## Analysis of the possibility of analog detectors calibration by exploiting Stimulated Parametric Down Conversion

Optics Express, Vol. 16, Issue 17, pp. 12550-12558 (2008)

http://dx.doi.org/10.1364/OE.16.012550

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### Abstract

Spontaneous parametric down conversion (SPDC) has been largely exploited as a tool for absolute calibration of photon-counting detectors, i.e detectors registering very small photon fluxes. In [J. Opt. Soc. Am. B 23, 2185 (2006)] we derived a method for absolute calibration of analog detectors using SPDC emission at higher photon fluxes, where the beam is seen as a continuum by the detector. Nevertheless intrinsic limitations appear when high-gain regime of SPDC is required to reach even larger photon fluxes. Here we show that stimulated parametric down conversion allow one to avoid this limitation, since stimulated photon fluxes are increased by the presence of the seed beam.

© 2008 Optical Society of America

## 1. Introduction

*G*. On the other hand, the SPDC non-classical correlation at single-photon level which enables the absolute calibration of counting detectors survives, in some form, when a coherent seed is injected and the photon flux becomes macroscopic.

## 2. Analog detection and quantum efficiency

17. K. Soda, I. Nishio, and A. Wada, “Analysis of noises in photocurrent time-correlation spectroscopy of scattered light,” J. Appl. Phys. **47**, 729–735 (1976). [CrossRef]

*q*〉 is the average charge produced in a detection event. We have assumed the response function for the two detectors to be the same,

_{j}*f*

_{1}(

*t*)=

*f*

_{2}(

*t*)=

*f*(

*t*).

*j*=

*k*where 〈

*F̂*(

_{j}*t*)

_{n}*F̂*(

_{j}*t*)〉 is the auto-correlation function of the photon flux at detector

_{m}*j*, and for

*j*≠

*k*where 〈

*F̂*(

_{j}*t*)

_{n}*F̂*(

_{k}*t*)〉 is the cross-correlation between the fluxes incident on the two different detectors. By the substitution

_{m}*F̂*≡〈

_{j}*F̂*〉+

_{j}*δF̂*, it is convenient to express them as

_{j}*η*of detector

_{j}*D*, defined as the number of pulses generated per incident photon. In [4

_{j}4. G. Brida, M. Chekhova, M. Genovese, A. Penin, and I. Ruo-Berchera, “The possibility of absolute calibration of analog detectors by using parametric down-conversion: a systematic study,” J. Opt. Soc. Am. B **23**, 2185–2193 (2006). [CrossRef]

*η*=1) preceded by a beam splitter with the intensity transmission coefficient equal to the quantum efficiency of the real detector [20

20. D. N. Klyshko and A. V. Masalov, “Photon noise: observation, squeezing, interpretation,” Phys. Uspekhi **38**, 1203–1230 (1995). [CrossRef]

*𝓕*(

*τ*)=

*∫dtf*(

*t*)

*f*(

*t*+

*τ*).

## 3. Correlation functions of stimulated PDC

**q**and -

**q**, and with conjugate frequencies,

*ω*

_{1}=

*ω*/2-Ω and

_{pump}*ω*

_{2}=

*ω*/2+Ω, are coupled such that the energy and momentum conservation hold. The equations describing the down conversion process are the input-output transformations

_{pump}*â*

^{out}_{1}and

*â*

^{out}_{2}after the non-linear interaction to the fields

*â*

^{in}_{1}and

*â*

^{in}_{2}before the interaction started. The coefficients

*U*(

_{k}**q**,Ω) and

*V*(

_{k}**q**,Ω) must satisfy the properties

*U*and

_{k}*V*define the strength of the process and at the same time the range of transverse momentum and frequencies in which it takes place and thus are named gain functions. In fact, the finite length of the crystal introduces a partial relaxation of phase matching condition concerning the third component of the momenta. By selecting a certain frequency, the transverse momentum uncertainty, i.e. the angular dispersion of the emission direction, is not null (Δ

_{k}*~(*

_{q}*l*tan

*)*ϑ ¯

^{-1}for the non-collinear PDC, where

*l*is the crystal length and

*is the central angle of propagation with respect to the pump direction). On the contrary, by fixing the transverse momentum*ϑ ¯

**q**, or equivalently a direction of propagation

*ϑ*, the spectral bandwidth ΔΩ is proportional to

*l*

^{-1/2}, and typically 1/ΔΩ~10

^{-13}s for type I.

