## Quantum noise evolution under optical Kerr effects and two-photon absorption in a semiconductor waveguide

Optics Express, Vol. 16, Issue 5, pp. 3167-3171 (2008)

http://dx.doi.org/10.1364/OE.16.003167

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### Abstract

We theoretically study evolution of quantum noise of ultrashort pulsed light that propagates a semiconductor waveguide where nonlinear optical interaction occurs. Optical quantum noise is simulated by statistical (pseudo-)random distribution of phasors in a phase space with Gaussian probability weight, and each phasor evolution is governed by beam propagation method. It is shown that Kerr effects squeeze quantum noise of coherent light in a phase space such that photon-number noise is unchanged while phase noise increasing with uncertainty area invariant. However, two-photon absorption alters the photon-number statistics of light unlike Kerr effects.

© 2008 Optical Society of America

## 1. Introduction

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

3. E. S. Polzik, J. Carri, and H. J. Kimble, “Atomic spectroscopy with squeezed light for sensitivity beyond the vacumm-state limit,” Appl. Phys. BB55279–290 (1992). [CrossRef]

4. M. Hillery, “Quantum cryptography with squeezed states,” Phys. Rev. A **61**, 022309–022316 (2000). [CrossRef]

6. H. -A. Bachor, “Quantum noise, quantum measurement, and squeezing,” J. Opt. B: Quantum and semiclassical optics , **6**S626–S633 (2004). [CrossRef]

6. H. -A. Bachor, “Quantum noise, quantum measurement, and squeezing,” J. Opt. B: Quantum and semiclassical optics , **6**S626–S633 (2004). [CrossRef]

## 2. Simulation

_{0.15}Ga

_{0.85}As waveguide as a nonlinear medium where Kerr effects occur during propagation of 1 ps pulsed light at just below half the band gap energy corresponding to 1550 nm wavelength. Tuning incident photon energy across half the band gap enables TPA to turn on or off.

*X*1 and

*X*2 in a complex phase space, where

*X*1 and

*X*2 are the real and the imaginary part of a complex field envelop

*A*, respectively. The propagation of light for each phasor is calculated by

*γ*=

*ω*

_{0}

*n*

_{2}/

*ca*

_{eff}is the Kerr coefficient. Here

*a*

_{eff}is the effective cross-sectional area for third-order nonlinearity and

*n*

_{2}is the nonlinear index parameter inducing coupling of light intensity (

*I*) with its phase via the intensity dependence of a refractive index:

*n*

_{2}used in the simulation is given by the formula with the band-gap (1.61 eV) and the wavelength (1550 nm) in use [10

10. M. Sheik-Bahae, D. J. Hagan, and E. W. Van Styland. “Dispersion and Band-gap scaling of the electronic Kerr effect in solids assoicated with two-photon absorption,” Phys. Rev. Lett. **65**96–99 (1990). [CrossRef] [PubMed]

11. M. Sheik-Bahae, D. C. Hutchings, D. J. Hagan, and E. W. Van Stryland, “Dispersion of bound electronic nonlinear refraction in solids,” IEEE J. Quantum. Electron. **27**1296–1309 (1991). [CrossRef]

*L*) waveguide.

*X*1 axis as shown in Fig.1. This allows the fluctuations of the

*X*1 and the

*X*2 of an initial state to be determined by the amplitude fluctuation given by the photon-number variance of the coherent state.

*A*(

*t*=

*t*) at the waveguide output, for the various pulse energies of incident light. Here

_{c}*t*is the temporal pulse center. For each pulse energy, the distribution denoted by blue dots is compared with the reference distribution (red dots) that represents the coherent state (

_{c}*C*-state) with the same average photon-number as that of the squeezed state at the waveguide output. The photon-number noise change is given by the Fanofactor

*F*≡(Δ

*N*)

^{2}/(Δ

*N*)

_{c}^{2}, where the (Δ

*N*)

_{c}^{2}is the photon-number variance of the

*C*-state.

^{2}increases for these pulse energies with respect to that of the corresponding

*C*-state, (ΔΦ

_{c})

^{2}.

8. R. G. Ispasoiu and T. Goodson III, “Photon-number squeezing by two-photon absorption in an organic polymer,” Opt. Commun. **178**371–376 (2000). [CrossRef]

9. H. Cao, W. S Warren, A. Dogariu, and L. J. Wang, “Reduction of optical intensity noise by means of two-photon absorption,” J. Opt. Soc. Am. B **20**560–563 (2003). [CrossRef]

*λ*=1540 nm) enables TPA to turn on in the semiconductor waveguide. To include TPA effects in the simulation, equation (1) is extended by adding a term -

*α*

_{2}|

*A*|

^{2}

*A*/2, where

*α*

_{2}≡

*κ*

_{2}/

*a*

_{eff}is the TPA coefficient. Here

*κ*

_{2}is the TPA parameter in the unit of m/W, which is determined by the formula in [10

10. M. Sheik-Bahae, D. J. Hagan, and E. W. Van Styland. “Dispersion and Band-gap scaling of the electronic Kerr effect in solids assoicated with two-photon absorption,” Phys. Rev. Lett. **65**96–99 (1990). [CrossRef] [PubMed]

11. M. Sheik-Bahae, D. C. Hutchings, D. J. Hagan, and E. W. Van Stryland, “Dispersion of bound electronic nonlinear refraction in solids,” IEEE J. Quantum. Electron. **27**1296–1309 (1991). [CrossRef]

12. L. Giles and P. L. Knight, “Two-photon absorption and nonclassical states of light,” Phys. Rev. A **48**1582–1593 (1993). [CrossRef]

