## Pulse shaping with birefringent crystals: a tool for quantum metrology |

Optics Express, Vol. 21, Issue 19, pp. 21889-21896 (2013)

http://dx.doi.org/10.1364/OE.21.021889

Acrobat PDF (1392 KB)

### Abstract

A method for time differentiation based on a Babinet-Soleil-Bravais compensator is introduced. The complex transfer function of the device is measured using polarization spectral interferometry. Time differentiation of both the pulse field and pulse envelope are demonstrated over a spectral width of about 100 THz with a measured overlap with the objective mode greater than 99.8%. This pulse shaping technique is shown to be perfectly suited to time metrology at the quantum limit.

© 2013 OSA

## 1. Introduction

1. J. Abadie, “A gravitational wave observatory operating beyond the quantum shot-noise limit,” Nat. Phys. **7**, 962–965 (2011). [CrossRef]

2. V. Giovannetti, S. Lloyd, and L. Maccone, “Quantum metrology,” Phys. Rev. Lett. **96**, 010401 (2006). [CrossRef] [PubMed]

3. C. M. Caves, “Quantum-mechanical noise in an interferometer,” Phys. Rev. D **23**, 1693–1708 (1981). [CrossRef]

5. S. L. Braunstein and C. M. Caves, “Statistical distance and the geometry of quantum states,” Phys. Rev. Lett. **72**, 3439–3443 (1994). [CrossRef] [PubMed]

2. V. Giovannetti, S. Lloyd, and L. Maccone, “Quantum metrology,” Phys. Rev. Lett. **96**, 010401 (2006). [CrossRef] [PubMed]

*N*is the number of photons, can be reached using coherent states. In this regime, getting to the standard quantum limit is already a challenge and thus quantum improvement is realistically limited to the use of gaussian states such as squeezed states [6

6. B. M. Escher, R. L. de Matos Filho, and L. Davidovich, “General framework for estimating the ultimate precision limit in noisy quantum-enhanced metrology,” Nat. Phys. **7**, 406–411 (2011). [CrossRef]

7. V. Delaubert, N. Treps, C. Fabre, H.-A. Bachor, and P. Réfrégier, “Quantum limits in image processing,” Europhys. Lett. **81**, 44001 (2008). [CrossRef]

8. O. Pinel, J. Fade, D. Braun, P. Jian, N. Treps, and C. Fabre, “Ultimate sensitivity of precision measurements with intense Gaussian quantum light: a multimodal approach,” Phys. Rev. A **85**,(2012). [CrossRef]

9. B. Lamine, C. Fabre, and N. Treps, “Quantum improvement of time transfer between remote clocks,” Phys. Rev. Lett. **101**, 123601 (2008). [CrossRef] [PubMed]

10. A.M. Weiner, “Femtosecond pulse shaping using spatial light modulators,” Rev. Sci. Instr. **71**, 1929–1960 (2000). [CrossRef]

12. A.M. Weiner, “Ultrafast optical pulse shaping: A tutorial review,” Opt. Comm. **284**, 3669–3692 (2011). [CrossRef]

13. F. Frei, A. Galler, and T. Feurer, “Space-time coupling in femtosecond pulse shaping and its effects on coherent control,” J. Chem. Phys. **130**, 034302 (2009). [CrossRef] [PubMed]

*ω*. More generally, perturbation linked to the first order decomposition of the index of refraction of the medium lead to a required pulse shape mode being, in frequency domain, a multiplication by a linear function of

*ω*[14

14. P. Jian, O. Pinel, C. Fabre, B. Lamine, and N. Treps, “Real-time displacement measurement immune from atmospheric parameters using optical frequency combs,” Opt. Express **20**, 27133–27146 (2012). [CrossRef] [PubMed]

