Arbitrary waveform synthesis by multiple harmonics generation and phasing in aperiodic optical superlattices

Wei-Hsun Lin; A. H. Kung

doi:10.1364/OE.17.016342

Arbitrary waveform synthesis by multiple harmonics generation and phasing in aperiodic optical superlattices

Wei-Hsun Lin and A. H. Kung

Optics Express
Vol. 17,
Issue 18,
pp. 16342-16351
(2009)
•https://doi.org/10.1364/OE.17.016342

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Wei-Hsun Lin and A. H. Kung, "Arbitrary waveform synthesis by multiple harmonics generation and phasing in aperiodic optical superlattices," Opt. Express 17, 16342-16351 (2009)

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Related Topics
Table of Contents Category
- Photonic Crystals and Devices
Optics & Photonics Topics
?

The topics in this list come from the OSA Optics and Photonics Topics applied to this article.
Previously assigned OCIS codes
- Multiharmonic generation (190.4160)
- Photonic crystals (230.5298)
- Pulse shaping (320.5540)

About this Article
History
- Original Manuscript: June 19, 2009
- Revised Manuscript: July 28, 2009
- Manuscript Accepted: July 28, 2009
- Published: August 28, 2009

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Open Access

Abstract

The process of generating periodic optical waveforms includes the generation and phasing of several harmonics of a fundamental frequency. In this work, we show that simultaneous generation and phasing of the harmonics can be performed in a monolithic aperiodic optical superlattice (AOS). Stable periodic waveforms can thus be delivered to a predetermined location by simply sending a laser beam through a properly designed and fabricated AOS crystal. A detailed mathematical description for generating the domain pattern in such an AOS crystal is given and the process is numerically demonstrated. The waveform that is generated from a monolithic AOS is highly reproducible and phase stable. We also use propagation in air as an example to show how any predictable phase and amplitude modifications such as air dispersion that will alter the desired waveform can be pre-compensated in the design phase of the AOS crystal.

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References

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Author
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Publication

Z. Jiang, D. S. Seo, S.-D. Yang, D. E. Leaird, R. V. Roussev, C. Langrock, M. M. Fejer, and A. M. Weiner, “Four-User, 2.5-Gb/s, Spectrally Coded OCDMA System Demonstration Using Low-Power Nonlinear Processing,” J. Lightwave Technol. 23(1), 143–158 (2005).
[Crossref]
M. C. Stowe, F. C. Cruz, A. Marian, and J. Ye, “High resolution atomic coherent control via spectral phase manipulation of an optical frequency comb,” Phys. Rev. Lett. 96(15), 153001 (2006).
[Crossref] [PubMed]
A. Baltuska, Th. Udem, M. Uiberacker, M. Hentschel, E. Goulielmakis, Ch. Gohle, R. Holzwarth, V. S. Yakovlev, A. Scrinzi, T. W. Hänsch, and F. Krausz, “Attosecond control of electronic processes by intense light fields,” Nature 421(6923), 611–615 (2003).
[Crossref] [PubMed]
A. M. Weiner, J. P. Heritage, and E. M. Kirschner, “High-resolution femtosecond pulse shaping,” J. Opt. Soc. Am. B 5(8), 1563–1572 (1988).
[Crossref]
J. K. Ranka, R. S. Windeler, and A. J. Stentz, “Visible continuum generation in air-silica microstructure optical fibers with anomalous dispersion at 800 nm,” Opt. Lett. 25(1), 25–27 (2000).
[Crossref]
P. B. Corkum, “Plasma perspective on strong field multiphoton ionization,” Phys. Rev. Lett. 71(13), 1994–1997 (1993).
[Crossref] [PubMed]
S. E. Harris and A. V. Sokolov, “Subfemtosecond Pulse Generation by Molecular Modulation,” Phys. Rev. Lett. 81(14), 2894–2897 (1998).
[Crossref]
T. W. Hänsch, “A proposed sub-femtosecond pulse synthesizer using separate phase-locked laser oscillators,” Opt. Commun. 80(1), 71–75 (1990).
[Crossref]
I. V. Shutov and A. S. Chirkin, “Consecutive high-order harmonic generation and formation of subfemtosecond light pulses in aperiodical nonlinear photonic crystals,” Phys. Rev. A 78(1), 013827 (2008).
[Crossref]
M. Y. Shverdin, D. R. Walker, D. D. Yavuz, G. Y. Yin, and S. E. Harris, “Generation of a single-cycle optical pulse,” Phys. Rev. Lett. 94(3), 033904 (2005).
[Crossref] [PubMed]
W. J. Chen, Z. M. Hsieh, S. W. Huang, H. Y. Su, C. J. Lai, T. T. Tang, C. H. Lin, C. K. Lee, R. P. Pan, C. L. Pan, and A. H. Kung, “Sub-single-cycle optical pulse train with constant carrier envelope phase,” Phys. Rev. Lett. 100(16), 163906 (2008).
[Crossref] [PubMed]
Z. Jiang, C.-B. Huang, D. E. Leaird, and A. M. Weiner, “Optical arbitrary waveform processing of more than 100 spectral comb lines,” Nat. Photonics 1(8), 463–467 (2007).
[Crossref]
N. Bloembergen, Nonlinear Optics (World Scientific, 1996).
S. N. Zhu, Y. Y. Zhu, and N. B. Ming, “Quasi–Phase-Matched Third-Harmonic Generation in a Quasi-Periodic Optical Superlattice,” Science 278(5339), 843–846 (1997).
[Crossref]
D. H. Jundt, M. M. Fejer, and R. L. Byer, “Optical Properties of Lithium-Rich Lithium Niobate Fabricated by Vapor Transport Equilibration,” IEEE J. Quantum Electron. 26(1), 135–138 (1990).
[Crossref]
E. Peck and K. Reeder, “Dispersion of Air,” J. Opt. Soc. Am. 62(8), 958–962 (1972).
[Crossref]

