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Nov 15, 2010 - Department of Quantum Electronics, Vilnius University, Sauletekio Avenue 9, Building 3, LT-10222 Vilnius, Lithuania. (Received 16 June 2010; ...
PHYSICAL REVIEW A 82, 053817 (2010)

Generation of coherent waves by frequency up-conversion and down-conversion of incoherent light A. Piskarskas, V. Pyragaite,* and A. Stabinis Department of Quantum Electronics, Vilnius University, Sauletekio Avenue 9, Building 3, LT-10222 Vilnius, Lithuania (Received 16 June 2010; published 15 November 2010) It is revealed that the generation of a coherent wave by frequency conversion of incoherent waves is a characteristic feature of three-wave interaction in a nonlinear medium when angular dispersion of input waves is properly chosen. In this case the combining action of the pairs of spectral components of incoherent waves may result in the cumulative driving of a single plane monochromatic wave in up-conversion and down-conversion processes. As a fundamental result we point out an enhancement of the spectral radiance of the generated wave in comparison with incoherent waves. DOI: 10.1103/PhysRevA.82.053817

PACS number(s): 42.25.Kb, 42.65.Ky

I. INTRODUCTION

The excitation of a coherent optical wave by an uncorrelated pump source is of particular practical importance. It was pointed out in Refs. [1,2] that such excitation is possible for parametric interactions when a single signal wave is coupled to a set of pump waves through a corresponding set of idler waves. In this case the parametric gain is independent of the phases of the pump waves involved, and in the field of the uncorrelated pump waves the signal wave can be amplified coherently. The pump incoherence is transferred to the idler waves. That was demonstrated experimentally in an optical parametric oscillator by Byer up-conversion and downconversion [3]. This feature of parametric interaction also takes place in parametric oscillators and amplifiers pumped by several intersecting beams of the same frequency [4–7]. When two- or three-beam pumped optical parametric amplifier (OPA) configuration was discussed for the first time, it was designed to reduce the spatial and spectral bandwidths of a signal wave excited from a quantum noise level [5,6]. The cumulative pump action is also typical for parametric processes excited by Bessel (more generally by conical) beams [8]. The wave propagating along the pump cone axis can be phase-matched with an infinite set of pump plane waves of the same frequency lying on the pump cone through a corresponding set of idler plane waves lying on the idler cone. The major benefit emerging from the use of a multiple-beam pump is that incoherent low-power pump sources can be used to amplify a single signal beam thus potentially increasing the output power of optical parametric chirped pulse amplifiers, see Ref. [9] and references therein. The multiple pump beams of significantly different frequencies also can provide efficient energy combining either in parametric generation [10] or amplification [11] processes. The generation of a coherent wave in the process of down-conversion of an incoherent plane pump wave with a continuous spectrum was first analyzed by Picozzi et al. [12]. It was shown that in the situation where the idler and pump velocities are equal, the signal excitation is independent of pump phase fluctuations, and the generation of a signal wave with slow phase variations, i.e., with a high degree of coherence, becomes possible. This interaction process survives in the regime of strong

*

II. PHASE-MATCHING CONDITIONS

Let us consider the noncollinear phase-matching conditions of a three-wave interaction in which the generation of a plane monochromatic wave is supported by two incoherent waves with continuous spatial-temporal spectrum. In general, the phase-matching conditions for interaction of three plane monochromatic waves can be written as ω1 + ω2 = ω3 , k1 cos θ1 + k2 cos θ2 = k3 cos θ3 ,

(1) (2)

k1 sin θ1 + k2 sin θ2 = k3 sin θ3 ,

(3)

where ωj and kj , j = 1,2,3, stand for frequencies and wave vectors of interacting waves, respectively, and θj is an angle