*ℰ*(

*t*) =

*t*)exp(−

*iω*

_{0}

*t*), where

*t*) is the pulse envelope and

*ω*

_{0}is the center frequency. Let us first consider two specific kinds of pulse shapes which are of interest for optimal measurements. Firstly the time derivative of the electric field, which is associated to time delay in vacuum: secondly the pulse corresponding to the time derivative of the envelope, which is associated to time of flight measurement: The time constants

*T*

_{1}and

*T*

_{2}have been introduced for sake of dimensional homogeneity. In frequency domain, we thus have to fulfill the relations and Such pulses can be generated using linear filters with transfer functions equal respectively to

*R*

_{1}(

*ω*) = −

*iωT*

_{1}and

*R*

_{2}(

*ω*) = −

*i*(

*ω*−

*ω*

_{0})

*T*

_{2}. Note however that, in the case of passive linear filters, the transfer function must have a magnitude smaller than or equal to 1, which will require a compromise between accuracy and efficiency. Especially in the latter case where the transfer function increases in the wings of the spectrum, the transfer function will have to be clipped in order to maintain a reasonable overall efficiency of the linear filter.

15. M. A. Preciado and M. A. Muriel, “Design of an ultrafast all-optical differentiator based on a fiber Bragg grating in transmission,” Opt. Lett. **33**, 2458–2460 (2008). [CrossRef] [PubMed]

17. F. X. Li, Y. W. Park, and J. Azana, “Complete temporal pulse characterization based an phase reconstruction using optical ultrafast differentiation (PROUD),” Opt. Lett. **32**, 3364–3366 (2007). [CrossRef] [PubMed]

## 2. Time differentiation using birefringence

*ℰ*(

*t*) relies on the destructive interference between two replica of the incident pulse separated by a time delay

*τ*much smaller than the optical cycle: Note that the 1/2 factor is introduced to account for the fact that the energy transmission of each arm of the interferometer is 1/4, when taking into account the beam splitter and the beam recombiner. This equation yields the desired time derivative

*ℰ*

_{1}(

*t*), with

*T*

_{1}=

*τ*/2. The time derivative of the pulse envelope can be similarly obtained by choosing

*τ*equal to a multiple of the optical cycle,

*i.e.*

*τ*= 2

*nπ*/

*ω*

_{0}, where

*n*is an integer number. Provided that

*τ*is much smaller than the pulse duration, we obtain

*ℰ*

_{2}(

*t*), with

*T*

_{2}=

*τ*/2.

*o⃗*) and extraordinary (

*e⃗*) axes make a 45° angle with respect to the vertical. We consider an initial pulse polarized along the vertical axis (

*x⃗*),

*L*, the transmitted electric field then reads or, in the initial reference frame where

*δk*(

*ω*) =

*k*(

_{e}*ω*) −

*k*(

_{o}*ω*) and

*φ*(

*ω*) = (

*k*(

_{e}*ω*) +

*k*(

_{o}*ω*))

*L*/2. Assuming that the thickness

*L*is small enough so that

*δk*(

*ω*)

*L*is always much smaller than

*π*/2, we can expand the above expression up to first order Finally, we may expand the wavevector difference as a function of frequency around the center frequency,

*δk*(

*ω*) =

*δk*(

*ω*

_{0}) + (

*ω*−

*ω*

_{0})

*δk′*(

*ω*

_{0}) = (

*ω*−

*ω*

_{1})

*δk′*(

*ω*

_{0}), where where

*δn*(

*ω*) and

*δn*(

_{g}*ω*) are respectively the differences in phase index and group index between the extraordinary and ordinary axes.

