## Abstract

The evolution of the difference frequency generation between a planar pump wave and a focused signal wave has been numerically investigated in this paper. We show that, at the difference frequency wave, various beam patterns such as ring and moon-like, are resulted due to the nonuniform distribution of phase velocity in the focused signal wave. The subluminal and superluminal regions can be identified by the intersection of two generated beam profiles that correspond to a pair of phase-mismatches with equal value but opposite signs.

© 2007 Optical Society of America

## 1. Introduction

Second-harmonic generation (SHG) and difference frequency generation (DFG) are typical quadratic nonlinear processes that involve three-wave interactions. Conversion efficiency is the primary concern, which requires phase matching (Δk=0) if monochromatic plane waves are assumed. For focused Gaussian beams, however, quadratic nonlinear processes generally require quite different phase matching (PM) conditions [1–4]. For example, maximized SHG occurs at an optimal phase mismatch (Δk_{opt}>0) along the beam axis that compromises the PM conditions among on-axis SHG and off-axis frequency mixing with diverging (or converging) portions of the beam [2]. In this paper we will pay attention to a different aspect of quadratic nonlinear processes with Gaussian beams, where the generated beam patterns resulted from large phase mismatch (Δk≠0) are the major concerns, regardless of the conversion efficiencies. Quadratic nonlinear processes are thought to be effective means for generating various beam patterns. For this purpose, noncollinear PM configurations were commonlly adopted. The ring-shaped SHG [5], conical second-harmonic beam [6], line-shaped pattern [7], hollow difference-frequency (DF) beam [8], etc., have been observed. These investigations are important in the applications such as manipulating and guiding cold atoms [9].

Equivalent to the noncollinear PM configurations, phase-velocity distribution due to beam focusing may also result in various beam patterns in a quadratic nonlinear process. The phase velocity in a focused beam, e.g., Gaussian beam, is nonuniformly distributed, which is pronouced if the beam waist is comparable to the wavelength [10–11]. In such a case, the PM condition varies transversely, which leads to different conversion efficiencies across the beam hence a patterned output [12–18]. The generated beam pattern contains the information of the phase-velocity distribution of the focused beam and may be changed by varying the nominal phase-mismatch ΔkL. Thus it may provide us a means to retrieval the phase-velocity distribution of a focused beam based on the pattern analyses, which would be useful in the applications where phase velocity plays a key role, e.g., the electron acceleration [19–21].

In particular, here we study the evolution of the beam pattern in the difference-frequency (DF) mixing between a planar pump wave and a focused signal wave. As expected, the generated beam patterns are very sensitive to the phase-velocity distribution at a large phase mismatch. Various patterns of the DF beam, e.g., ring-like and moon-like, may be obtained during the nonlinear process along the crystal. The DF beam profiles with different PM conditions are also studied, which shows that the subluminal and superluminal regions can be identified by the intersection of the two DF beam profiles that correspond to a pair of phase mismatches with the same value but opposite signs.

## 2. Preliminaries

#### 2.1. Phase-velocity distribution of a focused Gaussian beam

Previous investigations have shown that the phase-velocity is nonuniformly distributed across nonuniform beams [10–11], which is more pronounced for small-sized or focused beams. For a Gaussian beam, by following Ref [11] and from some straightforward calculations, we write the phase-velocity in the *z* direction:

where *ω* and *k* denote the angular-frequency and the wave-number, respectively. In addition, *φ*(*z*) = tan^{-1} (*z*/*z _{R}*),

*R*(

*z*) =

*z*[1 + (

*z*/

_{R}*z*)

^{2}], and

*z*is the Rayleigh length.

_{R}As shown in Fig. 1, the phase-velocity *v _{pz}* of a Gaussian beam is nonuniformly distributed in both transverse and longitudinal directions. In the longitudinal region around the beam waist (|

*z*|<

*z*), the phase-velocity decreases rapidly with the radial position. Subluminal phase-velocity occurs approximately in the region

_{R}*z*|<

*z*with

_{R}*r*>

*w*

_{0}, where

*w*

_{0}is the radius of the beam waist. In the application of electron acceleration, when relativistic electrons are injected into this subluminal region, they can be trapped in the acceleration phase for a sufficiently long time and receive considerable account of energy from the strong laser field. This is the so called capture and acceleration scenario (CAS) [20]. CAS crucially relies on the existence of a subluminal region. However, the subluminal region is difficult to be identified by using traditional interferometric methods and has not been experimentally demonstrated since the phase-velocity is nonuniformly distributed and differs only slightly from the speed of light

*c*.

