## Optical parametric fluorescence spectra in periodically poled media

Optics Express, Vol. 10, Issue 19, pp. 990-995 (2002)

http://dx.doi.org/10.1364/OE.10.000990

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### Abstract

A theoretical method and an original numerical procedure to calculate the light spectra generated by optical parametric fluorescence (OPF) in a periodically polled medium is presented. This efficient procedure allows us to precisely study the generation in a periodically poled lithium niobate crystal. As an example, the evolution of the OPF spectra as a function of the pump frequency is presented as an animation. Furthermore, we show that OPF spectra can be generated when the pump frequency goes below the degeneracy.

© 2002 Optical Society of America

## 1 Introduction

1. J.A. Armstrong, N. Blombergen, J. Ducuing, and P.S. Pershan, “Interaction between light waves in a nonlinear dielectric,” Phys. Rev. **127**, 1918–1937 (1962). [CrossRef]

3. P. Baldi, et al, “Modelling and experimental observation of parametric fluorescence in periodically poled lithiulm niobate waveguides,” IEEE J. Quantum Electron. **31**, 997–1008 (1995). [CrossRef]

4. A.S. Chirkin, V.V. Volkov, G.D. Laptev, and E.Yu. Morozov., “Consecutive three-wave interactions in nonlinear optics of periodically inhomogeneous media,” Quantum Electronics **30**, 847–858 (2000). [CrossRef]

## 2 Mathematical description

*ω*are transformed into two photons at

*ω*- Ω and

*ω*+ Ω, creating the signal and the idler waves, in a quadratic (

*χ*

^{(2)}) nonlinear medium. In this article, we suppose the pump radiation to be coherent, i.e. its spectrum is a

*δ*-function with the peak at 2

*ω*. Since the phase matching conditions tolerate a small violation, we admit the spectrum of the generated light not to be a

*δ*-function. Since the power of the generated light is small compared to that of the pump [3

3. P. Baldi, et al, “Modelling and experimental observation of parametric fluorescence in periodically poled lithiulm niobate waveguides,” IEEE J. Quantum Electron. **31**, 997–1008 (1995). [CrossRef]

*ω*into thin frequency slots of equal width ∆

*ω*. We suppose that the spectral density and the frequency remain the same for the whole slot, and we suppose that its frequency is that of the slot center. We also suppose that

*ω*is at the boundary of two slots, and the number of slots above this frequency and below is the same and is equal to

*N*·

*N*and ∆

*ω*are chosen to cover the interested frequency range.

*p*(respectively -

*p*) is the number of a slot above (respectively below)

*ω*, and there are

*N*symmetrical pairs of such slots. Thus the process of OPF is divided into

*N*independent processes 2

*ω*→ (

*ω*- (

*p*- 1/2)∆

*ω*)+(

*ω*+ (

*p*- 1/2)∆

*ω*). In the following mathematical description we consider only one such process for a given

*p*. The obtained results can be easily applied to the rest by choosing the relevant index

*p*.

*∂*

_{z}denotes the derivative

*d*/

*d*

_{z},

*n*(

*ω*) is the refractive index at

*ω*,

*κ*is the nonlinear coupling coefficient, defined as

*κ*= (

*iωχ*

^{(2)}

*A*

_{pump})/(2

*n*(

*ω*)

*c*),

*A*

_{pump}being the pump amplitude,

*c*the velocity of the light in vacuum,

*χ*

^{(2)}the nonlinear coefficient of the crystal, and

*κ*considered the same for all frequencies

*ω*

_{p}(

*ω*

_{-p}) since we study the behavior of the system far from the resonance,

*A*

_{±p}is the amplitude at

*ω*

_{±p}=

*ω*± (

*p*- 1/2)∆

*ω*, ∆

*β*

_{p}is the phase mismatch defined by the condition

*k*(

*ω*′) denotes the wave vector at the frequency

*ω′*. The function

*g*(

*z*) is equal to 1 within the unreversed domains type and is equal to -1 within the reversed domains. Eqs. (1), along with the initial conditions at the beginning of interaction,

## 3 Solving the set of differential equations

*K*= (

*ω*

_{-p}

*ω*

_{p}/

*ω*

^{2})

^{1/2}· (

*n*(

*ω*)

^{2}/

*n*(

*ω*

_{-p})

*n*(

*ω*

_{p}))

^{1/2}·

*κ*.

