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Optics Express

Optics Express

  • Editor: Michael Duncan
  • Vol. 10, Iss. 19 — Sep. 23, 2002
  • pp: 990–995
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Optical parametric fluorescence spectra in periodically poled media

Vladislav Beskrovnyy and Pascal Baldi  »View Author Affiliations


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

Optical parametric fluorescence (OPF) is a quadratic nonlinear process which is nowadays commonly used as a source of correlated photons over a wide range of wavelengths. This process also allows to characterize the nonlinear properties of various materials. Relevant characteristics are the tuning curve showing the central frequencies, the generated waves spectrum, and the gain of the nonlinear process.

Nowadays, periodically - poled media (PPM) are widely used in nonlinear optics as they allow to compensate for phase mismatch at any wavelength in the transparency range of the material [1

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]

,2

2. 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.

]. This technique allows using the largest nonlinear coefficient of the crystal by choosing appropriate propagation directions and wave polarization [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]

]. PPM also offers to run two nonlinear processes simultaneously [4

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]

].

This article is devoted to the numerical study of the light spectra generated by OPF using coherent continuous wave. We present an original numerical procedure that allows solving this nonlinear problem for the noisy input responsible of OPF. This procedure is shown to be as accurate as but much faster than standard methods, e.g. the Runge-Kutta method.

2 Mathematical description

OPF is a nonlinear optical process in which some photons of the pump radiation at the frequency 2ω 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 2ω. 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]

], we suppose that the generated light does not affect the pump, i.e. the pump amplitude remains unchanged.

To define the spectrum of the generated light, we divide the spectral range around the frequency 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.

In the following, 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 2ω → (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.

The process under consideration is described by the set of differential equations

zAp=g(z)(ωpn(ω)/ωn(ωp))κAp*exp(iΔβpz),
zAp=g(z)(ωpn(ω)/ωn(ωp))κAp*exp(iΔβpz),
(1)

where z denotes the derivative d/dz , n(ω) is the refractive index at ω, κ is the nonlinear coupling coefficient, defined as κ = (iωχ (2) Apump )/(2n(ω)c), Apump 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 ωp (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, ∆βp is the phase mismatch defined by the condition

Δβp=Δβp=k(2ω)k(ωp)k(ωp)=2ωn(2ω)cωpn(ωp)cωpn(ωp)c,
(2)

where 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,

Ap(z=0)=Ap,0Ap(z=0)=Ap,0
(3)

define the evolution of the waves in the crystal along the propagation direction.

3 Solving the set of differential equations

The set of Eqs. (1) can be replaced by a set of two linear differential equations of the first order by the substitution

Ap=Bpωpωn(ω)n(ωp)exp(iΔβpz2),Ap=Bpωpωn(ω)n(ωp)exp(iΔβpz2),
(4)

which gives

zBp*=iΔβpBp*/2+g(z)K*Bp,
zBp=iΔβpBp/2+g(z)KBp*,
zBp*=g(z)K*Bp+iΔβpBp*/2,
zBp=g(z)KBp*iΔβpBp/2,
(5)

where we have introduced K = (ω -p ωp /ω 2)1/2 · (n(ω)2/n(ω -p)n(ωp ))1/2 · κ.

To simplify the notation, we write this set of equations in the matrix form

zC=MC,
(6)

with the initial condition 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

Cc1c2c3c4=Bp*BpBp*Bp,C0=Bp,0*Bp,0Bp,0*Bp,0,
(7)

M is a 4 × 4 matrix which defines the evolution

M=iΔβp/200g(z)K*0iΔβp/2g(z)K00g(z)K*iΔβp/20g(z)K00iΔβp/2.
(8)

Since the function g(z) takes either the values +1 or -1, we can introduce two new matrices M± for the areas with those values:

M±=iΔβp/200±K*0iΔβp/2±K00±K*iΔβp/20±K00iΔβp/2
(9)

4 Propagation of the light in a periodically poled medium

Let the PPM have domains of equal length Λ/2. This is the simplest but practically quite often used case of poling. To specify the medium let us suppose the first domain be positive, i.e. g(z) = 1. The solution for C in such a medium is written as

C(z)=exp(M+z)C0,
(10)

where the matrix exponential for the matrix A is introduced as exp(A) = Σk=0 (Ak /k!).

The wave state at the end of the first domain is 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

C(Λ/2)=exp(MΛ/2)C(Λ/2)=exp(MΛ/2)exp(M+Λ/2)C0.
(11)

Repeating this procedure we receive the light state after passing L domain pairs

C=(exp(MΛ/2)exp(M+Λ/2))LC0=QC0,
(12)

where we introduced a new matrix 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

We have thus shown that knowing the matrix 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.