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1050-2947/2010/82(5)/053817(7)

pump depletion and leads to the formation of a three-wave soliton composed of two incoherent waves and one coherent wave [12]. The group-velocity mismatch (GVM) of the signal wave with respect to the pump and idler waves is the key parameter that governs the evolution of the signal wave as well as its coherence properties [12,13]. A GVM induces the phase-locking mechanism in which the incoherence of the pump is absorbed by the idler wave, allowing the signal wave to grow efficiently with a high degree of coherence. The coherence acquired by the signal wave may increase as the coherence of the pump wave decreases if the group-velocity dispersion (GVD) parameter of the pump matches the GVD parameter of the idler wave [14]. These phenomena may take place also in the case of down-conversion of incoherent monochromatic beams when the transverse spatial dynamic in a three-wave interaction is governed by spatial walk-off and diffraction [15]. In the present paper we show that the generation of a coherent wave by two incoherent waves with a continuous spatial-temporal spectrum is a characteristic feature of threewave interaction in a quadratic nonlinear medium when the angular dispersion of both incoherent waves is properly chosen. This paper is organized as follows. In Sec. II we analyze the phase-matching conditions which are needed for excitation of a coherent wave by two incoherent ones. As an example, in Secs. III and IV the results of analytical analysis and numerical simulation of coherent second-harmonic generation by an incoherent wave are presented, respectively. In Sec. V conclusions are drawn.

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©2010 The American Physical Society

A. PISKARSKAS, V. PYRAGAITE, AND A. STABINIS

PHYSICAL REVIEW A 82, 053817 (2010) 



k3 →

k2 θ2

g

into Taylor series kj = kj 0 + ujj0 + 2j 0 2j , j = 1,2, where uj 0 and gj 0 are group velocity and GVD coefficient at central frequency ωj 0 , respectively, and paraxial approximation sin θj ≈ θj , cos θj ≈ 1 − θj2 /2 yield

z θ3 →

θ1 k1

2 1 1 1 dβj = + gj 0 j − , Vj uj 0 2kj 0 dj

x

FIG. 1. Schematic depiction of noncollinear phase-matching of three plane monochromatic waves in a nonlinear medium.

− → between the wave vector kj and the direction of wave propagation z, Fig. 1. We assume that axis z is the direction of collinear phase-matching of three plane monochromatic waves at frequencies ω10 , ω20 , ω30 and wave vectors k10 , k20 , k30 , respectively, which obey the relations ω10 + ω20 = ω30 and k10 + k20 = k30 . The frequency ωj can be written as ωj = ωj 0 + j , where j is a frequency shift of the j wave with respect to the central frequency ωj 0 . Further we analyze the phase-matching conditions which support the generation of a plane monochromatic wave in the process of sum-frequency generation (SFG) as well as in parametric down-conversion.

Gj = gj 0 −

1 + 2 = 0,

(4)

k1 cos θ1 + k2 cos θ2 = k30 , k1 sin θ1 + k2 sin θ2 = 0.

(5) (6)

Obviously, an excitation of the plane monochromatic sumfrequency wave is most effective if all components of the spatial-temporal spectrum of first and second waves are simultaneously phase-matched with a third wave. That is possible by the use of first and second waves with an appropriate angular dispersion. The relation dω1 /dω2 = −1 and the differentiation of Eq. (5) yield d n (k1 cos θ1 ) d n (k2 cos θ2 ) = (−1)n−1 , dω1n dω2n

n = 1,2,3, . . .

(7)

(11)

where βj ≈ kj 0 θj is a transverse wave vector of the j wave. As a result, at 1 + 2 = 0, Eq. (4), β1 + β2 = 0, Eq. (6), and V1 = V2 , Eq. (8), the angular dispersion curves of first and second waves for g10 + g20 > 0 are hyperbolas     21,2 1 2k10 k20 1 2 ± 1,2 + (g10 + g20 ) , β1,2 = − k30 u10 u20 2 (12) where (+) and (−) stand for indices 1 and 2, respectively. The insertion of Eq. (12) into Eqs. (10) and (11) gives 1 1 = + G1,2 1,2 , V1,2 V0

A. Sum-frequency generation

Let us suppose that a sum-frequency wave is a plane monochromatic wave, ω3 = ω30 and θ3 = 0. Then phasematching conditions (1)–(3) yield

2 1 d 2 βj , 2kj 0 d2j

(10)

where 1 = V0



k10 k20 + u10 u20

 k30 ,

(13)

(14)

and G1,2 = ±(k10 g10 − k20 g20 )/k30 .