*C*=

*C*

_{0}, where

*C*is the column matrix 4 × 1 of the evolving amplitudes, and

*C*

_{0}is the matrix

*C*at

*z*= 0

*M*is a 4 × 4 matrix which defines the evolution

*g*(

*z*) takes either the values +1 or -1, we can introduce two new matrices

*M*± for the areas with those values:

## 4 Propagation of the light in a periodically poled medium

*g*(

*z*) = 1. The solution for

*C*in such a medium is written as

*A*is introduced as exp(

*A*) =

*A*

^{k}/

*k*!).

*C*(Λ/2) = exp(

*M*

_{+}Λ/2)

*C*

_{0}- To obtain the wave state at the output of the second domain we have to chose the matrix

*M*_ for

*M*, and suppose the input for the first medium to be the output of the first one. Thus, the wave state at the output of the second domain is

*L*domain pairs

*Q*= (exp(

*M*_Λ/2) exp(

*M*

_{+}Λ/2))

^{L}, which defines the transformation of the light at the input into the light at the output of the medium.

## 5 State of the light at the output for a noisy input

*Q*allows calculating the complex wave amplitudes at the output for those at the input into the medium. This matrix also gives the possibility to calculate statistical values at the output when statistical values at the input are known. Here we show how to find the power of the output for OPF, i.e. light for the quantum noise input.

*ω*

_{p}=

*ω*+ (

*p*+ 1/2) ∆

*ω*is

*S*=

*Q*

^{*}

*T*

_{4,4}

*Q*. Knowing the matrix

*S*allows us to calculate the square of the module of the amplitude of all interacting waves at the output by using Eq. (14). Indeed, after averaging over the initial state and returning to the old variables at the input into the medium (7), Eq. (14) takes the form

*A*

_{0,±p}

*A*

_{0,±p}〉 = 0, independence of the noise amplitudes at different frequencies 〈

*A*

_{0,-p}

*A*

_{0,±p}〉 = 〈

*A*

_{0,±p}〉 = 0, and Planck distribution of the noise at the input into the medium 〈

*A*

_{0,±p}〉 =

*ħω*

_{±p}∆

*ω*, where

*ħ*= 1.05 × 10

^{-34}J. sec is the Plank’s constant.

*ω*

_{p}. We note here that we could have chosen a matrix

*T*

_{3,3}instead of

*T*

_{4,4}in the Eq. (13) to obtain the same result. The value at the frequency

*ω*

_{-p}is obtained by choosing the matrix

*T*

_{1,1}(or

*T*

_{2,2}) instead of the matrix

*T*

_{4,4}in the expression (13).

## 6 Spectra of optical parametric fluorescence

*p*from 1 to

*N*and compiling the data from Eq. (16) for

*ω*

_{-p}and

*ω*

_{p}, we can obtain the spectrum of the generated light at

*ω*. As an illustration, we have studied OPF in a bulk periodically poled lithium niobate (PPLN) crystal. The length of a domain is Λ/2 = 7.572

*μ*m so that the poling period is Λ = 15.144

*μ*m. The PPLN is supposed to consist of 100 domain pairs.

3. P. Baldi, et al, “Modelling and experimental observation of parametric fluorescence in periodically poled lithiulm niobate waveguides,” IEEE J. Quantum Electron. **31**, 997–1008 (1995). [CrossRef]

*τ*is in degrees Kelvin and the wavelength is in nanometers, and is substituted into Eq. (2).

^{15}rad ∙ s

^{-1}(this frequency corresponds to the wavelength of 714nm in the vacuum) to 2.720×10

^{15}rad ∙ s

^{-1}(693nm) with a step equal to 0.016×10

^{15}rad ∙ s

^{-1}and were compiled into an animation presented under the link of the Fig. 1.

*ω*- ∆

*ω*and

*ω*+ ∆

*ω*, which did not exceed 10

^{-12}. Furthermore, the frequency slot width ∆

*ω*does not affect the accuracy, but only the resolution of the plot. The main advantage of that procedure is its rapidity, as the transformation matrices have to be calculated only once for all the noisy inputs. It took us 1.5 minute to obtain the animation of Fig. 1 using a standard personal computer, while the traditional Runge-Kutta method we tried was 100 times longer.

