## Noise figure and photon statistics in coherent anti-Stokes Raman scattering

Optics Express, Vol. 14, Issue 23, pp. 11418-11432 (2006)

http://dx.doi.org/10.1364/OE.14.011418

Acrobat PDF (220 KB)

### Abstract

Coherent anti-Stokes Raman scattering (CARS) is a well-known Raman scattering process that occurs when Stokes, anti-Stokes and pump waves are properly phase-matched. Using a quantum-field approach with Langevin noise sources, we calculate the noise figure for wavelength conversion between the Stokes and anti-Stokes waves in CARS and show its dependence on phase mismatch. Under phase matched conditions, the minimum noise figure is approximately 3 dB, with a correction that depends on the pump frequency, Stokes shift, refractive indices, and nonlinear susceptibilities. We calculate the photon statistics of CARS and show that the photon number distribution is non-Gaussian. Our findings may be significant for currently pursued applications of CARS including wavelength conversion in data transmission and spectroscopic detection of dilute biochemical species.

© 2006 Optical Society of America

## 1. Introduction

1. P.D. Maker and R.W. Terhune, “Study of optical effects due to an induced polarization third order in the electric field strength,” Phys. Rev. **137**, A801 (1965). [CrossRef]

3. A.M. Zheltikov and P. Radi, “Non-linear Raman spectroscopy 75 years after the Nobel Prize for the discovery of Raman scattering and 40 years after the first CARS experiments,” J. of Raman Spectrosc. **36**, 92–94 (2005). [CrossRef]

4. G. Beadie, Z.E. Sariyanni, Y. V. Rostovtsev, T. Opatrny, J. Reintjes, and M.O. Scully, “Towards a FAST CARS anthrax detector: coherence preparation using simultaneous femtosecond laser pulses,” Opt. Commun. **244**, 423–430 (2005). [CrossRef]

10. R. Claps, V. Raghunathan, D. Dimitropoulos, and B. Jalali, “Anti-Stokes Raman conversion in Silicon waveguides,” Opt. Express **11**, 2862–2872 (2003). [CrossRef] [PubMed]

11. J. Perina, “Photon statistics in Raman scattering with frequency mismatch,” Optica Acta **28**, 1529 (1981). [CrossRef]

14. P.L. Voss and P. Kumar, “Raman-noise-induced noise-figure limit for *χ*^{(3)} parametric amplifiers,” Opt. Lett. **29**, 445 (2004). [CrossRef] [PubMed]

16. K.K. Das, G. S. Agarwal, Y.M. Golubev, and M.O. Scully, “Langevin analysis of fundamental noise limits in coherent anti-Stokes Raman spectroscopy,” Phys. Rev. A **71**, 013802 (2005). [CrossRef]

4. G. Beadie, Z.E. Sariyanni, Y. V. Rostovtsev, T. Opatrny, J. Reintjes, and M.O. Scully, “Towards a FAST CARS anthrax detector: coherence preparation using simultaneous femtosecond laser pulses,” Opt. Commun. **244**, 423–430 (2005). [CrossRef]

11. J. Perina, “Photon statistics in Raman scattering with frequency mismatch,” Optica Acta **28**, 1529 (1981). [CrossRef]

16. K.K. Das, G. S. Agarwal, Y.M. Golubev, and M.O. Scully, “Langevin analysis of fundamental noise limits in coherent anti-Stokes Raman spectroscopy,” Phys. Rev. A **71**, 013802 (2005). [CrossRef]

14. P.L. Voss and P. Kumar, “Raman-noise-induced noise-figure limit for *χ*^{(3)} parametric amplifiers,” Opt. Lett. **29**, 445 (2004). [CrossRef] [PubMed]

14. P.L. Voss and P. Kumar, “Raman-noise-induced noise-figure limit for *χ*^{(3)} parametric amplifiers,” Opt. Lett. **29**, 445 (2004). [CrossRef] [PubMed]

## 2. Equations of motion

*â*

_{S}and

*â*

_{AS}respectively), including the effect of this excess noise (using the Langevin noise operators

*N̂*(

*x*),

*N̂*+ (

*x*) discussed below) are described by the following two equations:

*â*

_{S}(

*x*),

*x*)]=[

*â*

_{S}(

*x*′),

*x*′)]. This is achieved in the equations through the presence of the terms involving the “Langevin noise source operators”

*N̂*(

*x*),

*N̂*

^{+}(

*x*), which introduce fluctuations in the Stokes and anti-Stokes fields, as described in more detail below.