*i.e.*the reference field that will interfere with the signal field in order to extract information. Thus, it is the relative pulse shape of the local oscillator versus the signal which is important. In the present case, we can decide that the signal field corresponds to the field polarized along the

*x*axis and the shaped local oscillator field corresponds to the field polarized along the −

*y*axis,

*i.e. ℰ⃗′*=

*ℰx⃗*−

*ℰ*

_{1}

*y⃗*. We then obtain a transfer function equal to

*R*

_{1}(

*ω*) = −

*i*(

*ω*−

*ω*

_{1})

*δk′*(

*ω*

_{0})

*L*/2. For a non dispersive material, the phase index and group index would be identical and the device would produce a pure time shift, as desired. As expected, the value of the time shift is

*τ*= 2

*T*

_{1}=

*δk′*(

*ω*

_{0})

*L*,

*i.e*the difference in group delay between the two polarizations.

*δn*≈ 8.9×10

^{−3}and

*δn*≈ 9.5×10

_{g}^{−3}so that

*ω*

_{1}≈ 0.063

*ω*

_{0}, corresponding to a frequency of 24 THz only. This is reasonably close to zero so that the achieved transfer function can be considered satisfactory.

*L*such that the optical path difference is a multiple of the center wavelength:

*δk*(

*ω*

_{0})

*L*/2 =

*nπ*, where

*n*is an integer number. Expanding

*δk*(

*ω*) around the center frequency at first order,

*i.e.*

*δk*(

*ω*) =

*δk*(

*ω*

_{0}) + (

*ω*−

*ω*

_{0})

*δk′*(

*ω*

_{0}), Eq. (8) yields Again, choosing

*ℰ*along the

*x*axis and

*ℰ*

_{2}along the −

*y*axis yields the transfer function

*R*(

*ω*) = −

*i*(

*ω*−

*ω*

_{0})

*T*

_{2}, with

*T*

_{2}=

*δk′*(

*ω*

_{0})

*L*/2, which corresponds to a time shift

*τ*= 2

*T*

_{2}equal to the group delay difference between the two polarizations. Greater values of the order

*n*are associated with greater thicknesses

*L*and thus a better transmission, at the cost of accuracy since the expansion performed above will become less appropriate. A greater accuracy can also be achieved by use of the half-integer order value

*n*= 1/2. This corresponds to a case where the phase shift is close to

*π*/2, so that the cosine in Eq. (8) can be linearized whereas the sine can be approximated with 1. At first order, using the relation cos(

*π*/2 +

*α*) ≈ −

*α*, we thus obtain Choosing

*ℰ*now along the −

*y*axis and

*ℰ*

_{2}along the

*x*axis allows getting the desired transfer function, with a greater accuracy at the cost of a smaller efficiency.

## 3. Experimental setup

^{−5}. A quartz BSB compensator (Fichou), rotated by 45°, is used to control the effective length

*L*of birefringent material. An analyzer aligned along the horizontal or vertical direction then allows the measurement of the appropriate amplitude spectra, using an integrating sphere (FOIS-1, Ocean Optics) and a spectrometer (Acton SP2300, Princeton Instruments).

18. C. Dorrer and F. Salin, “Characterization of spectral phase modulation by classical and polarization spectral interferometry,” J. Opt. Soc. Am. B **15**, 2331–2337 (1998). [CrossRef]

20. L. Lepetit, G. Chériaux, and M. Joffre, “Linear techniques of phase measurement by femtosecond spectral interferometry for applications in spectroscopy,” J. Opt. Soc. Am. B **12**, 2467–2474 (1995). [CrossRef]

21. C. Dorrer, N. Belabas, J. P. Likforman, and M. Joffre, “Spectral resolution and sampling issues in Fourier-transform spectral interferometry,” J. Opt. Soc. Am. B **17**, 1795–1802 (2000). [CrossRef]

_{4}crystal whose neutral axis is vertical. Taking the difference between the spectral phase obtained with the BSB inserted in the path and the spectral phase obtained when the BSB is not present allows subtracting the difference in spectral phase between the two axes of the YVO

_{4}crystal.