2008 (2)

I. V. Shutov and A. S. Chirkin, “Consecutive high-order harmonic generation and formation of subfemtosecond light pulses in aperiodical nonlinear photonic crystals,” Phys. Rev. A 78(1), 013827 (2008).
[Crossref]

W. J. Chen, Z. M. Hsieh, S. W. Huang, H. Y. Su, C. J. Lai, T. T. Tang, C. H. Lin, C. K. Lee, R. P. Pan, C. L. Pan, and A. H. Kung, “Sub-single-cycle optical pulse train with constant carrier envelope phase,” Phys. Rev. Lett. 100(16), 163906 (2008).
[Crossref] [PubMed]

2007 (1)

Z. Jiang, C.-B. Huang, D. E. Leaird, and A. M. Weiner, “Optical arbitrary waveform processing of more than 100 spectral comb lines,” Nat. Photonics 1(8), 463–467 (2007).
[Crossref]

2006 (1)

M. C. Stowe, F. C. Cruz, A. Marian, and J. Ye, “High resolution atomic coherent control via spectral phase manipulation of an optical frequency comb,” Phys. Rev. Lett. 96(15), 153001 (2006).
[Crossref] [PubMed]

2005 (2)

Z. Jiang, D. S. Seo, S.-D. Yang, D. E. Leaird, R. V. Roussev, C. Langrock, M. M. Fejer, and A. M. Weiner, “Four-User, 2.5-Gb/s, Spectrally Coded OCDMA System Demonstration Using Low-Power Nonlinear Processing,” J. Lightwave Technol. 23(1), 143–158 (2005).
[Crossref]

M. Y. Shverdin, D. R. Walker, D. D. Yavuz, G. Y. Yin, and S. E. Harris, “Generation of a single-cycle optical pulse,” Phys. Rev. Lett. 94(3), 033904 (2005).
[Crossref] [PubMed]

2003 (1)

A. Baltuska, Th. Udem, M. Uiberacker, M. Hentschel, E. Goulielmakis, Ch. Gohle, R. Holzwarth, V. S. Yakovlev, A. Scrinzi, T. W. Hänsch, and F. Krausz, “Attosecond control of electronic processes by intense light fields,” Nature 421(6923), 611–615 (2003).
[Crossref] [PubMed]

2000 (1)

J. K. Ranka, R. S. Windeler, and A. J. Stentz, “Visible continuum generation in air-silica microstructure optical fibers with anomalous dispersion at 800 nm,” Opt. Lett. 25(1), 25–27 (2000).
[Crossref]

1998 (1)

S. E. Harris and A. V. Sokolov, “Subfemtosecond Pulse Generation by Molecular Modulation,” Phys. Rev. Lett. 81(14), 2894–2897 (1998).
[Crossref]

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(5339), 843–846 (1997).
[Crossref]

1993 (1)

P. B. Corkum, “Plasma perspective on strong field multiphoton ionization,” Phys. Rev. Lett. 71(13), 1994–1997 (1993).
[Crossref] [PubMed]

1990 (2)

D. H. Jundt, M. M. Fejer, and R. L. Byer, “Optical Properties of Lithium-Rich Lithium Niobate Fabricated by Vapor Transport Equilibration,” IEEE J. Quantum Electron. 26(1), 135–138 (1990).
[Crossref]

T. W. Hänsch, “A proposed sub-femtosecond pulse synthesizer using separate phase-locked laser oscillators,” Opt. Commun. 80(1), 71–75 (1990).
[Crossref]

1988 (1)

A. M. Weiner, J. P. Heritage, and E. M. Kirschner, “High-resolution femtosecond pulse shaping,” J. Opt. Soc. Am. B 5(8), 1563–1572 (1988).
[Crossref]

1972 (1)

E. Peck and K. Reeder, “Dispersion of Air,” J. Opt. Soc. Am. 62(8), 958–962 (1972).
[Crossref]

Baltuska, A.

