## Enhanced degree of temporal coherence through temporal and spatial phase coupling within a focused supercontinuum

Optics Express, Vol. 17, Issue 22, pp. 20140-20148 (2009)

http://dx.doi.org/10.1364/OE.17.020140

Acrobat PDF (910 KB)

### Abstract

In the diffraction of a supercontinuum source, a redistribution of amplitude and phase at the focal region is incurred by the coupling between the supercontinuum and the spatial phase caused by the lens diffraction, making it extremely difficult to predict the focal behaviour. We show that the coupling between the temporal phase of a SC source and the spatial phase from the diffraction by a low numerical aperture (NA) lens causes dramatic alterations in the spectra and the temporal coherence near the focal region, and that this effect is maximized in points of singularity. Furthermore, we show that such an enhancement in temporal coherence can be controlled by the pulse evolution through the photonic crystal fiber, in which nonlinear and disperive effects such as the soliton fission process provides the key phase evolution necessary for dramatically changing the coherence time of the focused electromagnetic wave.

© 2009 Optical Society of America

## 1. Introduction

1. J. C. Knight, J. Broeng, T. A. Birks, and P. St. J. Russell, “Photonic band cap guidance in optical fibers,” Science **282**, 1476–1478 (1998).
[CrossRef] [PubMed]

2. 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**, 25–27 (2000).
[CrossRef]

3. I. Hartl, X. D. Li, C. Chudoba, R. K. Ghanta, T. H. Ko, J. G. Fujimoto, J. K. Ranka, and R. S. Windeler, “Ultrahigh-resolution optical coherence tomography using continuum generation in an air-silica microstructure fiber,” Opt. Lett. **26**, 608–610 (2001).
[CrossRef]

6. J. E. Morris, A. E. Carruthers, M. Mazilu, P. J. Reece, T. Cizmar, P. Fischer, and K. Dholakia, “Optical micro-manipulation using supercontinuum Laguerre Gaussian and Gaussian beams,” Opt. Express **16**, 1011–10129 (2008).
[CrossRef]

1. J. C. Knight, J. Broeng, T. A. Birks, and P. St. J. Russell, “Photonic band cap guidance in optical fibers,” Science **282**, 1476–1478 (1998).
[CrossRef] [PubMed]

2. 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**, 25–27 (2000).
[CrossRef]

7. W. H. Reeves, D. V. Skryabin, F. Biancalana, J. C. Knight, P. St. J. Russell, F. G. Omenetto, A. Efimov, and A.J. Taylor, “Transformation and control of ultrashort pulses in dispersion-engineered photonic crystal fibers,” Nature **474**, 511–515 (2003).
[CrossRef]

8. A. V. Husakou and J. Herrmann, “Supercontinuum generation higher-order solutions by fission in photonic crystal fibers,” Phys. Rev. Lett. **87**, 203901 (2001).
[CrossRef] [PubMed]

10. F. De Martini, C. H. Townes, T. K. Gustafson, and P. L. Kelley, “Self-Steepening of light pulses,” Phys. Rev. **164**, 312–323 (1967).
[CrossRef]

11. C.V. Raman, “A change of wave-length in light scattering [8],” Nature **121**, 619- (1928).
[CrossRef]

3. I. Hartl, X. D. Li, C. Chudoba, R. K. Ghanta, T. H. Ko, J. G. Fujimoto, J. K. Ranka, and R. S. Windeler, “Ultrahigh-resolution optical coherence tomography using continuum generation in an air-silica microstructure fiber,” Opt. Lett. **26**, 608–610 (2001).
[CrossRef]

4. H. N. Paulsen, K. M. Hilligsøe, J. Thøgersen, S. R. Keiding, and J. J. Larsen, “Coherent anti-Stokes Raman scattering microscopy with a photonic crystal fiber based light source,” Opt. Lett. **28**, 1123–1125 (2003).
[CrossRef] [PubMed]