## 4. Time differentiation of the pulse electric field

*ℰ*

_{1}(

*t*) of the electric field

*ℰ*(

*t*). We set the BSB compensator to a differential quartz thickness of

*δL*= 5.4

*μ*m, which corresponds to a time delay

*τ*= 0.17 fs. We then measured the spectra along the

*x*and

*y*polarizations, corresponding respectively to |

*ℰ*(

*ω*)|

^{2}and |

*ℰ*

_{1}(

*ω*)|

^{2}, as shown in Fig. 2(a). As predicted by theory, the overall spectral amplitude of the shaped pulse is significantly smaller, corresponding to about 3.6% as compared to the other polarization. Figure 2(b) shows the square root of the ratios of the two spectra, corresponding to |

*ℰ*

_{1}(

*ω*)|/|

*ℰ*(

*ω*)| = |

*R*

_{1}(

*ω*)|. Here again, as expected, the result is very close to the objective

*ωT*

_{1}.

_{4}crystal as explained in the previous section. The spectral fringes obtained without and with the BSB are shown in Fig. 3(a) on a restricted spectral range. It is clear that the two sets of fringes are in quadrature, as expected for the purely imaginary transfer function

*R*

_{1}(

*ω*) = −

*iωT*

_{1}. Figure 3(b) shows the spectral phase produced after applying the FTSI processing, after subtracting the reference phase obtained in absence of the BSB so that dispersion in the YVO

_{4}crystal cancels out. The result is indeed frequency independent and takes the expected value of −

*π*/2.

*iωT*

_{1}. A complete calculation based on the Sellmeier relations for quartz taking into account the difference between the phase index and the group index predicts more than 99.99% overlap with the objective mode, while we measured 99.83%.

## 5. Time differentiation of the pulse envelope

*δL*in the BSB until we observe a zero transmission in the spectrum measured along the

*x*axis, corresponding to an order

*n*= 1/2. We then adjust the point of zero transmission to the center frequency

*ω*

_{0}, which corresponds in quartz to a differential thickness of

*δL*= 45

*μ*m. Figure 4(a) shows the measured spectra with the analyzer along the

*y*axis (|

*ℰ*(

*ω*)|

^{2}) and along the

*x*axis (|

*ℰ*

_{2}(

*ω*)|

^{2}). Note that the two axes have been exchanged in order to account for the half-integer value of the order

*n*. Figure 4(b) shows the amplitude ratio which is clearly proportional to |

*ω*−

*ω*

_{0}|.

*π*phase shift expected for the change of sign of the function

*ω*−

*ω*

_{0}when crossing the center frequency. However, the desired transfer function

*R*

_{2}(

*ω*) = −

*i*(

*ω*−

*ω*

_{0})

*T*

_{2}would yield a −

*π*phase jump instead of the observed +

*π*phase jump. This discrepancy is of no importance and merely relates to a different sign convention on the horizontal axis, which could occur for example after a reflection on a mirror, or a

*π*/2 rotation of the analyzer. Another issue is the small overshoot in the phase jump, which is however not associated with a great error in the complex transfer function since the amplitude is close to zero in this spectral range. Here the complete calculation predicts 99.99% overlap with the objective mode, while we measure 99.86%.

## 6. Conclusion

*ω*

_{1}either to zero or to the carrier frequency in order to provide an even more exact transfer function.

9. B. Lamine, C. Fabre, and N. Treps, “Quantum improvement of time transfer between remote clocks,” Phys. Rev. Lett. **101**, 123601 (2008). [CrossRef] [PubMed]

## Acknowledgments

## References and links

1. | J. Abadie, “A gravitational wave observatory operating beyond the quantum shot-noise limit,” Nat. Phys. |

2. | V. Giovannetti, S. Lloyd, and L. Maccone, “Quantum metrology,” Phys. Rev. Lett. |

3. | C. M. Caves, “Quantum-mechanical noise in an interferometer,” Phys. Rev. D |

4. | C. W. Helstrom, |

5. | S. L. Braunstein and C. M. Caves, “Statistical distance and the geometry of quantum states,” Phys. Rev. Lett. |