(15)

The GVM νSFG between the third and first (or second) wave is   1 1 k30 k10 k20 k30 . − = − − (16) νSFG = u30 V0 u30 u10 u20 The typical dependences β1,2 = f (1,2 ) are shown in Fig. 2. For calculation of the refractive indices and related quantities we adopted the same Sellmeier equations as in [18].

As a result, the dispersion properties of first and second waves should be strictly related in order to create a plane monochromatic sum-frequency wave in a nonlinear medium. The group velocity Vj−1 = d(kj cos θj )/dωj , j = 1,2, of first and second waves propagating in the z direction should be equal, i.e., V1 = V2 . For GVD coefficients Gj =

d(Vj−1 )/dωj

G1 = −G2 .

(8) we obtain (9)

We note that localized (nondiffracting) incoherent waves (V1 = V2 = const, d n (kj cos θj )/dωjn = 0, n = 2,3, . . .) [16,17] can be used for generation of coherent waves in the SFG process. Further we determine the required angular dispersion of first and second waves. An expansion of the wave vectors

FIG. 2. Angular dispersion curves of first (black line) and second (grey line) waves. Beta barium borate (BBO) crystal, type I phasematching. λ30 = 0.53 µm. λ10 (in µm): 0.8 (1), 1.06 (2).

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GENERATION OF COHERENT WAVES BY FREQUENCY UP- . . .

For second-harmonic generation (SHG) at ω10 = ω20 = ω0 = ω30 /2, k10 = k20 = k0 = k30 /2, u10 = u20 = u0 , and g10 = g20 = g0 , Eqs. (12)–(16) take the form 2 = k0 g0 21,2 , β1,2

V1,2 = V0 = u0 ,

(17) 1 1 − . u30 u0 The phase-matching for SHG is possible only in the media with a positive value of the GVD coefficient g0 , and in this case an angular dispersion is linear  (18) β1,2 = ± k0 g0 1,2 , G1,2 = 0,

νSHG =

where β1 + β2 = 0 and 1 + 2 = 0. We note that a plane monochromatic SFG (or SHG) wave is simultaneously phasematched with those spectral components of first and second waves whose disposition on angular dispersion curves in the (β,) plane is antisymmetric with respect to point (0,0), Fig. 2. For example, the pairs of components (β1 ,1 and β2 = −β1 , 2 = −1 ) create a coherent SHG wave at frequency 2ω0 with a wave vector 2k0 which is parallel to axis z (β3 = 0). So, at the proper angular dispersion the combining action of mutually incoherent components of spatial-temporal spectra of summed waves may result in the cumulative driving of the plane monochromatic sum-frequency wave. It should be pointed out that the angular dispersion of incoherent waves which is needed for the generation of a coherent sum-frequency (or second-harmonic) wave is different from the angular dispersion for wideband achromatic up-conversion in frequency doubling and sum-frequency mixing [19–21]. B. Parametric down-conversion

Further we analyze the phase-matching conditions which support the generation of a plane monochromatic idler wave in a down-conversion of the pump wave. We suppose for the idler wave in Eqs. (1)–(3), 2 = 0 and θ2 = 0. In this case Eqs. (1)–(3) take the form 1 = 3 ,

(19)

k1 cos θ1 + k20 = k3 cos θ3 , k1 sin θ1 = k3 sin θ3 ,

(20) (21)

PHYSICAL REVIEW A 82, 053817 (2010)

Taylor series and paraxial approximation yield Eqs. (10) and (11), where now j = 1,3. As a result, at 1 = 3 = , β1 = β3 = β, and V1 = V3 = V , the angular dispersion curves of signal and pump waves for g30 − g10 > 0 are coinciding ellipses:    1 2k10 k30 1  + (g10 − g30 )2 /2 . (23) β2 = − k20 u10 u30 The insertion of Eq. (23) into Eqs. (10) and (11) yields 1 1 = + G, V V0 where 1 = V0



k30 k10 − u30 u10

 k20 ,

(24)