*A*,

*B*,

*C*and

*D*coefficients are presented in full in Appendix B.

## 3. Wavelength conversion noise figure

*x*=

*L*of the wavelength converter, when there is anti-Stokes (Stokes) field present at the input

*x*=

*0*. Let us consider, for example, the case where there are no photons at the Stokes frequency and the anti-Stokes wave at the input is a coherent state. The assumption of zero photons in the Stokes input is a good one, since at room temperature the mean photon number of the thermal optical radiation is practically zero. In our notation, |〉

_{S},|〉

_{AS}denotes the states of the Stokes and anti-Stokes inputs, respectively.

*x*in the waveguide, the state of the reservoir is denoted by |〉

_{R,x}(the states on which the operators

*N̂*(

*x*) and

*N̂*

^{+}(

*x*) act). To simplify the notation we will use the combined state

*β*≫

*g*

_{S}(where Stokes and anti-Stokes are uncoupled) when there is no input at

*x*=

*0*. This gives the Stokes spontaneous emission rate, which is proportional to the mean number of reservoir quanta plus one. Since we know that the Stokes spontaneous emission rate is also proportional to the mean phonon number plus one, we can identify the reservoir frequency with the vibrational mode frequency. We will assume next that the noise reservoir is in the ground state (

*kT*≪

*ℏω*

_{VIB}).

*a*. This situation corresponds to the state |0〉

_{S}|

*a*〉

_{AS}|0〉

_{R}and the mean input photon number in the anti-Stokes field is |a|

^{2}.

*N*

_{S,TOT}=

*dx*′|

*N*

_{S}(

*x*)|2. For the fluctuations in photon number we find:

*f*, then |

*a*|

^{2}=

*R*/Δ

*f*, where

*R*is the input photon rate. The optical (signal-tonoise ratio)

*SNR*for an ideal photodetector is:

*F*of the wavelength conversion process is:

*N*

_{AS,TOT}=

*dx*′|

*N*

_{AS}(

*x*)|

^{2}. In the limit of a large input signal, this expression reduces to:

*A*(

*L*)|

^{2}originates from the zero-point fluctuations of the Stokes input, which propagates from x=0 to x=L while mixing with the converted signal. On the other hand, the term proportional to

*N*

_{AS,TOT}quantifies the noise from the material vibrations that are coupled to the Stokes wave and the converted signal.

*β*=0) with

*ψ*

_{P}=0; the relevant coefficients are given in the Appendix B in Eqs. (B9) and (B10). There are two regimes in which we are interested. The first case is when

*g*

_{S}

*x*≫1 with

*g*

_{S}(

*r*

^{2}-1)

*x*≪1, which gives |

*A*(

*x*)|

^{2}≈(1+

*g*

_{S}

*x*)

^{2}, |

*B*(

*x*)|

^{2}≈(

*rg*

_{S}

*x*)

^{2}and

*N*

_{S,TOT}≈2

*g*

_{S}

*x*. In this case, the minimum noise figure becomes:

*g*

_{S}(

*r*

^{2}-1)

*x*≫1 and the conversion efficiency saturates. In this case,

*β*=0 for Stokes to anti-Stokes conversion, we have the following limiting values for the noise figure:

_{S}=(30 cm/GW)×I

_{P}and r=1.2 (typical parameters for semiconductor Raman media). We note that the curves for the two processes are qualitatively similar, converging in the weak pump limit, and asymptotically approaching the limiting value near 3 dB in the strong pump regime. When the intensity is low the noise figure is heavily impaired due to the low conversion gain, but at high intensity the quantum noise approaches a limit similar to the 3 dB minimum in amplifiers.