The square of the module of the amplitude at frequency ωp = ω + (p + 1/2) ∆ω is

Bd*Bd=c4+c4=C*T4,4C,
(13)

Bd*Bd=(C0*Q*)T4,4(QC0)=C0*(Q*T4,4Q)C0=C0*SC0,
(14)

where 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

C0*SC0=p=1,r=14,4sp,rC0,1*C0,1=
(s1,1+s2,2)ωωp·n(ωp)n(ω)A0,p*A0,p+(s3,3+s4,4)ωωp·n(ωp)n(ω)A0,p*A0,p,
(15)

where we took into consideration a uniform phase distribution 〈A 0,±p A 0,±p〉 = 0, independence of the noise amplitudes at different frequencies 〈A 0,-p A 0,±p〉 = 〈A0,p* A 0,±p〉 = 0, and Planck distribution of the noise at the input into the medium 〈A0,±p* A 0,±p〉 = ħω ±pω, where ħ = 1.05 × 10-34J. sec is the Plank’s constant.

Thus, we receive the expression for the square of the module of the amplitude at the output of the media

Bd*Bd=C0*SC0=(s1,1+s2,2)n(ωp)n(ω)ħωΔω+(s3,3+s4,4)n(ωp)n(ω)ħωΔω.
(16)

At this point we know how to calculate the square of the module of the amplitude at the frequency ωp . 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

By running p from 1 to N and compiling the data from Eq. (16) for ω -p and ωp , 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 μm so that the poling period is Λ = 15.144 μm. The PPLN is supposed to consist of 100 domain pairs.

All interacting waves, the pump, the idler and the signal waves, have been chosen to be extraordinary ones in order to use the largest nonlinear coefficient [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]

]. The refractive index for such type of interaction is defined by the formula [5

5. 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.

]

ne2=4.5567+2.605·107τ2+0.970·105+2.70·102τ2λ2(2.01·102+5.4·105τ2)22.24·108λ2,
(17)
Fig. 1. OPF amplification spectrum η = Pout /(ħωp ∆ω) - 1 at the pump frequency of 2.68×1015 rad·s-1 (703.3 nm). The animation (1.46 MB) is the evolution of the OPF amplification spectrum when the pump frequency changes. Each frame corresponds to a a particular pump frequency. [Media 1]

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

The corresponding nonlinear coupling coefficient of Eq. (1) is κ = 10-6 μm-1.

We have tested the accuracy of our procedure by calculating the difference between the amplitudes at 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.

Snapshots from Fig. 1 for different pump frequencies are in good qualitative agreement with previously reported experimental results obtained on PPLN waveguides [6

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

].

Fig. 1 shows that when the pump frequency is far above its value at degeneracy, defined as the value for which the process 2ω = 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 1.3387×1015 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.

Meanwhile other smaller peaks, corresponding to higher orders of phase mismatch, exist. As the main peak disappears, two secondary peaks closest to the main one become dominant in the spectrum. These peaks have analogous scenario of joining each other and disappearing to make the second secondary peak dominant. However, the higher the order of the peak is, the weaker it is, so higher order peaks are unlikely to be observed.

Fig. 2 shows the tuning curve which defines the frequencies of the maximums and minimums of the spectra. Crosses indicate the frequencies at which a local maximum in the spectra of the generated light was found for a particular pump frequency. Circles indicate analogous minimums in the spectra. The upper and the bottom part of the curves correspond to the signal and idler waves respectively. The thick solid line indicates the frequencies of the main maximums. Thin solid lines are for maximums of the higher orders, while thin dashed lines are for minimums.

Fig. 2. Tuning curves for the main peak (thick solid curve) and minimums (dashed curves) and secondary maximums (thin solid curves). Crosses represent maximums found in the spectra for different pump frequencies and circles represent minimums.

7 Conclusion

We have presented a theoretical method which allows calculating the spectra of optical parametric fluorescence (OPF) in periodically - poled media. This model allowed us to study in details the generated light spectrum for different pump frequencies.

In particular, we have shown that, unlike what is generally thought, the generated OPF does not disappear suddenly when the pump frequency goes below the degeneracy value, but it is continuously getting weaker until it disappears.

Furthermore, there are secondary peaks in the generated spectrum, weaker than the main peak. At pump frequencies where the main peak does not exist, the secondary peaks become even dominant, though they are not as powerful as the main one.

Acknowledgments

This research was supported for V.Beskrovnyy by the French Ministry for Researches under its “Accueil de jeunes chercheurs étrangers en séjour de recherche post-doctoral”.

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. 127, 1918–1937 (1962). [CrossRef]

2.

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.

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]

5.

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.

6.

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

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

  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]
  2. 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.
  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, E. Yu. Morozov, �??Consecutive three-wave interactions in nonlinear optics of periodically inhomogeneous media,�?? Quantum Electron. 30, 847-858 (2000). [CrossRef]
  5. 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.
  6. 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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