(25)

and G=

k30 g30 − k10 g10 . k20

(26)

The GVM νOPA between the idler and signal (or pump) wave is   1 1 k10 k20 k30 k20 . νOPA = − = + − (27) u20 V0 u10 u20 u30 In the case of a degenerate parametric amplifier we obtain   1 1 , (28) − νOPA = 2 u0 u30 where u0 = u10 = u20 . The angular dependence β = f () for signal and pump waves is shown in Fig. 3. A plane monochromatic idler (i) wave is phase-matched only with those spectral components of signal (s) and pump (p) waves whose frequencies  and transverse wave vectors β are equal, Fig. 3. So, the plane monochromatic idler wave can be excited coherently by mutually incoherent pump and signal waves with controlled angular dispersion. We note that for parametric interaction of plane waves (θ1 = θ2 = θ3 = 0), Eqs. (19)–(21) can be simplified 1 = 3 , k1 + k20 = k3 ,

(29)

where indices 1 and 3 correspond to signal and pump waves, respectively. The obtained phase-matching conditions (19)–(21) stand for coherent as well as incoherent signal and pump waves. The relation dω3 /dω1 = 1 and the differentiation of Eq. (20) yield d n (k1 cos θ1 ) d n (k3 cos θ3 ) = , n = 1,2,3, . . . n dω1 dω3n

(22)

So, the group velocities, GVD coefficients, and higher-order dispersion parameters of signal and pump waves for the most effective excitation of a plane monochromatic idler wave should be equal (V1 = V3 , G1 = G3 , . . .). That is possible only by employment of an appropriate angular dispersion of signal and pump waves. We note that nondiffracting signal and pump waves (V1 = V3 = const) also can be used. Further we determine the required angular dispersion β = f () of signal and pump waves for noncritical phasematching. An expansion of wave vectors k1 and k3 into

FIG. 3. Angular dispersion curve of pump and signal waves. Lithium triborate (LBO) crystal (xy plane). Type I noncritical phase-matching (T = 142◦ C), λ1 = 1.06 µm, λ3 = 0.53 µm.

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PHYSICAL REVIEW A 82, 053817 (2010)

and Eq. (22) takes the form n

n

d k1 d k3 , n = 1,2,3, . . . n = dω1 dω3n

(30)

This means that in a second-order dispersion approximation, group velocities and GVD coefficients should be matched (equal) in order to create the idler wave of high degree of coherence, as it was pointed out in [13].

The spectral amplitudes a10 (,β) of the stationary Gaussian stochastic process A1 (t,x) are governed by Gaussian statistics and δ-correlated. The substitution of T (z) into Eq. (38) and calculation of the mean value T (z1 )T ∗ (z2 ) yield  ∞ ∞ sin2 F σ2 |a3 (,β,z)|2  = |a10 (1 ,β1 )|2  8π 4 −∞ −∞ F 2 × |a10 ( − 1 ,β − β1 )|2  d1 dβ1 , (39) where