*r*=1.2,

*g*

_{S}= 0.25

*cm*

^{-1}(which translates to a gain coefficient of 0.5 cm

^{-1}) and L=2 cm (see Fig. 2). Interestingly, the noise figure rapidly becomes large as the phase mismatch is increased, yet also shows periodic undulations corresponding with the well-known relationship between CARS efficiency and phase mismatch.

## 4. Photon number distribution

*N̂*,

*N̂*

^{+}]=1. The quantity |

*N*

_{S}|

^{2}equals the quantity

*N*

_{S,TOT}in Section 4 of the paper and |

*N*

_{AS}|

^{2}equals

*N*

_{AS,TOT}. The quantities

*A,B,C,D*are the same quantities that arppear in equations (2a) and (2b). Formulated in this way, the lumped description of (13a) and (13b) gives the same results as the description of (1a) and (1b).

*q*=〈

*n*〉

_{R}/(1+〈

*n*〉

_{R}) and 〈

*n*〉

_{R}is the thermal occupation number of the reservoir. The probability distribution for both Stokes to anti-Stokes conversion and anti-Stokes to Stokes conversion has the following form:

*n*

_{0}and |

*a*|

^{2}is the mean photon number of the input coherent state. For Stokes to anti-Stokes conversion

*µ*/

*λ*=|

*D*|

^{2}/(

*u*+|

*D*|

^{2}) and

*λ*=(

*u*+|

*D*|

^{2})/(1+

*u*+|

*D*|

^{2}) with

*u*=|

*N*

_{AS}|

^{2}

*q*/(1-

*q*), whereas, for anti-Stokes to Stokes conversion

*µ*/

*λ*=|

*B*|

^{2}/(

*u*′+|

*B*|

^{2}) and

*λ*=(

*u*′+|

*B*|

^{2})/(1+

*u*′+|

*B*|

^{2}) with

*u*′=|

*N*

_{S}|

^{2}/(1-

*q*). The distribution has the mean value: 〈

*n*

_{0}〉=(

*λ*+

*µ*|

*a*|

^{2})/(1-

*λ*). A derivation of this result for the simplified case where the reservoir is in the ground state is presented in Appendix C. Notice that when there is no input signal (

*a*=0) the photon distribution in Eq. (15) becomes a thermal distribution; the spontaneously emitted Stokes and anti-Stokes fields in CARS have thermal distributions with a temperature that is determined by the parameter

*λ*.

## 5. Conclusions

## Appendix A

*A*

_{j}→(

*ω*

_{j}/

*n*

_{j})1/2

*â*

_{j}. This replacement is similar in purpose to substitutions found in standard textbooks (see for example [18]), but in this case the modes are enumerated in frequency rather than wavevector. This substitution yields the following differential equations for the field operators:

*φ*

_{P}is the phase of the pump wave,

*â*

_{S},

*â*

_{AS},

*â*

_{S},

*d*[

*â*

_{j},

*dx*≠0. But this is inconsistent with basic quantum theory. The commutator of the field operators determines the noise limit imposed by the uncertainty principle. More specifically, 〈Δ

*â*

_{j,I},

*â*

_{j,Q}]〉

^{2}, where

*â*

_{j,I}=(1/2)(

*â*

_{j}+

*â*

_{j,Q}=(1/2

*i*)(

*â*

_{j}-

*d*[

*â*

_{j,I},

*â*

_{j,Q}]/

*dx*=0. The analysis presented here is inspired by the analysis of an optical amplifier in [19]. Since all the terms in the propagation equations are due to coupling of the optical fields with the damped vibrational modes in the material, we need to introduce only one ‘noise reservoir’ into the propagation equations. As we will show we can obtain consistent results with this approach.