6. | B. M. Escher, R. L. de Matos Filho, and L. Davidovich, “General framework for estimating the ultimate precision limit in noisy quantum-enhanced metrology,” Nat. Phys. |

7. | V. Delaubert, N. Treps, C. Fabre, H.-A. Bachor, and P. Réfrégier, “Quantum limits in image processing,” Europhys. Lett. |

8. | O. Pinel, J. Fade, D. Braun, P. Jian, N. Treps, and C. Fabre, “Ultimate sensitivity of precision measurements with intense Gaussian quantum light: a multimodal approach,” Phys. Rev. A |

9. | B. Lamine, C. Fabre, and N. Treps, “Quantum improvement of time transfer between remote clocks,” Phys. Rev. Lett. |

10. | A.M. Weiner, “Femtosecond pulse shaping using spatial light modulators,” Rev. Sci. Instr. |

11. | A. Monmayrant, S. Weber, and B. Chatel, “A newcomer’s guide to ultrashort pulse shaping and characterization,” J. Phys. B |

12. | A.M. Weiner, “Ultrafast optical pulse shaping: A tutorial review,” Opt. Comm. |

13. | F. Frei, A. Galler, and T. Feurer, “Space-time coupling in femtosecond pulse shaping and its effects on coherent control,” J. Chem. Phys. |

14. | P. Jian, O. Pinel, C. Fabre, B. Lamine, and N. Treps, “Real-time displacement measurement immune from atmospheric parameters using optical frequency combs,” Opt. Express |

15. | M. A. Preciado and M. A. Muriel, “Design of an ultrafast all-optical differentiator based on a fiber Bragg grating in transmission,” Opt. Lett. |

16. | Y. Park, J. Azana, and R. Slavik, “Ultrafast all-optical first- and higher-order differentiators based on interferometers,” Opt. Lett. |

17. | F. X. Li, Y. W. Park, and J. Azana, “Complete temporal pulse characterization based an phase reconstruction using optical ultrafast differentiation (PROUD),” Opt. Lett. |

18. | C. Dorrer and F. Salin, “Characterization of spectral phase modulation by classical and polarization spectral interferometry,” J. Opt. Soc. Am. B |

19. | D. Brida, C. Manzoni, and G. Cerullo, “Phase-locked pulses for two-dimensional spectroscopy by a birefringent delay line,” Opt. Lett. |

20. | L. Lepetit, G. Chériaux, and M. Joffre, “Linear techniques of phase measurement by femtosecond spectral interferometry for applications in spectroscopy,” J. Opt. Soc. Am. B |

21. | C. Dorrer, N. Belabas, J. P. Likforman, and M. Joffre, “Spectral resolution and sampling issues in Fourier-transform spectral interferometry,” J. Opt. Soc. Am. B |

**OCIS Codes**

(320.5540) Ultrafast optics : Pulse shaping

(320.7100) Ultrafast optics : Ultrafast measurements

(320.7085) Ultrafast optics : Ultrafast information processing

**ToC Category:**

Ultrafast Optics

**History**

Original Manuscript: July 17, 2013

Manuscript Accepted: August 18, 2013

Published: September 10, 2013

**Citation**

Guillaume Labroille, Olivier Pinel, Nicolas Treps, and Manuel Joffre, "Pulse shaping with birefringent crystals: a tool for quantum metrology," Opt. Express **21**, 21889-21896 (2013)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-19-21889