III. GENERATION OF COHERENT SECOND HARMONIC BY INCOHERENT FUNDAMENTAL WAVE

As an example, here we present an analytical consideration of the excitation of a coherent second harmonic by an incoherent wave at low conversion efficiency. In a paraxial approximation the variation of the amplitude envelopes Aj with propagation in a nonlinear medium is described by the coupled equations g10 ∂ 2 A1 ∂A1 i ∂ 2 A1 =i − + σ A∗1 A3 , (31a) 2 ∂z 2 ∂t 2k10 ∂x 2 ∂A3 ∂A3 g30 ∂ 2 A3 i ∂ 2 A3 = −ν +i − + σ A21 , (31b) 2 ∂z ∂t 2 ∂t 2k30 ∂x 2 which are written in the reference frame of a fundamental wave; t is time, ν = u130 − u110 is the GVM parameter, and σ is the coupling coefficient; and for simplicity the diffraction of the waves is taken into account in the xz plane. As the initial condition of Eqs. (31) at z = 0 we take a stationary Gaussian stochastic process for the envelope A1 (t,x) of the fundamental wave with zero mean A1 (t,x) = 0. At low conversion efficiency the nonlinear term in Eq. (31a) can be In this case the Fourier transformation aj (,β) = neglected. ∞ ∞ A −∞ −∞ j (t,x) exp[−i(t − βx)] dt dx of Eqs. (31) yields ∂a1 = −i 1 a1 , ∂z

(32)

∂a3 = −i 3 a3 + σ T , (33) ∂z     1 β2 1 β2 , 3 = ν + , g10 2 − g30 2 − 1 = 2 k10 2 k30 (34) and

 T =





−∞

∞ −∞

A21 (t,x) exp[−i(t − βx)] dt dx.

(35)

The solutions of Eqs. (32) and (33) have the form a1 (z) = a10 exp(−i 1 z), where a10 = a1 (0), and



a3 (z) = σ exp(−i 3 z)

z

exp(i 3 z )T (z ) dz .

F =

For normalized spectral radiances S1 = 4π1 2 |a10 |2  and S3 = 1 |a3 |2  we obtain 4π 2   σ 2 z2 ∞ ∞ sin2 F S1 (1 ,β1 ) S3 = 2π 2 −∞ −∞ F 2 (41) × S1 ( − 1 ,β − β1 ) d1 dβ1 . Further we take into consideration the nonlinear length Ln defined as Ln = (σ A10 )−1 where A10 is the averaged amplitude of the fundamental wave at z = 0, A10 = |A1 (t,x)|2 1/2 . Then the spectral radiance S3 can be written as  2  ∞  ∞ sin2 F z S3 = 2 S1 (1 ,β1 ) 2 Ln −∞ −∞ F  × S1 ( − 1 ,β − β1 ) d1 dβ1  ∞  ∞  S1 (,β) d dβ , (42) −∞



z 2 g10 1 + (1 − )2 2 − ν − g30 2 /2 2

z 2 − β1 + (β1 − β)2 (2k10 ) + β 2 /(2k30 ) . (43) + 2 We assume that spectral radiance of the fundamental wave at the input of a nonlinear crystal is Gaussian,   2 (β + p)2 , (44) S1 = S0 exp − − 21 β12 F =

where 1 and β1 characterize the widths of the temporal and spatial spectra, respectively, and p is a parameter of linear angular dispersion. By the use of normalized variables and parameters ξ = 1 /ω10 , δ = /ω10 , η = β1 /k10 , µ = β/k10 , a = 1 /ω10 , b = β1 /k10 , m = pω10 /k10 , and 2 m0 = (ω10 g10 /k10 )1/2 , Eq. (42) takes the form S3 (,β,z) S0   2  z 2 δ2 (µ + mδ)2 = exp − 2 − π ab Ln 2a 2b2    ∞ ∞ 2 2 sin F ξ (η + mξ )2 dξ dη, × exp −2 2 − 2 2 a b2 −∞ −∞ F (45)

(37)

0

0

0

(38)

−∞

where

(36)

The mean value of |a3 (z)|2 is  z z 2 2 |a3 |  = σ exp[i(z1 − z2 ) 3 ]T (z1 )T ∗ (z2 ) dz1 dz2 .

z [ 1 (1 ,β1 ) + 1 ( − 1 ,β − β1 ) − 3 (,β)]. 2 (40)

where F = α[m20 ξ 2 − η2 + h], α = m20 g30 ( 2 g10

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πzn10 , λ10

and h = − νcδ − n10

− 0.5)δ 2 . Here λ10 and n10 are the wavelength of

GENERATION OF COHERENT WAVES BY FREQUENCY UP- . . .