*F̂*

_{G}:

*F̂*

_{G},

*â*

_{S},

*g*

_{S}. The noise operator and the field operator do not commute. The inhomogeneous solution of the propagation equation gives:

*C*is a proportionality constant. In order to satisfy Eq. (A5), we must have

*C*=-2

*g*

_{S}and the noise sources must obey the following commutation relation:

*F̂*

_{G}(

*x*) acts as a creation operator and

*x*) as an annihilation operator for “noise reservoir” excitations. To make the correspondence more explicit we rename the noise source operator:

*N̂*(

*x*) and

*N̂*

^{+}(

*x*) are now standard annihilation and creation operators with [

*N̂*(

*x*),

*N̂*

^{+}(

*x*′)]=

*δ*(

*x*-

*x*′) and exp(

*jθ*

_{S}) is an undetermined phase factor whose purpose will become clear below. In a similar manner, we can also show that for the decoupled anti-Stokes mode, the noise source

*d*[

*â*

_{S},

*dx*=0 and

*d*[

*â*

_{S},

*â*

_{AS}]/

*dx*=0. It is not difficult to see that the first condition is satisfied using the previous expressions. The purpose of the explicit phase factors included above becomes clear when we examine the second condition. Although the values of these phase factors have no impact on the previous condition (as long as

*θ*

_{S}and

*θ*

_{AS}are real), they must be constrained to satisfy the second condition. Inserting the renamed noise operators into the equations of propagation and requiring

*d*[

*â*

_{S},

*â*

_{AS}]/

*dx*=0, we obtain the condition:

*P*

_{S}(0)=

*P*

_{AS}(0)=1. The functions

*P*

_{S}(

*x*) and

*P*

_{AS}(

*x*) are calculated explicitly in Appendix B (see equations (B9) and (B10)). Inserting (A10) and (A11) in (A9) we obtain:

*α*

_{AS}=

*r*

^{2}

*g*

_{S}and

*κ*=

*rg*

_{S}in the last step, with

*θ*

_{S}=-

*π*/2 and

*θ*

_{AS}=-Δ

*β*·

*x*-2

*ψ*

_{P}-(3

*π*/2) without loss of generality. Our final equations are:

## Appendix B

*θ*=exp(

*j*2

*ψ*

_{P}). It is easy to see that the linear combinations:

*C*(±)=(

*g*

_{S}-

*i*(Δ

*β*/2)-Λ

_{(±)}

*θ*/(

*rg*

_{S}) or

## Appendix C

*e*

^{-jkn}where

*n*is the photon number. The probability distribution is expressed in terms of the characteristic function as:

*â*

_{S}(

*x*))

^{n}(

*x*))

^{n}〉.The expectation value is given by:

*η*=|

*A*|

^{2}and

*ξ*′=|

*Ba*|

^{2}/

*η*into the previous expression, we can rewrite it as:

*∂*/

*∂ξ*′ and

*ξ*′ as operators we note that [1+

*∂*/

*∂ξ*′,

*ξ*′]=1 so that they have the same commutation relationship as

*â*and

*â*

^{+}. We define a new variable

*ξ*″ as

*η*-

*ηe*

^{ξ}=1-

*e*

^{ξ}″, which we use to rewrite the characteristic function:

*ξ*′(1+

*∂*/

*∂ξ*′), which gives:

*ξ*″ and setting

*ξ*=

*jk*we obtain the characteristic function in closed form:

*x*)

*â*

_{S}(

*x*) has a value

*n*

_{o}given a coherent state with amplitude

*a*as the input is given by the Fourier transform of the characteristic function. To compute the transform, we rewrite the integral obtained from Eqs. (C2) and (C6) by changing to the variable

*z*=

*e*

^{jk}:

*Z*

_{o}=(

*η*-1)/

*η*and the integrand has a pole at

*z*=

*z*

_{o}. To evaluate this integral, we expand the exponential and carry out the integration in the complex plane of every term in the expansion. The result is:

## Acknowledgements

## References and links

1. | P.D. Maker and R.W. Terhune, “Study of optical effects due to an induced polarization third order in the electric field strength,” Phys. Rev. |