*f*, placed at distance

*f*from the crystal. The spatial distribution of the far field is, in this case, the Fourier transform of the field just at the output face of the crystal. This special imaging configuration is convenient to show the basic of calculation, nonetheless the validity of the results is more general. Thus, any transverse mode

**q**is associated with a single point

**x**in the detection (focal) plane according to the geometric transformation

**q**→2

*π*

**x**/(

*λf*). The far field operator in the space-temporal domain is therefore

*Î*(

_{k}**x**,

*t*)≡

*B̂*

^{†}

_{k}(

**x**,

*t*)

*B̂*(

_{k}**x**,

*t*) is

_{0}and a certain distribution of the transverse momentum

**q**. Here we assume that, for the modes that are stimulated by the seed, the spontaneous component of the emission is negligible with respect to the stimulated one. This corresponds to the assumption |

*V*(

_{k}**q**,Ω

_{0})|

^{2}≪1 and, at the same time, |

*α*(

**q**)|

^{2}≫1, that is a typical experimental situation in which a few millimeters length crystal is pumped by a continuous pump and the seed has a power just around the microwatt or more. The intensities (Eq. (10)) of the two stimulated beams after the crystal can be evaluated by using Eq. (7), with the substitution

**q**→2

*π*

**x**/(λ

*f*). This leads to

*G*≡

*max*|

*V*(

_{k}**q**,Ω

_{0})|

^{2}. We notice that, aside from the spontaneous emission that is neglected, the generated beams conserve the original momentum distribution of the seed but weighted according to the gain function.

*L*, to a continuous set of modes for

*L*→∞. Thus, in the following, the Ω-sum and the Kronecker function

*δ*

_{xx′}in Eqs. (17) and (18) are substituted respectively with the integral over Ω and the Dirac function

*δ*(

**x**-

**x**′). The fact that the second-order correlation function has an unphysical spatial behaviour, dominated by a delta function, comes from the starting assumption of a plane-wave pump, having infinite transverse dimension. In a realistic case the delta function is replaced by the Fourier transform of the pump transverse profile.

*R*

_{1}and

*R*

_{2}. By applying relations (Eq. (8)) we have

*R*

_{1}×

*R*

_{1}and

*R*

_{2}×

*R*

_{2}respectively, we have the following auto-correlation functions for the photon fluxes

*G*the gain functions |

*Ũ*(

_{j}**x**,Ω)|~1, according to the relations (Eq. (8)). Then, integrating Eq. (18) over

*R*

_{1}×

*R*

_{2}gives the following cross-correlation:

_{0}, the temporal width of both autocorrelation and cross-correlation in a fixed point

**x**still depends on the spectral bandwidth ΔΩ of the gain functions. This explains why they originate from the correlation between the monochromatic seed and the spontaneous emission that is broadband. Therefore, the coherence time of the stimulated emission is still of the order of

*τ*=1/ΔΩ, analogously to the spontaneous case.

_{coh}## 4. Quantum efficiency estimation

*δi*

_{1}(

*t*)=

*i*

_{1}(

*t*)-〈

*i*

_{1}〉 can be calculated by introducing the result in Eq. (21) in Eq. (6), obtaining

*G*≪1) and large intensity of the seed (|

*α*|

^{2}≫1) the fluctuation of the current at detector

*D*

_{1}is dominated by the shot noise component.

*τ*≫

_{p}*τ*. We stress that it is the usual situation, the coherence time of SPDC being on the order of picoseconds or less and the typical resolving time of detectors on the order of nanosecond. In this case any fluctuations in the light power are averaged over

_{coh}*τ*.