PHYSICAL REVIEW A 82, 053817 (2010)

FIG. 4. Dependence of the spectral radiance of the second harmonic excited by an incoherent wave on (a) the parameter of the linear angular dispersion m and (b) the propagation length in a nonlinear medium. Analytical consideration. BBO crystal, type I phase-matching, λ10 = 1.06 µm, a = 0.3, Ln = 3 mm. (a) z/Ln = 0.5 (1), 1 (2), 2 (3). (b) m = m0 , b = 0.0001 (1), 0.001 (2), 0.0025 (3), 0.005 (4); m = 0, b = 0.0025 (5).

the fundamental wave and corresponding refractive index, respectively. The parameters a and b characterize the temporal and spatial coherence of the fundamental wave, respectively, and parameter m0 is determined by the GVD coefficient of the fundamental wave. If the spatial coherence of the fundamental wave is rather high, parameter b is small, b a 1, and the integral in Eq. (45) can be simplified. For a properly chosen angular dispersion (m = m0 ), an integration in Eq. (45) yields  2 S3 z = exp[−δ 2 /(2a 2 ) − (µ + mδ)2 /(2b2 )] S0 Ln × sin2 (αh)/(αh)2 .

(46)

At a  0.1, h ≈ −νcδ/n10 . Then, after removal of the angular 2 dispersion (m = 0) and by use of approximation sinu2 u ≈ exp(−χ 2 u2 ), χ = 0.6, the temporal spectrum of the second harmonic takes the form    2  χ 2 z2 ν 2 21 z 2 S3 (,0,z) . 1 + = exp − S0 Ln 2 2 21 (47) √ Taking into the consideration the correlation time τc = 2/ 1 of the fundamental wave and the correlation length of the interaction Lc = τc /|ν| for the width of the temporal spectrum of the second harmonic we obtain √ 3 = 2[1 + χ 2 (z/Lc )2 ]−1/2 . (48) 1 So, at z Lc , 3 ≈ 3.3(|ν|z)−1 . It means that under propagation in a nonlinear medium, the width of the temporal spectrum of the second harmonic is decreasing. At these conditions the degree of the coherence of the second harmonic can be considerably increased. By use of Eq. (45) the spectral radiance at central frequency S30 = S3 (0,0,z) in the case of small values b  10−4 can be written as     2 1 z 2 ∞ sin2 F S30 = exp(−2ξ 2 /a 2 ) dξ, (49) 2 S0 π a Ln F −∞

where F = α(m20 − m2 )ξ 2 . The dependence of spectral radiance S30 on the ratio m/m0 is shown in Fig. 4(a). So, under propagation in a nonlinear medium the spectral radiance of the second harmonic S30 can be much higher than the spectral radiance of the fundamental wave S0 . That depends on the length of propagation z as well as on the ratio m/m0 which characterizes the suitability of the employed angular dispersion. If the angular dispersion is absent (m = 0), the spectral radiance of the second harmonic is much smaller. The spectral radiance S30 significantly depends on the width of the spatial spectrum b = β1 /k10 (spatial coherence) of the fundamental wave, Fig. 4(b). The results were obtained by numerical calculation of integral (45). For b > 10−4 the spectral radiance S30 decreases with increase of parameter b. In this case the phase-matching conditions cannot be exactly fulfilled for all components of the spatial-temporal spectrum of interacting waves. We note that the influence of parameter b on spectral radiance S30 becomes negligible at b  10−4 , compare to Fig. 4(a). Further we shall calculate the average intensity of the second harmonic  ∞ ∞ 1 |A3 (t,x)|2  = S3 (,β) d dβ (50) 4π 2 −∞ −∞ and compare it at m = m0 with the average intensity of the fundamental wave. Using Eq. (46), for intensity ratio γ we obtain  2  −1/2 z |A3 (t,x)|2  = 2 γ = 1 + χ 2 z2 /L2c . (51) 2 |A1 (t,x)|  Ln Here for the BBO crystal at a = 0.3 and λ1 = 1.06 µm, Lc = 31.5 µm. At z Lc the dispersion of interacting waves can be neglected, and ratio γ is ≈ 2( Lzn )2 [22]. At z Lc we find γ ≈