12. K. B. Shi, P. Li, S. Z. Yin, and Z. W. Liu, “Chromatic confocal microscopy using supercontinuum light,” Opt. Express **12**, 2096–2101 (2004).
[CrossRef] [PubMed]

13. K. Isobe, W. Watanabe, S. Matsunaga, T. Higashi, K. Fukui, and K. Itoh, “Multi-spectral two-photon excited fluorescence microscopy using supercontinuum light source,” Jpn. J. Appl. Phys. **44**, L167–L169 (2005).
[CrossRef]

6. J. E. Morris, A. E. Carruthers, M. Mazilu, P. J. Reece, T. Cizmar, P. Fischer, and K. Dholakia, “Optical micro-manipulation using supercontinuum Laguerre Gaussian and Gaussian beams,” Opt. Express **16**, 1011–10129 (2008).
[CrossRef]

15. G. Gbur, T. D. Visser, and E. Wolf, “Anomalous behavior of spectra near phase singularities of focused waves,” Phys. Rev. Lett. **88**, 013901 (2002).
[CrossRef] [PubMed]

17. W. Wang, N. Ishii, S. G. Hanson, Y. Miyamoto, and M. Takeda, “Phase singularities in analytic signal of white-light speckle pattern with application to micro-displacement measurement,” Opt. Comm. **248**, 59–68 (2005).
[CrossRef]

## 2. Supercontinuum diffraction

*u*and

*v*are given by

*r*respectively, where

*r*and

*z*are the radial and axial coordinated of the lens image space. The parameter a and b are the aperture radius and the integral lower bound for the lens, NA is the numerical aperture,

*J*

_{0}is a zero order Bessel function of the first kind,

*ω*is the angular frequency and c is the speed of light. If

*b*=0, the diffraction is for the complete aperture and for a non zero

*b*is a diaphragm.

*U*(

*ω*) is the Fourier transform of the SC wave using the dispersion parameters, nonlinear parameters and the method described by

*Chick et. al*. [19

19. B. J. Chick, J. W. M. Chon, and M. Gu, “Polarization effects in a highly birefringent nonlinear photonic crystal fiber with two-zero dispersion wavelengths,” Opt. Express **16**, 20099–20080 (2008).
[CrossRef] [PubMed]

*U*

_{1}(

*u,v,ω*) is used to obtain the temporal profile

*U*

_{1}(

*u,v,t*).

*v*=0) for the diffraction field [Eq. (1)] can be obtained by the following equation:

*a*

^{2}

*u*/2=±2

*nπ*and if

*b*=0, the equation is equal to

*ω*and

*z*there lays a region of singularity.

*z*and occur at

*u*=±4

*nπ*(where n is an integer, for the radial direction is the zero of a zero order Bessel function of the first kind). The parameters

*u*

_{0}and

*v*

_{0}(Fig. 1a) are defined as the normalized axial and radial coordinates of the optical system and are given by

*z*and

*r*, where NA is the numerical aperture,

*z*and

*r*are the axial and radial dimensions (in

*µ*m), and λ

_{0}is the center wavelength of the original pulse coupled to the nonlinear photonic crystal fiber (the input pulse is a hyperbolic secant pulse which is used to represent a mode-locked laser pulse).

*S*(

*t,z*

_{0})) and evolves with time. The intuitive observation however would be to view the focal plane from the side, where the intensity is both temporally and axially dependent since the leading intensities of the pulse are modified by the diffraction for an axial position and differs from the trailing intensities (

*S*(

*t, z*)), which is referred to as a nonstationary observer.

*U*(

*ω*) is the intensity distribution for the stationary observation frame where the intensity for the nonstationary observation frame is obtained by taking the diagonal of the the matrix

*U*

_{1}(

*u, t*) for different v.

## 3. Temporal coherence and mean frequency

*g*

^{1}(

*u*

_{0},

*v*

_{0},

*τ*)) which is generalized through the correlation between two points and is calculated by [20]

*z*and

*t*are the axial and temporal coordinates.