Byer, R. L.

Chen, W. J.

Chirkin, A. S.

Corkum, P. B.

P. B. Corkum, “Plasma perspective on strong field multiphoton ionization,” Phys. Rev. Lett. 71(13), 1994–1997 (1993).
[Crossref] [PubMed]

Cruz, F. C.

Fejer, M. M.

Gohle, Ch.

Goulielmakis, E.

Hänsch, T. W.

T. W. Hänsch, “A proposed sub-femtosecond pulse synthesizer using separate phase-locked laser oscillators,” Opt. Commun. 80(1), 71–75 (1990).
[Crossref]

Harris, S. E.

M. Y. Shverdin, D. R. Walker, D. D. Yavuz, G. Y. Yin, and S. E. Harris, “Generation of a single-cycle optical pulse,” Phys. Rev. Lett. 94(3), 033904 (2005).
[Crossref] [PubMed]

S. E. Harris and A. V. Sokolov, “Subfemtosecond Pulse Generation by Molecular Modulation,” Phys. Rev. Lett. 81(14), 2894–2897 (1998).
[Crossref]

Hentschel, M.

Heritage, J. P.

A. M. Weiner, J. P. Heritage, and E. M. Kirschner, “High-resolution femtosecond pulse shaping,” J. Opt. Soc. Am. B 5(8), 1563–1572 (1988).
[Crossref]

Holzwarth, R.

Hsieh, Z. M.

Huang, C.-B.

Z. Jiang, C.-B. Huang, D. E. Leaird, and A. M. Weiner, “Optical arbitrary waveform processing of more than 100 spectral comb lines,” Nat. Photonics 1(8), 463–467 (2007).
[Crossref]

Huang, S. W.

Jiang, Z.

Z. Jiang, C.-B. Huang, D. E. Leaird, and A. M. Weiner, “Optical arbitrary waveform processing of more than 100 spectral comb lines,” Nat. Photonics 1(8), 463–467 (2007).
[Crossref]

Jundt, D. H.

Kirschner, E. M.

A. M. Weiner, J. P. Heritage, and E. M. Kirschner, “High-resolution femtosecond pulse shaping,” J. Opt. Soc. Am. B 5(8), 1563–1572 (1988).
[Crossref]

Krausz, F.

Kung, A. H.

Lai, C. J.

Langrock, C.

Leaird, D. E.

Z. Jiang, C.-B. Huang, D. E. Leaird, and A. M. Weiner, “Optical arbitrary waveform processing of more than 100 spectral comb lines,” Nat. Photonics 1(8), 463–467 (2007).
[Crossref]

Lee, C. K.

Lin, C. H.

Marian, A.

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(5339), 843–846 (1997).
[Crossref]

Pan, C. L.

Pan, R. P.

Peck, E.

E. Peck and K. Reeder, “Dispersion of Air,” J. Opt. Soc. Am. 62(8), 958–962 (1972).
[Crossref]

Ranka, J. K.

Reeder, K.

E. Peck and K. Reeder, “Dispersion of Air,” J. Opt. Soc. Am. 62(8), 958–962 (1972).
[Crossref]

Roussev, R. V.

Scrinzi, A.

Seo, D. S.

Shutov, I. V.

Shverdin, M. Y.

M. Y. Shverdin, D. R. Walker, D. D. Yavuz, G. Y. Yin, and S. E. Harris, “Generation of a single-cycle optical pulse,” Phys. Rev. Lett. 94(3), 033904 (2005).
[Crossref] [PubMed]

Sokolov, A. V.

S. E. Harris and A. V. Sokolov, “Subfemtosecond Pulse Generation by Molecular Modulation,” Phys. Rev. Lett. 81(14), 2894–2897 (1998).
[Crossref]

Stentz, A. J.

Stowe, M. C.

Su, H. Y.

Tang, T. T.

Udem, Th.

Uiberacker, M.

Walker, D. R.