6. S. Tanzilli, et al, “PPLN waveguide for quantum communication,” Eur. Phys. J. D **18**, 155–160 (2002) and reference 23 therein. [CrossRef]

*ω*=

*ω*+

*ω*is quasi-phase-matched, two distinguished peaks corresponding to the idler and the signal waves are generated. When approaching the degeneracy value, these peaks come closer to each other and become wider. When they are rather close, they overlap. There is one extremely wide flat peak when the degeneracy value of the frequency equal to 1.3387×10

^{15}rad·s

^{-1}(1408nm) is reached. We emphasize that for pump frequencies below the degeneracy value, this peak does not disappear immediately. As the pump frequency decreases the peak slowly becomes lower and thinner, until it disappears.

^{15}rad · s

^{-1}(703.3 nm) is chosen. It corresponds to the central vertical grid line in Fig. 2. The fact that this grid line crosses the thick line two times means that there are two main peaks in the spectrum, as one can see in Fig. 1. The circle between the two crosses on the bold curve in Fig. 2 corresponds to the central minimum in Fig. 1. Other circles and crosses which are on the grid line correspond to secondary minimums and maximums of the Fig. 1. For seek of clarity, only the generated frequencies within the range of 1.1-1.6×10

^{15}rad · s

^{-1}are represented in Fig. 2, while the generated spectrum in Fig. 1 covers the range 0.8-1.8× 10

^{15}rad · s

^{-1}.

## 7 Conclusion

## Acknowledgments

## References and links

1. | J.A. Armstrong, N. Blombergen, J. Ducuing, and P.S. Pershan, “Interaction between light waves in a nonlinear dielectric,” Phys. Rev. |

2. | M.M. Fejer, “Nonlinear frequency conversion : materials requirement, engineerd materials, and quasi-phasematching”, in |

3. | P. Baldi, et al, “Modelling and experimental observation of parametric fluorescence in periodically poled lithiulm niobate waveguides,” IEEE J. Quantum Electron. |

4. | A.S. Chirkin, V.V. Volkov, G.D. Laptev, and E.Yu. Morozov., “Consecutive three-wave interactions in nonlinear optics of periodically inhomogeneous media,” Quantum Electronics |

5. | A. Rauber, “Chemistry and physics of lithium niobate” in |

6. | S. Tanzilli, et al, “PPLN waveguide for quantum communication,” Eur. Phys. J. D |

**OCIS Codes**

(000.3860) General : Mathematical methods in physics

(190.2620) Nonlinear optics : Harmonic generation and mixing

(190.4410) Nonlinear optics : Nonlinear optics, parametric processes

**ToC Category:**

Research Papers

**History**

Original Manuscript: July 17, 2002

Revised Manuscript: September 6, 2002

Published: September 23, 2002

**Citation**

Vladislav Beskrovnyy and Pascal Baldi, "Optical parametric fluorescence spectra in periodically poled media," Opt. Express **10**, 990-995 (2002)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-10-19-990

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### References

- J.A. Armstrong, N. Blombergen, J. Ducuing, and P.S. Pershan, �??Interaction between light waves in a nonlinear dielectric,�?? Phys. Rev. 127, 1918 - 1937 (1962). [CrossRef]
- M.M. Fejer, �??Nonlinear frequency conversion : materials requirement, engineerd materials, and quasi-phasematching�??, in Beam shaping and control with nonlinear optics, edited by F. Kajzar and R. Reinisch, Plenum Press, 1997, p.375-406.
- P.Baldi, et al, �??Modelling and experimental observation of parametric fluorescence in periodically poled lithiulm niobate waveguides,�?? IEEE J. Quantum Electron. 31, 997-1008 (1995). [CrossRef]
- A.S. Chirkin, V.V. Volkov, G.D. Laptev, E. Yu. Morozov, �??Consecutive three-wave interactions in nonlinear optics of periodically inhomogeneous media,�?? Quantum Electron. 30, 847-858 (2000). [CrossRef]
- A. Rauber, �??Chemistry and physics of lithium niobate�?? in Current topics in Materials Science, edited by E.L.Kaldis, North - Holland Publishing Company, 1978, p.529.
- S. Tanzilli, et al, �??PPLN waveguide for quantum communication,�?? Eur. Phys. J. D 18, 155-160 (2002) and reference 23 therein. [CrossRef]

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