2 τc z , χ L2n |ν|

(52)

and the mean intensity of the second harmonic increases with propagation linearly. With increase of ratio z/Ln the

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PHYSICAL REVIEW A 82, 053817 (2010)

depletion of the fundamental wave intensity due to conversion into the second harmonic should be taken into account. We note that the same dependence of the second-harmonic intensity on propagation coordinate z as in Eq. (52) is also typical for the second harmonic generated in a powder of nonlinear crystal when the fundamental wave is coherent [23]. The correlation time τc of the incoherent fundamental wave corresponds to the mean value of the crystallite size which characterizes the spatial correlation in the medium of randomly orientated particles. Such a coincidence originates from the fundamental properties of the three-wave interaction. The second-harmonic generation in the crystal by an incoherent wave is analogous to the generation of a second harmonic by a coherent wave in the medium composed of particles which are randomly distributed. IV. NUMERICAL SIMULATIONS

At larger conversion efficiency of the fundamental wave, the nonlinear term in Eq. (31a) cannot be neglected. In this case the numerical simulations of Eqs. (31) were performed by means of the symmetrized split-step Fourier method [24]. The input fundamental wave with a Gaussian spectral radiance profile (44) was generated by the method described in [25] generalized for the two-dimensional case. In the space-time domain the amplitude of the fundamental wave can be written as follows:

FIG. 5. The dependence of spectral radiance of the second harmonic excited by an incoherent wave on propagation length in a nonlinear medium. Numerical simulation of Eqs. (31) with [curves (3) and (4)] and without [curves (1) and (2)] the nonlinear term in Eq. (31a). m = m0 (1,3); m = 0 (2,4). N = 200 realizations. BBO crystal, type I phase-matching, λ10 = 1.06 µm, a = 0.3, b = 0.0025, Ln = 3 mm. V. CONCLUSIONS

where ωs /ω10 and ks /k10 are normally distributed √ random √ numbers with variances σω = a/ 2 and σk = b/ 2, respectively. ϕs is a random phase and |A1 (t,x)|2  = A210 . Quantity Ns has to be sufficiently large. In our calculations Ns = 50. Equations (31) were simulated N = 200 times in order to fix the averaged values. In Fig. 5 the obtained spectral radiances for angularly dispersed [curves (1) and (3)] as well as for nondispersed [curves (2) and (4)] fundamental waves are depicted. We compared the results when the nonlinear term in Eq. (31a) was neglected [curves (1) and (2)] and included [curves (3) and (4)]. The inclusion of the nonlinear term in Eq. (31a) determines the slower growth of spectral radiance at z/Ln  1.2, compare curves (1) and (2) to (3) and (4), respectively.

The phase-matching conditions of a three-wave interaction in which the generation of a plane monochromatic wave is supported by two incoherent waves with continuous spatialtemporal spectrum were determined. This phenomenon can take place in a nonlinear medium in the process of sumfrequency generation as well as in parametric down-conversion if an angular dispersion of incoherent waves is properly chosen. The physical point of this phenomenon is the combining action of the pairs of spectral components of incoherent waves which result in cumulative driving of the generated wave with a high degree of coherence. The analytical and numerical consideration of excitation of a coherent second-harmonic wave by an incoherent one was provided. It is shown that the spectral radiance of secondharmonic radiation significantly depends on the angular dispersion of the fundamental wave as well as on its temporal and spatial coherence. As a result, a significant enhancement of spectral radiance of the second harmonic in comparison with the fundamental wave is possible. In general, the spectral radiance of a second-harmonic wave decreases with a decrease of spatial coherence of the fundamental wave radiation. In this case the phase-matching conditions cannot be exactly fulfilled for all components of the spatial-temporal spectrum of interacting waves.

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A1 (t,x,z = 0) ∼

Ns 

exp[iωs (t + px) − iks x + iϕs ],

(53)

s=1

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