M. Y. Shverdin, D. R. Walker, D. D. Yavuz, G. Y. Yin, and S. E. Harris, “Generation of a single-cycle optical pulse,” Phys. Rev. Lett. 94(3), 033904 (2005).
[Crossref] [PubMed]

Weiner, A. M.

Z. Jiang, C.-B. Huang, D. E. Leaird, and A. M. Weiner, “Optical arbitrary waveform processing of more than 100 spectral comb lines,” Nat. Photonics 1(8), 463–467 (2007).
[Crossref]

A. M. Weiner, J. P. Heritage, and E. M. Kirschner, “High-resolution femtosecond pulse shaping,” J. Opt. Soc. Am. B 5(8), 1563–1572 (1988).
[Crossref]

Windeler, R. S.

Yakovlev, V. S.

Yang, S.-D.

Yavuz, D. D.

M. Y. Shverdin, D. R. Walker, D. D. Yavuz, G. Y. Yin, and S. E. Harris, “Generation of a single-cycle optical pulse,” Phys. Rev. Lett. 94(3), 033904 (2005).
[Crossref] [PubMed]

Ye, J.

Yin, G. Y.

M. Y. Shverdin, D. R. Walker, D. D. Yavuz, G. Y. Yin, and S. E. Harris, “Generation of a single-cycle optical pulse,” Phys. Rev. Lett. 94(3), 033904 (2005).
[Crossref] [PubMed]

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(5339), 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(5339), 843–846 (1997).
[Crossref]

IEEE J. Quantum Electron. (1)

J. Lightwave Technol. (1)

J. Opt. Soc. Am. (1)

E. Peck and K. Reeder, “Dispersion of Air,” J. Opt. Soc. Am. 62(8), 958–962 (1972).
[Crossref]

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

A. M. Weiner, J. P. Heritage, and E. M. Kirschner, “High-resolution femtosecond pulse shaping,” J. Opt. Soc. Am. B 5(8), 1563–1572 (1988).
[Crossref]

Nat. Photonics (1)

Z. Jiang, C.-B. Huang, D. E. Leaird, and A. M. Weiner, “Optical arbitrary waveform processing of more than 100 spectral comb lines,” Nat. Photonics 1(8), 463–467 (2007).
[Crossref]

Nature (1)

Opt. Commun. (1)

T. W. Hänsch, “A proposed sub-femtosecond pulse synthesizer using separate phase-locked laser oscillators,” Opt. Commun. 80(1), 71–75 (1990).
[Crossref]

Opt. Lett. (1)

Phys. Rev. A (1)

Phys. Rev. Lett. (5)

M. Y. Shverdin, D. R. Walker, D. D. Yavuz, G. Y. Yin, and S. E. Harris, “Generation of a single-cycle optical pulse,” Phys. Rev. Lett. 94(3), 033904 (2005).
[Crossref] [PubMed]

P. B. Corkum, “Plasma perspective on strong field multiphoton ionization,” Phys. Rev. Lett. 71(13), 1994–1997 (1993).
[Crossref] [PubMed]

S. E. Harris and A. V. Sokolov, “Subfemtosecond Pulse Generation by Molecular Modulation,” Phys. Rev. Lett. 81(14), 2894–2897 (1998).
[Crossref]

Science (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(5339), 843–846 (1997).
[Crossref]

Other (1)

N. Bloembergen, Nonlinear Optics (World Scientific, 1996).

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Fig. 1 (left frame) Absolute values of the normalized effective second order nonlinear coefficients

| \tilde{d} (Δ k) |

calculated from Eq. (4) of the designed aperiodic optical superlattice and (right frame) simulation results of the electric field evolution in the designed AOS.

Download Full Size | PPT Slide | PDF

Fig. 2 (large frames) Simulation results of the phase evolution inside the designed AOS and (small frames) the corresponding waveforms (solid curve) and envelopes (dotted curve). (a) For AOS generation parameters of phase

ζ_{(211)} = ζ_{(321)} = ζ_{(431)} = ζ_{(541)} = 0

, simulation results agree with expected values

φ_{C E} = 0.5 π

and

φ_{m} = - 0.5 π

.For

ζ_{(211)} = ζ_{(321)} = ζ_{(431)} = ζ_{(541)} = 0.5 π

, 1.0π and 1.5 π from (b) to (d), the expected values are (b)

φ_{C E} = 0

φ_{m} = 0

, (c)

φ_{C E} = - 0.5 π

φ_{m} = 0.5 π

, (d)

φ_{C E} = - 1.0 π

φ_{m} = 1.0 π

, respectively. In a stable periodic waveform,

φ_{C E}

affects the shape of the waveform, and

φ_{m}

produces a timeshift. This is confirmed with the results shown in the small frame figures on the right where the envelopes shift sequentially by 1/4 period from (a) to (d), which is due to the 0.5π phase shift of

φ_{m}

from one figure to the next.