*u*

_{0}and

*v*

_{0}remain constant; hence Eq. (3) becomes an auto-correlation technique determined by

*g*

^{1}(

*u*

_{0},

*v*

_{0},

*τ*) depends on the position (u0,v0) of the detector.

*τ*(Figs. 3a-d) using Eq. (6). For the stationary observation frame, the coherence time changes around the region of the phase singularity (Fig. 3a) and in fact an enhancement of the coherence time occurs because of the spectral redistribution that modifies the bandwidth. Compared with the coherence time,

_{c}*τ*

_{0}=0.005

*ps*, of the SC before it is focused,

*τ*at the singularities is enhanced by a factor of 2. The coherence time in this situation is symmetric with respect to the focal plane, which is physically expected since it is contributed by a single axial position. In this case, the spatial phase contribution from the lens diffraction is unchanged during the correlation measurement, since the diffraction equation is symmetric with respect to the focal plane. This symmetry holds for larger NA lenses and thus the coherence time shows little variation with NA (Fig. 3c).

_{c}24. K. L. Corwin, N. R. Newbury, J. M. Dudley, S. Coen, S. A. Diddams, K. Weber, and R. S. Windler, “Fundamental noise limitations to supercontinuum generation in microstructured fiber,” Phys. Rev. Lett. **90**, 112904 (2003).
[CrossRef]

## 4. Conclusion

## Acknowledgements

## References and links

1. | J. C. Knight, J. Broeng, T. A. Birks, and P. St. J. Russell, “Photonic band cap guidance in optical fibers,” Science |

2. | 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. |

3. | I. Hartl, X. D. Li, C. Chudoba, R. K. Ghanta, T. H. Ko, J. G. Fujimoto, J. K. Ranka, and R. S. Windeler, “Ultrahigh-resolution optical coherence tomography using continuum generation in an air-silica microstructure fiber,” Opt. Lett. |

4. | H. N. Paulsen, K. M. Hilligsøe, J. Thøgersen, S. R. Keiding, and J. J. Larsen, “Coherent anti-Stokes Raman scattering microscopy with a photonic crystal fiber based light source,” Opt. Lett. |

5. | Th. Udem, R. Holzwarth, and T. W. Hänsch, “Optical frequency metrology,” Nature |

6. | J. E. Morris, A. E. Carruthers, M. Mazilu, P. J. Reece, T. Cizmar, P. Fischer, and K. Dholakia, “Optical micro-manipulation using supercontinuum Laguerre Gaussian and Gaussian beams,” Opt. Express |

7. | W. H. Reeves, D. V. Skryabin, F. Biancalana, J. C. Knight, P. St. J. Russell, F. G. Omenetto, A. Efimov, and A.J. Taylor, “Transformation and control of ultrashort pulses in dispersion-engineered photonic crystal fibers,” Nature |

8. | A. V. Husakou and J. Herrmann, “Supercontinuum generation higher-order solutions by fission in photonic crystal fibers,” Phys. Rev. Lett. |

9. | J. M. Dudley, G. Genty, and S. Coen, “Supercontinuum generation in photonic crystal fiber,” Mod. Phys. Rev. |

10. | F. De Martini, C. H. Townes, T. K. Gustafson, and P. L. Kelley, “Self-Steepening of light pulses,” Phys. Rev. |

11. | C.V. Raman, “A change of wave-length in light scattering [8],” Nature |

12. | K. B. Shi, P. Li, S. Z. Yin, and Z. W. Liu, “Chromatic confocal microscopy using supercontinuum light,” Opt. Express |

13. | K. Isobe, W. Watanabe, S. Matsunaga, T. Higashi, K. Fukui, and K. Itoh, “Multi-spectral two-photon excited fluorescence microscopy using supercontinuum light source,” Jpn. J. Appl. Phys. |