_{p}4. G. Brida, M. Chekhova, M. Genovese, A. Penin, and I. Ruo-Berchera, “The possibility of absolute calibration of analog detectors by using parametric down-conversion: a systematic study,” J. Opt. Soc. Am. B **23**, 2185–2193 (2006). [CrossRef]

*q*, i.e. 〈

*q*〉=

_{k}*q*and 〈

*q*

^{2}

*〉=*

_{k}*q*

^{2}. Therefore, according to Eqs. (24) and (25), the quantum efficiency can be evaluated as

*τ*[4

4. G. Brida, M. Chekhova, M. Genovese, A. Penin, and I. Ruo-Berchera, “The possibility of absolute calibration of analog detectors by using parametric down-conversion: a systematic study,” J. Opt. Soc. Am. B **23**, 2185–2193 (2006). [CrossRef]

*τ*. We would like to stress that in this case, the assumption

_{p}*f*

_{1}(

*t*)=

*f*

_{2}(

*t*)=

*f*(

*t*) is not necessary. Since

*∫*

*dτ𝓕*(

*τ*)=1 we obtain

## 5. Conclusion

## Acknowledgments

## References and links

1. | H. A. Bachor, |

2. | M. Genovese, “Research on hidden variable theories: A review on recent progresses,” Phys. Rep. |

3. |
See, for a general review,
N. Gisin, G. Ribordy, W. Tittel, and H. Zbinden, “Quantum cryptography,” Rev. Mod. Phys. |

4. | G. Brida, M. Chekhova, M. Genovese, A. Penin, and I. Ruo-Berchera, “The possibility of absolute calibration of analog detectors by using parametric down-conversion: a systematic study,” J. Opt. Soc. Am. B |

5. | B. Y. Zel’dovich and D. N. Klyshko, “Statistics of field in parametric luminescence,” Sov. Phys. JETP Lett. |

6. | D. C. Burnham and D. L. Weinberg, “Observation of Simultaneity in Parametric Production of Optical Photon Pairs,” Phys. Rev. Lett. |

7. | D. N. Klyshko, “Use of two-photon light for absolute calibration of photoelectric detectors,” Sov. J. Quant. Elect. |

8. | A. A. Malygin, A. N. Penin, and A. V. Sergienko, “Absolute Calibration of the Sensitivity of Photodetectors Using a Two-Photon Field,” Sov. Phys. JETP Lett. |

9. | D. N. Klyshko and A. N. Penin, “The prospects of quantum photometry,” Phys. Uspekhi |

10. | A. Migdall, “Correlated-photon metrology without absolute standards,” Phys. Today January, 41–46 (1999). [CrossRef] |

11. | G. Brida, M. Genovese, and C. Novero, “An application of two photons entangled states to quantum metrology,” J. Mod. Opt. |

12. | V. M. Ginzburg, N. G. Keratishvili, Y. L. Korzhenevich, G. V. Lunev, A. N. Penin, and V. I. Sapritsky, “Absolute meter of photodetector quantum efficiency based on the parametric down-conversion effect,” Opt. Eng. |

13. | G. Brida, M. V. Chekhova, M. Genovese, M. Gramegna, L. A. Krivitsky, and S. P. Kulik, “Conditioned Unitary Transformation on biphotons,” Phys. Rev. A |

14. | G. Brida, M. Chekhova, M. Genovese, M. Gramegna, L. Krivitsky, and M. L. Rastello, “Single-photon detectors calibration by means of conditional polarization rotation,” J. Opt. Soc. Am. B |

15. | G. Brida, M. Chekhova, M. Genovese, M. Gramegna, L. Krivitsky, and M. L. Rastello, “Absolute quantum efficiency measurements by means of conditioned polarization rotation,” IEEE Trans Instrum. Meas. |

16. | G. Brida et al., |

17. | K. Soda, I. Nishio, and A. Wada, “Analysis of noises in photocurrent time-correlation spectroscopy of scattered light,” J. Appl. Phys. |

18. | R. G. McIntyre, “Multiplication Noise in Uniform Avalanche Diodes,” IEEE Trans. Electron. Devices , |

19. | The condition is fulfilled if the electronics is the same for the two detector and has a pass band much smaller than the band of the detectors. |

20. | D. N. Klyshko and A. V. Masalov, “Photon noise: observation, squeezing, interpretation,” Phys. Uspekhi |

**OCIS Codes**

(120.1880) Instrumentation, measurement, and metrology : Detection

(120.3940) Instrumentation, measurement, and metrology : Metrology

(270.4180) Quantum optics : Multiphoton processes

**ToC Category:**

Instrumentation, Measurement, and Metrology

**History**

Original Manuscript: May 30, 2008

Revised Manuscript: July 9, 2008

Manuscript Accepted: July 10, 2008

Published: August 5, 2008

**Citation**

Giorgio Brida, Maria Chekhova, Marco Genovese, and Ivano Ruo-Berchera, "Analysis of the possibility of analog
detectors calibration by exploiting
stimulated parametric down
conversion," Opt. Express **16**, 12550-12558 (2008)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-17-12550