Download Full Size | PPT Slide | PDF

Fig. 3 Generated electromagnetic waveform (solid curve) and the corresponding envelope (dotted curve) which are (a) at the crystal exit and (b) at 1 m away from the crystal exit. The generation amplitude parameters are set to

a_{(211)} = 1.000

a_{(321)} = 0.996

a_{(431)} = 1.174

a_{(541)} = 1.294

, which are the same as those for calculating Fig. 2. The generation phase parameters are set to

ζ_{(211)} = 1.332 π

ζ_{(321)} = 0.030 π

ζ_{(431)} = 0.186 π

ζ_{(541)} = 1.942 π

to compensate the phase distortion due to dispersion in air.

Download Full Size | PPT Slide | PDF

Fig. 4 Plots of simulated waveforms (solid curve) and envelopes (dotted curve). The generation amplitude parameters are set to

a_{(211)} = 1.000

a_{(321)} = 2.188

a_{(532)} = 12.119

a_{(752)} = 21.416

and the generation phase parameters are set to

ζ_{(211)} = ζ_{(321)} = ζ_{(431)} = ζ_{(541)} = 0

, 0.25π, 0.5π, 0.75π from (a) to (d) respectively. From Eq. (8) and with

φ_{1} = 0

, it can be predicted that

φ_{C E}

will change from 0.5π to −0.25π. The expected values of the CEP are (a)

φ_{C E} = 0.5 π

φ_{m} = - 0.5 π

. (b)

φ_{C E} = 0.25 π

φ_{m} = - 0.25 π

. (c)

φ_{C E} = 0

φ_{m} = 0

. (d)

φ_{C E} = - 0.25 π

φ_{m} = 0.25 π

, which agree with above waveforms.

Download Full Size | PPT Slide | PDF

Tables (1)

Table 1 Phase mismatch of selected processes^a

View Table

Equations (8)

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

φ_{n} = φ_{C E} + n φ_{m},

\frac{\partial E_{r}}{\partial z} = - i \sum_{ω_{p} + ω_{q} = ω_{r}} \frac{ω_{r} d^{(2)} (z)}{c n_{r}} E_{p} E_{q} \exp (- i Δ k_{(r p q)} z),

\frac{\partial E_{r}}{\partial z} = - i \sum_{(r p q)} \frac{ω_{r}}{c n_{r}} {\tilde{d}}_{(r p q)} E_{p} E_{q},

{\tilde{d}}_{(r p q)} \equiv \tilde{d} (Δ k_{(r p q)}) = \frac{1}{L} \int d^{(2)} (z) \exp (- i Δ k_{(r p q)} z) d z .

\begin{array}{l} φ_{n}^{(f i n a l)} = φ_{n} + ϕ_{n} = - \frac{π}{2} + φ_{{\tilde{d}}_{(n (n - 1) 1)}} + φ_{n - 1} + φ_{1} + ϕ_{n}, \\ φ_{1}^{(f i n a l)} = φ_{1} + ϕ_{1}, \end{array}

\frac{π}{2} - φ_{{\tilde{d}}_{(211)}} + 2 ϕ_{1} - ϕ_{2} = \frac{π}{2} - φ_{{\tilde{d}}_{(321)}} + ϕ_{1} + ϕ_{2} - ϕ_{3} = \dots,

d^{(2)} (z) \propto sgn (\sum_{(r p q)}^{N} a_{(r p q)} \sin (Δ k_{(r p q)} z + ζ_{(r p q)})),

d^{(2)} (z) \propto sgn (\int_{⌊ z / l - 1 ⌋ l}^{⌊ z / l ⌋ l} \sum_{(r p q)}^{N} a_{(r p q)} \cos (Δ k_{(r p q)} z' + ζ_{(r p q)}) d z'),

Channel	$Δ k$ (1/μm)	$2 π / Δ k$ (μm)
$ω_{1} + ω_{1} \to ω_{2}$	0.22380283	28.074647
$ω_{1} + ω_{2} \to ω_{3}$	0.31866532	19.717192
$ω_{1} + ω_{3} \to ω_{4}$	0.53402295	11.765759
$ω_{1} + ω_{4} \to ω_{5}$	0.89204589	7.0435673
$ω_{2} + ω_{3} \to ω_{5}$	1.2022660	5.2261190
$ω_{2} + ω_{5} \to ω_{7}$	3.7292894	1.6848211