## Action on pulse position and momentum using dispersion and phase modulation

Optics Express, Vol. 8, Issue 12, pp. 664-669 (2001)

http://dx.doi.org/10.1364/OE.8.000664

Acrobat PDF (291 KB)

### Abstract

The timing jitter and frequency jitter of quantized optical pulses obey Heisenberg’s uncertainty principle. We show how one jitter may be reduced at the expense of the other, using dispersion and phase modulation.

© Optical Society of America

## 1 Introduction

2. L. Knöll, W. Vogel, and D.-G. Welsch, “Action of passive, lossless optical systems in quantum optics,” Phys. Rev. A **36**, 3803–3818 (1987). [CrossRef] [PubMed]

*χ*

^{(2)}) or third order (

*χ*

^{(3)}) nonlinearities[8

8. R. E. Slusher, L. W. Hollberg, B. Yurke, J. C. Mertz, and J. F. Valley, “Observation of squeezed states generated by 4-wave mixing in an optical cavity,” Phys. Rev. Lett. **55**, 2409–2412 (1985). [CrossRef] [PubMed]

14. M. Shirasaki and H. A. Haus, “Squeezing of pulses in a nonlinear interferometer,” J. Opt. Soc. Am. B **7**, 30 (1990). [CrossRef]

15. F. Hong-Yi and J. VanderLinde, “Squeezed-state wave functions and their relation to classical phase-space maps,” Phys. Rev. A **40**, 4785 (1989). [CrossRef]

17. F. DiFilippo, V. Natarajan, K. R. Boyce, and D. E. Pritchard, “Classical amplitude squeezing for precisionm easurements,” Phys. Rev. Lett. **68**, 2859 (1992). [CrossRef] [PubMed]

18. A. E. Siegman and D. J. Kuizenga, “Proposed method for measuring picosecond pulse widths and pulse shapes in CW mode-locked lasers,” IEEE J. Quantum Electron. **6**, 212–215 (1970). [CrossRef]

21. J. C. Twichell and R. Helkey, “Phase-encoded optical sampling for analog-to-digital converters,” Phot. Tech. Lett. **12**, 1237–1239 (2000). [CrossRef]

## 2 Quantization of the Optical Pulse

*â*(

*T*,

*x*), where

*T*and

*x*represent two time scales: the slow time scale

*T*of pulse evolution, and the fast time scale (expressed as a spatial scale)

*x*=

*υ*

_{g}

*t*, where

*υ*

_{g}is the group velocity. The operator obeys the commutation relation[22

22. H. A. Haus, “Steady-state quantum analysis of linear systems,” Proc. of the IEEE **58**, 1599–1611 (1970). [CrossRef]

23. H. A. Haus and J. A. Mullen, “Quantum noise in linear amplifiers,” Phys. Rev. **128**, 2407 (1962). [CrossRef]

_{0}in terms of which the pulses are described, and a perturbation operator Δ

*â*that characterizes the noise

*ψ*

_{0}is the zeroth-order Hermite-Gaussian of the set:

*b*=

*ω*

^{″},

*ω*

^{″}=

*d*

^{2}

*ω*/

*d*

*β*

^{2}is the group-velocity dispersion, and

*ϕ*=tan

^{-1}(

*z*/

*b*). This set expresses, in general, chirped pulses after a time

*T*starting with minimum width

*ξ*

_{0}. The pulsewidth changes when the pulse propagates in a dispersive medium according to the law

*Â*

_{n}can be expressed as sums of Hermitian operators

*A*

_{1}is obtained from the expansion of a pulse that has been displaced by Δ

*X*and frequency shifted by Δ

*ω*. The pulse is described by

*X*and Δ

*ω*, the perturbations are replaced by operators, and the result is equated to ΔÂ

_{1ψ̂1}(

*x*,

*T*) with the result

*n*〉=

*T*changes its position in a manner proportional to the carrier frequency change and the dispersion

*ω*

^{″}=

*d*

^{2}

*ω*/

*dβ*

^{2}. This leads to an equation relating the position after propagation over a time

*T*to its initial value Δ

*X̂*(0)

*iM*cosΩ

_{M}(

*t*-Δ

*X*/

*υ*

_{g}), where

*M*is the depth of modulation, and

*Ω*

_{M}is the modualation frequency. When expanded to first order in Δ

*X*, we find a first-order Hermite Gaussian in quadrature. This term produces a momentum perturbation

*X̂*and Δ

*P̂*between input and output experience transformations that can be described by ABCD matrices

## 3 Conclusions

*ABCD*matrix transformation. The uncertainty ellipse of position and momentum fluctuations can be transformed using a system containing phase modulators and dispersive propagation segments. The reduction of position fluctuations can be used to improve timing signals obtained from detected pulse trains. The system need not operate at the minimum uncertainty limit to be of use.