2. | H. Vogt, “Coherent and hyper-Raman techniques” in |

3. | A.M. Zheltikov and P. Radi, “Non-linear Raman spectroscopy 75 years after the Nobel Prize for the discovery of Raman scattering and 40 years after the first CARS experiments,” J. of Raman Spectrosc. |

4. | G. Beadie, Z.E. Sariyanni, Y. V. Rostovtsev, T. Opatrny, J. Reintjes, and M.O. Scully, “Towards a FAST CARS anthrax detector: coherence preparation using simultaneous femtosecond laser pulses,” Opt. Commun. |

5. | K.P. Knutsen, J.C. Johnson, A.E. Miller, P.B. Petersen, and R.J. Saykally, “High-spectral-resolution multiplex CARS spectroscopy using chirped pulses,” SPIE Proc. |

6. | K. Ishii and H. Hamaguchi, “Picosecond time-resolved multiplex CARS spectroscopy using optical Kerr gating,” Chem. Phys. Lett. |

7. | E.O. Potma, C.L. Evans, and X.S. Xie, “Heterodyne coherent anti-Stokes Raman scattering (CARS) imaging,” Opt. Lett. |

8. | F. Vestin, M. Afzelius, C. Brackmann, and P-E. Bengtsson, “Dual-broadband rotational CARS thermometry in the product gas of hydrocarbon flames,” Proc. of the Combustion Inst. |

9. | N. Djaker, P-F. Lenne, and H. Rigneault, “Vibrational imaging by coherent anti-Stokes Raman scattering (CARS) microscopy,” SPIE Proc. |

10. | R. Claps, V. Raghunathan, D. Dimitropoulos, and B. Jalali, “Anti-Stokes Raman conversion in Silicon waveguides,” Opt. Express |

11. | J. Perina, “Photon statistics in Raman scattering with frequency mismatch,” Optica Acta |

12. | J. Perina, “Photon statistics in Raman scattering of intense coherent light,” Optica Acta |

13. | M. Karska and J. Perina, “Photon statistics in stimulated Raman scattering of squeeze light,” J. Mod. Optics |

14. | P.L. Voss and P. Kumar, “Raman-noise-induced noise-figure limit for |

15. | R. Tang, P.L. Voss, J. Lasri, P. Devgan, and P. Kumar, “Noise-figure of fiber-optical parametric amplifiers and wavelength converters: experimental investigation,” Opt. Lett. |

16. | K.K. Das, G. S. Agarwal, Y.M. Golubev, and M.O. Scully, “Langevin analysis of fundamental noise limits in coherent anti-Stokes Raman spectroscopy,” Phys. Rev. A |

17. | R.W. Boyd, |

18. | A. Yariv, Quantum Electronics, 3 |

19. | H.A. Haus, |

20. | W.H. Louisell, |

**OCIS Codes**

(190.2620) Nonlinear optics : Harmonic generation and mixing

(190.4410) Nonlinear optics : Nonlinear optics, parametric processes

(190.5650) Nonlinear optics : Raman effect

(270.5290) Quantum optics : Photon statistics

**ToC Category:**

Quantum Optics

**History**

Original Manuscript: August 7, 2006

Revised Manuscript: October 13, 2006

Manuscript Accepted: October 13, 2006

Published: November 13, 2006

**Virtual Issues**

Vol. 1, Iss. 12 *Virtual Journal for Biomedical Optics*

**Citation**

D. Dimitropoulos, D. R. Solli, R. Claps, and B. Jalali, "Noise figure and photon statistics in coherent anti-Stokes Raman scattering," Opt. Express **14**, 11418-11432 (2006)