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### References

- J. Abadie, “A gravitational wave observatory operating beyond the quantum shot-noise limit,” Nat. Phys.7, 962–965 (2011). [CrossRef]
- V. Giovannetti, S. Lloyd, and L. Maccone, “Quantum metrology,” Phys. Rev. Lett.96, 010401 (2006). [CrossRef] [PubMed]
- C. M. Caves, “Quantum-mechanical noise in an interferometer,” Phys. Rev. D23, 1693–1708 (1981). [CrossRef]
- C. W. Helstrom, Quantum detection and estimation theory (Academic Press, 1976).
- S. L. Braunstein and C. M. Caves, “Statistical distance and the geometry of quantum states,” Phys. Rev. Lett.72, 3439–3443 (1994). [CrossRef] [PubMed]
- B. M. Escher, R. L. de Matos Filho, and L. Davidovich, “General framework for estimating the ultimate precision limit in noisy quantum-enhanced metrology,” Nat. Phys.7, 406–411 (2011). [CrossRef]
- V. Delaubert, N. Treps, C. Fabre, H.-A. Bachor, and P. Réfrégier, “Quantum limits in image processing,” Europhys. Lett.81, 44001 (2008). [CrossRef]
- O. Pinel, J. Fade, D. Braun, P. Jian, N. Treps, and C. Fabre, “Ultimate sensitivity of precision measurements with intense Gaussian quantum light: a multimodal approach,” Phys. Rev. A85,(2012). [CrossRef]
- B. Lamine, C. Fabre, and N. Treps, “Quantum improvement of time transfer between remote clocks,” Phys. Rev. Lett.101, 123601 (2008). [CrossRef] [PubMed]
- A.M. Weiner, “Femtosecond pulse shaping using spatial light modulators,” Rev. Sci. Instr.71, 1929–1960 (2000). [CrossRef]
- A. Monmayrant, S. Weber, and B. Chatel, “A newcomer’s guide to ultrashort pulse shaping and characterization,” J. Phys. B43, 103001 (2010). [CrossRef]
- A.M. Weiner, “Ultrafast optical pulse shaping: A tutorial review,” Opt. Comm.284, 3669–3692 (2011). [CrossRef]
- F. Frei, A. Galler, and T. Feurer, “Space-time coupling in femtosecond pulse shaping and its effects on coherent control,” J. Chem. Phys.130, 034302 (2009). [CrossRef] [PubMed]
- P. Jian, O. Pinel, C. Fabre, B. Lamine, and N. Treps, “Real-time displacement measurement immune from atmospheric parameters using optical frequency combs,” Opt. Express20, 27133–27146 (2012). [CrossRef] [PubMed]
- M. A. Preciado and M. A. Muriel, “Design of an ultrafast all-optical differentiator based on a fiber Bragg grating in transmission,” Opt. Lett.33, 2458–2460 (2008). [CrossRef] [PubMed]
- Y. Park, J. Azana, and R. Slavik, “Ultrafast all-optical first- and higher-order differentiators based on interferometers,” Opt. Lett.32, 710–712 (2007). [CrossRef] [PubMed]
- F. X. Li, Y. W. Park, and J. Azana, “Complete temporal pulse characterization based an phase reconstruction using optical ultrafast differentiation (PROUD),” Opt. Lett.32, 3364–3366 (2007). [CrossRef] [PubMed]
- C. Dorrer and F. Salin, “Characterization of spectral phase modulation by classical and polarization spectral interferometry,” J. Opt. Soc. Am. B15, 2331–2337 (1998). [CrossRef]
- D. Brida, C. Manzoni, and G. Cerullo, “Phase-locked pulses for two-dimensional spectroscopy by a birefringent delay line,” Opt. Lett.37, 3027–3029 (2012). [CrossRef] [PubMed]
- L. Lepetit, G. Chériaux, and M. Joffre, “Linear techniques of phase measurement by femtosecond spectral interferometry for applications in spectroscopy,” J. Opt. Soc. Am. B12, 2467–2474 (1995). [CrossRef]
- C. Dorrer, N. Belabas, J. P. Likforman, and M. Joffre, “Spectral resolution and sampling issues in Fourier-transform spectral interferometry,” J. Opt. Soc. Am. B17, 1795–1802 (2000). [CrossRef]

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