## 3. Conclusion

## Acknowledgments

## References and links

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

2. | W. Hai, X. Changde, P. Qing, X. Chenyang, Z. Yun, and P. Kunchi, “Optical measurement of weak absorption beyond shot-noise limit,” Laser Spectroscopy XIII Proc. 13th Int.Conf. Laser pectroscopy, Hangzhou, (1997). |

3. | E. S. Polzik, J. Carri, and H. J. Kimble, “Atomic spectroscopy with squeezed light for sensitivity beyond the vacumm-state limit,” Appl. Phys. BB55279–290 (1992). [CrossRef] |

4. | M. Hillery, “Quantum cryptography with squeezed states,” Phys. Rev. A |

5. | H-. A. Bachor, “A guide to experiments in quantum optics,” WILEY-VCH (1998). |

6. | H. -A. Bachor, “Quantum noise, quantum measurement, and squeezing,” J. Opt. B: Quantum and semiclassical optics , |

7. | M. B Ward, G. M. Schucan, K. Turner, T. Goodson III, K. J. Donovan, J. S. Aitchison, C. N. Ironside, A. M. Fox, and J. F. Ryan, “Photon-number squeezed light generation by multi-photon absorption,” CLEO/QELS Technical Digest QWE4, (1999). |

8. | R. G. Ispasoiu and T. Goodson III, “Photon-number squeezing by two-photon absorption in an organic polymer,” Opt. Commun. |

9. | H. Cao, W. S Warren, A. Dogariu, and L. J. Wang, “Reduction of optical intensity noise by means of two-photon absorption,” J. Opt. Soc. Am. B |

10. | M. Sheik-Bahae, D. J. Hagan, and E. W. Van Styland. “Dispersion and Band-gap scaling of the electronic Kerr effect in solids assoicated with two-photon absorption,” Phys. Rev. Lett. |

11. | M. Sheik-Bahae, D. C. Hutchings, D. J. Hagan, and E. W. Van Stryland, “Dispersion of bound electronic nonlinear refraction in solids,” IEEE J. Quantum. Electron. |

12. | L. Giles and P. L. Knight, “Two-photon absorption and nonclassical states of light,” Phys. Rev. A |

**OCIS Codes**

(190.0190) Nonlinear optics : Nonlinear optics

(190.3270) Nonlinear optics : Kerr effect

(270.2500) Quantum optics : Fluctuations, relaxations, and noise

(270.5290) Quantum optics : Photon statistics

**ToC Category:**

Nonlinear Optics

**History**

Original Manuscript: October 25, 2007

Revised Manuscript: February 18, 2008

Manuscript Accepted: February 18, 2008

Published: February 21, 2008

**Citation**

Heongkyu Ju and Eun-Cheol Lee, "Quantum noise evolution under optical Kerr effects and two-photon absorption in a semiconductor waveguide," Opt. Express **16**, 3167-3171 (2008)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-5-3167

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### References

- R. E. Slusher, L. W. Hollberg, B. Yurke, J. C. Mertz, and J. F. Valley, "Observation of squeezed states generated by four-wave mixing in an optical cavity," Phys. Rev. Lett. 55, 2409-2412 (1985). [CrossRef] [PubMed]
- W. Hai, X. Changde, P. Qing, X. Chenyang, Z. Yun, and P. Kunchi, "Optical measurement of weak absorption beyond shot-noise limit," Laser Spectroscopy XIII Proc. 13th Int. Conf. on Laser Spectroscopy, Hangzhou, (1997).
- E. S. Polzik, J. Carri, and H. J. Kimble, "Atomic spectroscopy with squeezed light for sensitivity beyond the vacumm-state limit," Appl. Phys. B B55, 279-290 (1992). [CrossRef]
- M. Hillery, "Quantum cryptography with squeezed states," Phys. Rev. A 61, 022309-022316 (2000). [CrossRef]
- H.- A. Bachor, A Guide to Experiments in Quantum Optics (Wiley-VCH, 1998).
- H. -A. Bachor, "Quantum noise, quantum measurement, and squeezing," J. Opt. B: Quantum Semiclassical Opt. 6, S626-S633 (2004). [CrossRef]
- M. B. Ward, G. M. Schucan, K. Turner, T. GoodsonIII, K. J. Donovan, J. S. Aitchison, C. N. Ironside, A. M. Fox and J. F. Ryan, "Photon-number squeezed light generation by multi-photon absorption," CLEO/QELS Technical Digest QWE4, (1999).
- R. G. Ispasoiu and T. GoodsonIII, "Photon-number squeezing by two-photon absorption in an organic polymer," Opt. Commun. 178, 371-376 (2000). [CrossRef]
- H. Cao, W. S. Warren, A. Dogariu, and L. J. Wang, "Reduction of optical intensity noise by means of two-photon absorption," J. Opt. Soc. Am. B 20, 560-563 (2003). [CrossRef]
- M. Sheik-Bahae, D. J. Hagan, and E. W. Van Styland. "Dispersion and Band-gap scaling of the electronic Kerr effect in solids assoicated with two-photon absorption," Phys. Rev. Lett. 65, 96-99 (1990). [CrossRef] [PubMed]
- M. Sheik-Bahae, D. C. Hutchings, D. J. Hagan, E. W. Van Stryland, "Dispersion of bound electronic nonlinear refraction in solids," IEEE J. Quantum. Electron. 27, 1296-1309 (1991). [CrossRef]
- L. Giles and P. L. Knight, "Two-photon absorption and nonclassical states of light," Phys. Rev. A 48, 1582-1593 (1993). [CrossRef]

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