#### 2.2. Theoretical model of DFG between a planar pump wave and a focused signal wave

Within the slowly varying amplitude approximation and neglecting the temporal variations, the equations that govern the evolutions of the interacting waves in a quadratic medium are

where *A _{p}*,

*A*and

_{s}*A*denote the complex amplitudes of the pump, signal and DF waves, respectively, and σ

_{g}*, σ*

_{p}*and σ*

_{s}*are the nonlinear coupling coefficients. The wave-vector mismatch is defined by Δ*

_{g}*k*=

*k*- (

_{p}*k*+

_{s}*k*). Typically, previous studies on DFG process were conducted in the situations where both the pump and signal were either planar waves or focused Gausian waves. In order to make the process sensitive to the phase-velocity distribution, we suggest that using different types of the input beams might be a proper choice. As a demostration, in this paper we study the type-II DFG process between a planar pump wave and a focused Gaussian signal wave. In the simulations, the nonlinear length L

_{g}_{NL}=(σ

^{2}

_{g}*A*(0)

_{p}*A*(0))

_{s}^{-1/2}is adopted as an overall quantity to define the laser and crystal parameters. While we adopt the parameters related to β-barium borate (BBO) crystal and a typical femtosecond Ti: sapphire laser (wavelengths

*λ*=

_{s}*λ*=2

_{g}*λ*= 800 nm) in the calculations, the approach is general for other laser and/or crystal parameters by using the nonlinear length L

_{p}_{NL}. In addition, the crystal is assumed to be thin, thus both the effects of spatial walk-off and temporal group-velocity mismatch (GVM) are neglected in Eq. (2) to (4).

In principle, the nonuniformly distributed phase-velocity may lead to different phase-matching conditions across the beam. Besides the nominal phase-mismatch Δ*k* (corresponding to the case if all the interacting waves are planar), an additional phase-mismatch *δk* will contribute. This additional phase-mismatch is also nonuniformly distributed and will vary the conversion efficiency transversely which results in a patterned DF beam. In order to reveal the influence of the phase-velocity distribution on the DF beam pattern, we assume that the gain experienced by the signal is small (i.e., the nonlinear length L_{NL} is assumed to be much large than the crystal length L). In such a case, the signal evolution can be approximated by its linear propagation and the approximation of pump nondepletion can also be adopted.

## 3. Numerical results

#### 3.1. Nonlinear evolution of the DF beam with a Gaussian signal beam

Figure 2 presents the evolution of the DF beam patterns inside the crystal with the parameters given in Tab. 1,

where w_{0s} is the beam waist of the signal in vacuum, z_{Rc} is the Rayleigh length of the signal
in the crystal. In Fig. 2(a), the phase-mismatch is taken as Δ*kL* = 8 . As expected, the
additional phase-mismatch *δk* plays an important role in generating the beam patterns. At the beginning, the DF beam has a bell-like pattern. With its nonlinear propagation in the crystal, a dip appears in the beam center. Then the central dip gradually spread outwards to form a ringlike pattern. And then a small bell-like pattern emerges at the center and the central dip turns into a dark ring and moves to the beam periphery. Finally, the DF beam returns to a bell-like form again but with a lower intensity. In Fig. 2(b) where the phase-mismatch is taken as
Δ*kL* = -8 , the evolution of the DF beam is opposite to that in Fig. 2(a): it has a bell-like pattern at the beginning, then comes a dark ring at the beam periphery, and then the dark ring moves to the beam center and turns into a central dip, at last the central dip disappears and the beam becomes a bell-like form again.

The properties shown in Fig. 2 can be understood in terms of the nonuniform distribution of the phase-velocity on the signal beam. The nonuniformly distributed phase-velocity, as shown in Fig. 1, leads to a phase-velocity difference between the focused beam and a planar wave, and the magnitude of the difference varies with the transverse coordinate *r* . The nonuniform distribution of the phase-velocity difference will cause the phase-mismatch to vary with transverse coordinate in the DFG process. As shown in Fig. 2, in the region | Δ*kL* |< 2*π*, the pattern of the DF beam is similar to that of the signal beam, then the additional phase mismatch resulted by the nonuniform distribution of the phase velocity is:

just as the phase velocity shown in Fig. 1, the additional phase mismatch *δk* varies with the radial position *r* especially in the region |*z*|< *z _{R}*, where it decreases rapidly with

*r*. In the DFG process, the conversion efficiency is influenced by the total phase mismatch

*Dk*, which can be written as

As a result, there should be a set of different beam patterns during the nonlinear evolution of the DF beam, because the conversion efficiency is influenced by the transversly varied *δk* .