## 4 Acknowledgments

## References and links

1. | L. S. Brown, |

2. | L. Knöll, W. Vogel, and D.-G. Welsch, “Action of passive, lossless optical systems in quantum optics,” Phys. Rev. A |

3. | R. J. Glauber and M. Lewenstein, “Quantum optics of dielectic media,” Phys Rev. A |

4. | H. Khosravi and R. Loudon, “Vacuum field fluctuations and spontaneous emission in a dielectric slab,” Proc. R. Soc. Lon don Ser. A |

5. | P. D. Drummond, “Electromagnetic quantization in dispersive inhomogeneous nonlinear dielectrics,” Phys. Rev. A |

6. | B. Huttner, J. J. Baumberg, and S. M. Barnett, “Canonical quantization of light in a linear dielectric,” Europhys. Lett. |

7. | P. W. Milonni, “Field quantization and radiative processes in dispersive dielectric media,” J. Mod. Opt. |

8. | R. E. Slusher, L. W. Hollberg, B. Yurke, J. C. Mertz, and J. F. Valley, “Observation of squeezed states generated by 4-wave mixing in an optical cavity,” Phys. Rev. Lett. |

9. | M. J. Potasek and B. Yurke, “Dissipative effects on squeezed light generated in systems governed by the nonlinear Schrödinger equation,” Phys. Rev. A |

10. | L. Wu, H. J. Kimble, J. L. Hall, and H. Wu, “Generation of squeezed states by parametric down conversion,” Phys. Rev. Lett. |

11. | M. Xiao, L. Wu, and H. J. Kimble, “Precision measurement beyond the shot-noise limit,” Phys. Rev. Lett. |

12. | M. W. Maeda, P. Kumar, and J. H. Shapiro, “Observation of squeezed noise produced by forward four-wave mixing in sodium vapor,” Op. Lett. |

13. | R. M. Shelby, M. D. Levenson, S. H. Perlmutter, R. G. DeVoe, and D. F. Walls, “Broad-band parametric deamplificationof quantum noise in an optical fiber,” Phys. Rev. B |

14. | M. Shirasaki and H. A. Haus, “Squeezing of pulses in a nonlinear interferometer,” J. Opt. Soc. Am. B |

15. | F. Hong-Yi and J. VanderLinde, “Squeezed-state wave functions and their relation to classical phase-space maps,” Phys. Rev. A |

16. | D. Rugar and P. Grutter, “Mechanical parametric amplification and thermomechanical noise squeezing,” Phys. Rev. Lett. |

17. | F. DiFilippo, V. Natarajan, K. R. Boyce, and D. E. Pritchard, “Classical amplitude squeezing for precisionm easurements,” Phys. Rev. Lett. |

18. | A. E. Siegman and D. J. Kuizenga, “Proposed method for measuring picosecond pulse widths and pulse shapes in CW mode-locked lasers,” IEEE J. Quantum Electron. |

19. | D. H. Auston, “Picosecond optoelectronic switching and gating in silicon,” Appl. Phys. Lett. |

20. | H. F. Taylor, “An electrooptic analog-to-digital converter-design and analysis,” IEEE J. Quantum. Electron. |

21. | J. C. Twichell and R. Helkey, “Phase-encoded optical sampling for analog-to-digital converters,” Phot. Tech. Lett. |

22. | H. A. Haus, “Steady-state quantum analysis of linear systems,” Proc. of the IEEE |

23. | H. A. Haus and J. A. Mullen, “Quantum noise in linear amplifiers,” Phys. Rev. |

24. | T. R. Clark, T. F. Carruthers, P. J. Matthews, and I. N. Duling III, “Phase noise measurements of ultrastable 10 GHz harmonically modelocked fibre laser,” Electron. Lett. |

**OCIS Codes**

(140.3460) Lasers and laser optics : Lasers

(270.0270) Quantum optics : Quantum optics

(270.6570) Quantum optics : Squeezed states

**ToC Category:**

Research Papers

**History**

Original Manuscript: April 24, 2001

Published: June 2, 2001

**Citation**

Matthew Grein, Hermann Haus, Leaf Jiang, and Erich Ippen, "Action on pulse position and momentum using dispersion and phase modulation," Opt. Express **8**, 664-669 (2001)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-8-12-664