14. | M. Born and E. Wolf, “Principles of Optics,” 7th ed. (Cambridge University Press, Cambridge, 1999). |

15. | G. Gbur, T. D. Visser, and E. Wolf, “Anomalous behavior of spectra near phase singularities of focused waves,” Phys. Rev. Lett. |

16. | D. W. Robinson and G. T. Reid, “Phase unwrapping methods,” Interferogram Analysis, 194–229 (1993). |

17. | W. Wang, N. Ishii, S. G. Hanson, Y. Miyamoto, and M. Takeda, “Phase singularities in analytic signal of white-light speckle pattern with application to micro-displacement measurement,” Opt. Comm. |

18. | M. Gu, “Advanced optical imaging theory,” (Springer Velag, Heidelberg, 2000). |

19. | B. J. Chick, J. W. M. Chon, and M. Gu, “Polarization effects in a highly birefringent nonlinear photonic crystal fiber with two-zero dispersion wavelengths,” Opt. Express |

20. | R. Loudon, “The quantum theory of light,” 2nd ed. (Oxford University Press, 1983). |

21. | M. Bertolotti, A. Ferrari, and L. Sereda, “Coherence properties of nonstationary polychromatic light sources,” J. Opt. Soc. Am. B |

22. | L. Sereda and M. Bertolotti, “Coherence properties of nonstationary light wave fields,” J. Opt. Soc. Am. A |

23. | G. P. Agrawal, “Nonlinear fiber optics,” 3rd ed. (Academic San Diego, Calif., 2002). |

24. | K. L. Corwin, N. R. Newbury, J. M. Dudley, S. Coen, S. A. Diddams, K. Weber, and R. S. Windler, “Fundamental noise limitations to supercontinuum generation in microstructured fiber,” Phys. Rev. Lett. |

25. | J. M. Dudley and S. Coen, “Coherence properties of supecontinuum sprectra generated in photonic crystal and tapered optical fibers,” Opt. Exp. |

**OCIS Codes**

(030.0030) Coherence and statistical optics : Coherence and statistical optics

(050.0050) Diffraction and gratings : Diffraction and gratings

(190.0190) Nonlinear optics : Nonlinear optics

(260.0260) Physical optics : Physical optics

**ToC Category:**

Coherence and Statistical Optics

**History**

Original Manuscript: September 25, 2009

Revised Manuscript: October 19, 2009

Manuscript Accepted: October 19, 2009

Published: October 20, 2009

**Citation**

Brendan J. Chick, James W. M. Chon, and Min Gu, "Enhanced degree of temporal coherence
through temporal and spatial phase
coupling within a focused
supercontinuum," Opt. Express **17**, 20140-20148 (2009)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-22-20140

