ti:LiNbO3 waveguides using pulsed and cw pumping,” Appl. Phys. B 73, 501–504 (2001).
[Crossref]

Suzuki, H.

Tadanaga, O.

Vecchi, M. P.

S. Kirkpatrick, C. D. Gelatt, and M. P. Vecchi, “Optimization by simulated annealing,” Science 220, 671–680 (1983).
[Crossref] [PubMed]

Yang, G. Z.

B. Y. Gu, B. Z. Dong, Y. Zhang, and G. Z. Yang, “Enhanced harmonic generation in aperiodic optical superlattices,” Appl. Phys. Lett. 75, 2175–2177 (1999).
[Crossref]

Zhang, Y.

B. Y. Gu, Y. Zhang, and B. Z. Dong, “Investigations of harmonic generations in aperiodic optical superlattices,” J. Appl. Phys. 87, 7629–7637 (2000).
[Crossref]

B. Y. Gu, B. Z. Dong, Y. Zhang, and G. Z. Yang, “Enhanced harmonic generation in aperiodic optical superlattices,” Appl. Phys. Lett. 75, 2175–2177 (1999).
[Crossref]

Zhu, S. N.

S. N. Zhu, Y. Y. Zhu, and N. B. Ming, “Quasi-phase-matched third-harmonic generation in a quasi-periodic optical superlattice,” Science 278, 843–846 (1997).
[Crossref]

Zhu, Y. Y.

S. N. Zhu, Y. Y. Zhu, and N. B. Ming, “Quasi-phase-matched third-harmonic generation in a quasi-periodic optical superlattice,” Science 278, 843–846 (1997).
[Crossref]

Appl. Phys. B (1)

G. Schreiber, H. Suche, Y. L. Lee, W. Grundkötter, V. Quiring, R. Ricken, and W. Sohler, “Efficient cascaded difference frequency conversion in periodically poled Ti:LiNbO3 waveguides using pulsed and cw pumping,” Appl. Phys. B 73, 501–504 (2001).
[Crossref]

Appl. Phys. Lett. (1)

B. Y. Gu, B. Z. Dong, Y. Zhang, and G. Z. Yang, “Enhanced harmonic generation in aperiodic optical superlattices,” Appl. Phys. Lett. 75, 2175–2177 (1999).
[Crossref]

IEEE J. Quantum Electron. (1)

M. M. Fejer, G. A. Magel, D. H. Jundt, and R. L. Byer, “Quasi-phase-matched second harmonic generation: tuning and tolerances,” IEEE J. Quantum Electron. 23, 2631–2654 (1992).
[Crossref]

J. Appl. Phys. (1)

B. Y. Gu, Y. Zhang, and B. Z. Dong, “Investigations of harmonic generations in aperiodic optical superlattices,” J. Appl. Phys. 87, 7629–7637 (2000).
[Crossref]

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

Opt. Exp. (3)

C. Y. Huang, C. H. Lin, Y. H. Chen, and Y. C. Huang, “Electro-optic Ti:PPLN waveguide as efficient optical wavelength filter and polarization mode converter,” Opt. Exp. 15, 2548–2554 (2007).
[Crossref]

C. H. Lin, Y. H. Chen, S. W. Lin, C. L. Chang, Y. C. Huang, and J. Y. Chang, “Electro-optic narrowband multi-wavelength filter in aperiodically poled lithium niobate,” Opt. Exp. 15, 9859–9866 (2007).
[Crossref]

S. T. Lin, G. W. Chang, Y. Y. Lin, Y. C. Huang, A. C. Chiang, and Y. H. Chen, “Monolithically integrated laser Bragg Q-switch and wavelength converter in a PPLN crystal,” Opt. Exp. 15, 17093–17098 (2007).
[Crossref]

Opt. Lett. (5)

Science (2)

S. Kirkpatrick, C. D. Gelatt, and M. P. Vecchi, “Optimization by simulated annealing,” Science 220, 671–680 (1983).
[Crossref] [PubMed]

S. N. Zhu, Y. Y. Zhu, and N. B. Ming, “Quasi-phase-matched third-harmonic generation in a quasi-periodic optical superlattice,” Science 278, 843–846 (1997).
[Crossref]

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Figures (5)

Fig. 1.
Fig. 1.

(a) Calculated transmission spectra of the cascade (solid green line) and single (dashed red line) APLN devices when they function as 4-channel EOA filters. (b) Calculated SH conversion spectra of the cascade (solid green line) and single (dashed red line) APLN devices when they function as 4-channel SH generators. The upper inset shows a schematic arrangement of the devices, where the APLN crystal is sandwiched in between a pair of cross polarizers that work in the telecom C-band.

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Fig. 2.
Fig. 2.

Measured (solid red circles) and calculated (solid green lines) transmission spectra of the (a) cascade and (b) single APLN devices at 37°C when they functioned as 4-channel EOA filters driven at 1027 V/mm.

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Fig. 3.
Fig. 3.

Measured (solid red circles) and calculated (solid blue lines) SH conversion spectra of the (a) cascade and (b) single APLN devices at 37oC when they functioned as 4-channel SH generators.

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Fig. 4.
Fig. 4.

Measured (solid red circles) and calculated (solid blue line) SH conversion spectra of the cascade APLN device. The transmission axis of the in-line polarizer at the input end of the crystal is switched to transmit TE waves in the telecom C band and the EOA PM conversion section is activated. The inset illustrates such system arrangement.

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Fig. 5.
Fig. 5.

Simulated (a) EOA filter transmission and (b) SH conversion spectra of the single APLN device when no error (solid green lines), an average 20% domain-width overpoled error (solid red lines) [9], or a temperature gradient of -0.1°C/cm descending from the crystal center to both ends (dashed blue lines) is introduced to the device.

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Equations (9)

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O F = { Σ i = 1 M [ η 0 ( λ i ) − η EO , SHG ( λ i ) ∣ ] w EO , SHG ( λ i ) }
+ β EO , SHG { max [ η EO , SHG ( λ i ) ] − min [ η EO , SHG ( λ i ) ] } ,
OF = { Σ i = 1 M [ η t ( λ i ) − η EO ( λ i ) w s , EO ( λ i ) − η SHG ( λ i ) w s , SHG ( λ i ) ∣ ] }
+ β s , EO { max [ η EO ( λ i ) ] − min [ η EO ( λ i ) ] }
+ β s , SHG { max [ η SHG ( λ i ) ] − min [ η SHG ( λ i ) ] } ,
η PPLN EO ∝ sin ⁡ 2 ( ∣ F ( D ) π 2 L ∣ 2 + ( Δ β EO 2 ) 2 L )
η PPLN SHG ∝ F 2 ( D ) sin ⁡ c 2 ( Δ β SHG 2 L )
η PPLN EO ∝ cos ⁡ 2 ( 1 π ( Δ β EO 2 L ) 2 )
η PPLN SHG ∝ sinc 2 ( Δ β SHG 2 L ) ,

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