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### References

- L. S. Brown, Quantum Field Theory (Cambridge University Press, 1992).
- L. Knöll, W. Vogel, and D.-G. Welsch, "Action of passive, lossless optical systems in quantum optics," Phys. Rev. A 36, 3803-3818 (1987). [CrossRef] [PubMed]
- R. J. Glauber and M. Lewenstein, "Quantum optics of dielectic media," Phys Rev. A 43, 467-491 (1991). [CrossRef] [PubMed]
- H. Khosravi and R. Loudon, "Vacuum field fluctuations and spontaneous emission in a dielectric slab," Proc. R. Soc. London Ser. A 436, 373-389 (1992). [CrossRef]
- P. D. Drummond, "Electromagnetic quantization in dispersive inhomogeneous nonlinear dielectrics," Phys. Rev. A 42, 6845-6857 (1990). [CrossRef] [PubMed]
- B. Huttner, J. J. Baumberg, and S. M. Barnett, "Canonical quantization of light in a linear dielectric," Europhys. Lett. 16, 177 (1991). [CrossRef]
- P. W. Milonni, "Field quantization and radiative processes in dispersive dielectric media," J. Mod. Opt. 42, 1991 (1995).
- R. E. Slusher, L. W. Hollberg, B. Yurke, J. C. Mertz, and J. F. Valley, "Observation of squeezed states generated by 4-wave mixing in an optical cavity," Phys. Rev. Lett. 55, 2409-2412 (1985). [CrossRef] [PubMed]
- M. J. Potasek and B. Yurke, "Dissipative effects on squeezed light generated in systems governed by the nonlinear Schrödinger equation," Phys. Rev. A 38, 1335-1348 (1988). [CrossRef] [PubMed]
- L. Wu, H. J. Kimble, J. L. Hall, and H. Wu, "Generation of squeezed states by parametric down conversion," Phys. Rev. Lett. 57, 2520 (1986). [CrossRef] [PubMed]
- M. Xiao, L. Wu, and H. J. Kimble, "Precision measurement beyond the shot-noise limit," Phys. Rev. Lett. 53, 278 (1987). [CrossRef]
- M. W. Maeda, P. Kumar, and J. H. Shapiro, "Observation of squeezed noise produced by forward four-wave mixing in sodium vapor," Opt. Lett. 12, 161 (1987). [CrossRef]
- R. M. Shelby, M. D. Levenson, S. H. Perlmutter, R. G. DeVoe, and D. F. Walls, "Broad-band parametric deamplification of quantum noise in an optical fiber," Phys. Rev. B 57, 691 (1986). [CrossRef]
- M. Shirasaki and H. A. Haus, "Squeezing of pulses in a nonlinear interferometer," J. Opt. Soc. Am. B 7, 30 (1990). [CrossRef]
- F. Hong-Yi and J. VanderLinde, "Squeezed-state wave functions and their relation to classical phase-space maps," Phys. Rev. A 40, 4785 (1989). [CrossRef]
- D. Rugar and P. Grutter, "Mechanical parametric amplification and thermomechanical noise squeezing," Phys. Rev. Lett. 67, 699 (1991). [CrossRef] [PubMed]
- F. DiFilippo, V. Natarajan, K. R. Boyce, and D. E. Pritchard, "Classical amplitude squeezing for precision measurements," Phys. Rev. Lett. 68, 2859 (1992). [CrossRef] [PubMed]
- A. E. Siegman and D. J. Kuizenga, "Proposed method for measuring picosecond pulse widths and pulse shapes in CW mode-locked lasers," IEEE J. Quantum Electron. 6, 212-215 (1970). [CrossRef]
- D. H. Auston, "Picosecond optoelectronic switching and gating in silicon," Appl. Phys. Lett. 26, 101-103 (1975). [CrossRef]
- H. F. Taylor, "An electrooptic analog-to-digital converter-design and analysis," IEEE J. Quantum. Electron. 15, 210-216 (1979). [CrossRef]
- J. C. Twichell and R. Helkey, "Phase-encoded optical sampling for analog-to-digital converters," Phot. Tech. Lett. 12, 1237-1239 (2000). [CrossRef]
- H. A. Haus, "Steady-state quantum analysis of linear systems," Proc. of the IEEE 58, 1599-1611 (1970). [CrossRef]
- H. A. Haus and J. A. Mullen, "Quantum noise in linear amplifiers," Phys. Rev. 128, 2407 (1962). [CrossRef]
- T. R. Clark, T. F. Carruthers, P. J. Matthews, and I. N. Duling III, "Phase noise measurements of ultrastable 10 GHz harmonically modelocked fibre laser," Electron. Lett. 35, 720-721 (1999). [CrossRef]

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