http://www.opticsinfobase.org/vjbo/abstract.cfm?URI=oe-14-23-11418

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### References

- P. D. Maker and R. W. Terhune, "Study of optical effects due to an induced polarization third order in the electric field strength," Phys. Rev. 137, A801 (1965). [CrossRef]
- H. Vogt, "Coherent and hyper-Raman techniques" in Topics in Applied Physics, M. Cardona and G. Guntherodt, eds., (Springer-Verlag, 1982) Vol. 50, p. 207.
- A. M. Zheltikov and P. Radi, "Non-linear Raman spectroscopy 75 years after the Nobel Prize for the discovery of Raman scattering and 40 years after the first CARS experiments," J. Raman Spectrosc. 36, 92-4 (2005). [CrossRef]
- G. Beadie, Z. E. Sariyanni, Y. V. Rostovtsev, T. Opatrny, J. Reintjes, and M. O. Scully, "Towards a FAST CARS anthrax detector: coherence preparation using simultaneous femtosecond laser pulses," Opt. Commun. 244, 423-430 (2005). [CrossRef]
- K. P. Knutsen, J. C. Johnson, A. E. Miller, P. B. Petersen, and R. J. Saykally, "High-spectral-resolution multiplex CARS spectroscopy using chirped pulses," in Multiphoton Microscopy in the Biomedical Sciences IV, A. Periasamy and P. T. So, eds, Proc. SPIE 5323, 230-239 (2004). [CrossRef]
- .K. Ishii and H. Hamaguchi, "Picosecond time-resolved multiplex CARS spectroscopy using optical Kerr gating," Chem. Phys. Lett. 367, 672-7 (2003). [CrossRef]
- .E.O. Potma, C.L. Evans, X.S. Xie, "Heterodyne coherent anti-Stokes Raman scattering (CARS) imaging," Opt. Lett. 31, 241-3 (2006). [CrossRef] [PubMed]
- F. Vestin, M. Afzelius, C. Brackmann, P-E. Bengtsson, "Dual-broadband rotational CARS thermometry in the product gas of hydrocarbon flames," Proc. of the Combustion Inst. 30, 1673-80 (2004). [CrossRef]
- N. Djaker, P-F. Lenne, H. Rigneault, "Vibrational imaging by coherent anti-Stokes Raman scattering (CARS) microscopy," in Femtosecond Laser Applications in Biology, S. Avrillier, and J.-M. Tualle, eds., SPIE Proc. 5463, 133-139 (2004). [CrossRef]
- R. Claps, V. Raghunathan, D. Dimitropoulos, and B. Jalali, "Anti-Stokes Raman conversion in Silicon waveguides," Opt. Express 11, 2862 - 2872 (2003). [CrossRef] [PubMed]
- J. Perina, "Photon statistics in Raman scattering with frequency mismatch," Optica Acta 28, 1529 (1981). [CrossRef]
- J. Perina, "Photon statistics in Raman scattering of intense coherent light," Optica Acta 28, 325 (1981). [CrossRef]
- M. Karska and J. Perina, "Photon statistics in stimulated Raman scattering of squeeze light," J. Mod. Optics 37, 195 (1990). [CrossRef]
- P. L. Voss and P. Kumar, "Raman-noise-induced noise-figure limit for χ(3) parametric amplifiers," Opt. Lett. 29, 445 (2004). [CrossRef] [PubMed]
- R. Tang, P. L. Voss, J. Lasri, P. Devgan, and P. Kumar, "Noise-figure of fiber-optical parametric amplifiers and wavelength converters: experimental investigation," Opt. Lett. 29, 2372 (2004). [CrossRef] [PubMed]
- K. K. Das, G. S. Agarwal, Y. M. Golubev, and M. O. Scully, "Langevin analysis of fundamental noise limits in coherent anti-Stokes Raman spectroscopy," Phys. Rev. A 71, 013802 (2005). [CrossRef]
- R. W. Boyd, Nonlinear Optics, 3rd ed. (Academic Press, 1992).
- A. Yariv, Quantum Electronics, 3rd ed. (Wiley: New York 1989), Chap. 5.
- H. A. Haus, Electromagnetic Noise and Quantum Optical Measurement (Springer 2000), p. 217 - 223.
- W. H. Louisell, Quantum Statistical Properties of Radiation (John Wiley & Sons 1973).

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