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### References

- J. C. Knight, J. Broeng, T. A. Birks and P. St. J. Russell, "Photonic band cap guidance in optical fibers," Science 282, 1476-1478 (1998). [CrossRef] [PubMed]
- 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, 25-27 (2000). [CrossRef]
- I. Hartl, X. D. Li, C. Chudoba, R. K. Ghanta, T. H. Ko, J. G. Fujimoto, J. K. Ranka, and R. S. Windeler, "Ultrahigh-resolution optical coherence tomography using continuum generation in an air-silica microstructure fiber," Opt. Lett. 26, 608-610 (2001). [CrossRef]
- H. N. Paulsen, K. M. Hilligsøe, J. Thøgersen, S. R. Keiding, and J. J. Larsen, "Coherent anti-Stokes Raman scattering microscopy with a photonic crystal fiber based light source," Opt. Lett. 28, 1123-1125 (2003). [CrossRef] [PubMed]
- Th. Udem, R. Holzwarth, and T. W. Hansch, "Optical frequency metrology," Nature 416, 233-237 (2002). [CrossRef] [PubMed]
- J. E. Morris, A. E. Carruthers, M. Mazilu, P. J. Reece, T. Cizmar, P. Fischer and K. Dholakia, "Optical micromanipulation using supercontinuum Laguerre Gaussian and Gaussian beams," Opt. Express 16, 1011-10129 (2008). [CrossRef]
- W. H. Reeves, D. V. Skryabin, F. Biancalana, J. C. Knight, P. St. J. Russell, F. G. Omenetto, A. Efimov and A.J. Taylor, "Transformation and control of ultrashort pulses in dispersion-engineered photonic crystal fibers," Nature 474, 511-515 (2003). [CrossRef]
- A. V. Husakou and J. Herrmann, "Supercontinuum generation higher-order solutions by fission in photonic crystal fibers," Phys. Rev. Lett. 87, 203901 (2001). [CrossRef] [PubMed]
- J. M. Dudley, G. Genty and S. Coen, "Supercontinuum generation in photonic crystal fiber," Mod. Phys. Rev. 78, 113-1184 (2006).
- F. De Martini, C. H. Townes, T. K. Gustafson, and P. L. Kelley, "Self-Steepening of light pulses," Phys. Rev. 164, 312-323 (1967). [CrossRef]
- C.V. Raman, "A change of wave-length in light scattering [8]," Nature 121, 619- (1928). [CrossRef]
- K. B. Shi, P. Li, S. Z. Yin and Z. W. Liu, "Chromatic confocal microscopy using supercontinuum light," Opt. Express 12, 2096-2101 (2004). [CrossRef] [PubMed]
- K. Isobe, W. Watanabe, S. Matsunaga, T. Higashi, K. Fukui and K. Itoh, "Multi-spectral two-photon excited fluorescence microscopy using supercontinuum light source," Jpn. J. Appl. Phys. 44, L167-L169 (2005). [CrossRef]
- M. Born and E. Wolf, "Principles of Optics," 7th ed. (Cambridge University Press, Cambridge, 1999).
- G. Gbur, T. D. Visser and E. Wolf, "Anomalous behavior of spectra near phase singularities of focused waves," Phys. Rev. Lett. 88, 013901 (2002). [CrossRef] [PubMed]
- D. W. Robinson and G. T. Reid, "Phase unwrapping methods," Interferogram Analysis, 194-229 (1993).
- W. Wang, N. Ishii, S. G. Hanson, Y. Miyamoto and M. Takeda, "Phase singularities in analytic signal of whitelight speckle pattern with application to micro-displacement measurement," Opt. Comm. 248, 59-68 (2005). [CrossRef]
- M. Gu, "Advanced optical imaging theory," (Springer Velag, Heidelberg, 2000).
- B. J. Chick, J. W. M. Chon and M. Gu, "Polarization effects in a highly birefringent nonlinear photonic crystal fiber with two-zero dispersion wavelengths," Opt. Express 16, 20099-20080 (2008). [CrossRef] [PubMed]
- R. Loudon, "The quantum theory of light," 2nd ed. (Oxford University Press, 1983).
- M. Bertolotti, A. Ferrari and L. Sereda, "Coherence properties of nonstationary polychromatic light sources," J. Opt. Soc. Am. B 12, 341-347 (1995). [CrossRef]
- L. Sereda and M. Bertolotti, "Coherence properties of nonstationary light wave fields," J. Opt. Soc. Am. A 15, 695-705 (1998). [CrossRef]
- G. P. Agrawal, "Nonlinear fiber optics," 3rd ed. (Academic San Diego, Calif., 2002).
- K. L. Corwin, N. R. Newbury, J. M. Dudley, S. Coen, S. A. Diddams, K. Weber and R. S. Windler, "Fundamental noise limitations to supercontinuum generation in microstructured fiber," Phys. Rev. Lett. 90, 112904 (2003). [CrossRef]
- J. M. Dudley and S. Coen, "Coherence properties of supecontinuum sprectra generated in photonic crystal and tapered optical fibers," Opt. Exp. 27, 1180-1182 (2002).

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