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### References

- H. A. Bachor, A guide to experimental quantum optics (Wiley-VCH, New York, 1998).
- M. Genovese, "Research on hidden variable theories: A review on recent progresses," Phys. Rep. 413, 319-398 (2005) and references therein. [CrossRef]
- See, for a general review, N. Gisin, G. Ribordy, W. Tittel, and H. Zbinden, "Quantum cryptography," Rev. Mod. Phys. 74, 145-195 (2000), and references therein. [CrossRef]
- G. Brida, M. Chekhova, M. Genovese, A. Penin, and I. Ruo-Berchera, "The possibility of absolute calibration of analog detectors by using parametric down-conversion: a systematic study," J. Opt. Soc. Am. B 23, 2185-2193 (2006). [CrossRef]
- B. Y. Zel�??dovich and D. N. Klyshko, "Statistics of field in parametric luminescence," Sov. Phys. JETP Lett. 9, 40-44 (1969).
- D. C. Burnham and D. L. Weinberg, "Observation of Simultaneity in Parametric Production of Optical Photon Pairs," Phys. Rev. Lett. 25, 84-87 (1970). [CrossRef]
- D. N. Klyshko, "Use of two-photon light for absolute calibration of photoelectric detectors," Sov. J. Quant. Elect. 10, 1112-1116 (1980). [CrossRef]
- A. A. Malygin, A. N. Penin, and A. V. Sergienko, "Absolute Calibration of the Sensitivity of Photodetectors Using a Two-Photon Field," Sov. Phys. JETP Lett. 33, 477-480 (1981).
- D. N. Klyshko and A. N. Penin, "The prospects of quantum photometry," Phys. Uspekhi 30, 716-723 (1987). [CrossRef]
- A. Migdall, "Correlated-photon metrology without absolute standards," Phys. Today 52, 41-46 (1999). [CrossRef]
- G. Brida, M. Genovese, and C. Novero, "An application of two photons entangled states to quantum metrology," J. Mod. Opt. 47, 2099-2104 (2000). [CrossRef]
- V. M. Ginzburg, N. G. Keratishvili, Y. L. Korzhenevich, G. V. Lunev, A. N. Penin, and V. I. Sapritsky, "Absolute meter of photodetector quantum efficiency based on the parametric down-conversion effect," Opt. Eng. 32, 2911-2916 (1993). [CrossRef]
- G. Brida, M. V. Chekhova, M. Genovese, M. Gramegna, L. A. Krivitsky, and S. P. Kulik, "Conditioned Unitary Transformation on biphotons," Phys. Rev. A 70, 032332 (2004). [CrossRef]
- G. Brida, M. Chekhova, M. Genovese, M. Gramegna, L. Krivitsky, and M. L. Rastello, "Single-photon detectors calibration by means of conditional polarization rotation," J. Opt. Soc. Am. B 22, 488-492 (2005). [CrossRef]
- G. Brida, M. Chekhova, M. Genovese, M. Gramegna, L. Krivitsky, and M. L. Rastello, "Absolute quantum efficiency measurements by means of conditioned polarization rotation," IEEE Trans Instrum. Meas. 54, 898-900 (2005). [CrossRef]
- G. Brida et al., J. Mod. Opt. to appear.
- K. Soda, I. Nishio, and A. Wada, "Analysis of noises in photocurrent time-correlation spectroscopy of scattered light," J. Appl. Phys. 47, 729-735 (1976). [CrossRef]
- R. G. McIntyre, "Multiplication Noise in Uniform Avalanche Diodes," IEEE Trans. Electron. Devices, ED-13, 164-168 (1966). [CrossRef]
- The condition is fulfilled if the electronics is the same for the two detector and has a pass band much smaller than the band of the detectors.
- D. N. Klyshko and A. V. Masalov, "Photon noise: observation, squeezing, interpretation," Phys. Uspekhi 38, 1203-1230 (1995). [CrossRef]

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