Because of the variation of *δk* with the transverse coordinate *r* , the evolution of the DF beam generated from a planar pump wave and a focused Gaussian signal wave can be divided into three steps. Firstly, the transverse distribution of the DF beam near the entrance is a
Gaussian one, just as that of the signal beam. In the region | Δ*kl* |< 2*π* , with the increase of the interaction length, the diffraction of the DF beam is similar to the signal beam, therefore it keeps a bell-like form that is similar to a Gaussian one.

Secondly, in the position around | Δ*kl* |= 2*π*, the distribution of the DF beam is very
sensitive to *δk* since its intensity is proportional to sinc^{2} (*Dkl* /2). It is no other than the sensitivity causes the evolution of the dark ring in Fig. 2. In Fig. 2(a) (Fig. 2(b)), since Δ*k* > 0 (Δ*k* < 0), the total phase mismatch | *Dkl* | at the beam center (periphery) reaches 2*π* first. As a result, a dip (dark ring) emerges at the beam center (periphery). With the increase of *l*, the dark ring moves to the beam periphery (center) because the transverse position corresponding to | *Dk* | *l* = 2*π* moves to the beam periphery (center).

Last, in the region | Δ*kl* |> 2*π*, the newly generated DF beam is not so sensitive to *δk* , and the pattern formed around the region | Δ*kl* |~2*π* is too weak to influence its intensity distribution, therefore the DF beam comes back to a bell-like form.

#### 3.2. Evolution of the DF beams with other types of signal beams

In fact, the previous theoretical investigation shows that, besides the Gaussian beam, other types of focused beam are also with nonuniform distribution of phase velocity [10]. Thus we may expect that there should be structure evolution in the DFG from other types of signal waves pumped by a planar wave.

In Fig. 3(a), we illustrate the evolution of the DFG between a focused elliptical Gaussian signal wave (EGSW) and a planar pump wave. In the numerical simulations, the ellipticity is
taken as *ρ* = 1.2 (the ellipticity is defined as *ρ* = *w*
_{0y}/*w*
_{0x}, where *w*
_{0x} (*w*
_{0y}) is the beam
waist in the *x* (*y*) direction). As expected, the DF beam first keeps an elliptical bell-like
form. In the position around Δ*kl* = 2*π*, an elliptical dip appears in the beam center; then it comes into a dark elliptical ring and moves toward the beam periphery. At last, the beam comes back to an elliptical bell-like form again.

The evolution of the DFG between a TEM_{10} Hermite-Gaussian signal beam (HGSB) and a planar pump wave is shown in Fig. 3(b). At the beginning, the DF beam has a pattern similar
to TEM_{10} Hermite Gaussian beam (HGB). With the increase of *l* , the two side-lobes evolve to the follow patterns: the gibbous moon-like patterns, the half moon-like patterns, and the old moon-like patterns. When the side-lobes evolve to the old moon-like patterns, two small gibbous moon-like patterns appear in the center. Then with the broadening of the interior gibbous moon-like patterns, the exterior old moon-like patterns become narrower and narrower. At last, the exterior old moon-like patterns disappear and the interior gibbous moonlike patterns become a pattern similar to TEM_{10} mode HGB signal beam again. This is caused by the evolution of the dark ring from the beam center to the periphery, which is induced by the phase-velocity of the HGB which is axially symmetric and decreases with *r* .

#### 3.3. DF beams generated from differently located crystals

In Fig. 4, we study the DFG generated in two different situations. In situation I (II), the entrance plane of the crystal is *z* = -*z _{Rc}* (

*z*=

*z*), and the exit plane is

_{Rc}*z*=

*z*(

_{Rc}*z*= -3

*z*) We find that at the exist plane, the two DF beams have the same amplitude and spatial distribution, but their phases are conjugated (i.e., these two beams are phase-conjugated waves), as clearly illustrated in Fig. 4(a). We have compared the DF beams generated for different values of crystal length and phase-mismatch Δ

_{Rc}*k*, the results show that the two DF beams are also phase-conjugated waves under the conditions of symmetric incident planes and equal crystal lengths.

Figure 4(b) explains the results shown in Fig. 4(a): In situation I, when the signal beam propagates over a short distance *h* from the plane *z* = -*z*
_{0} , according to Eq. (4), the locally
generated DF beam at the position *z* = -*z*
_{0}+ *h* is the phase-conjugated wave of the signal beam:

In obtaining Eq. (7), the diffraction is neglected because the propagating distance is very short. The DF beam given by Eq. (7) is equivalent to the signal beam at the plane *z* = *z*
_{0}. When the
DF wave generated at *z* = -*z*
_{0} propagates to the exist plane *z* = - *z*
_{0} + *L* , it is equivalent to
the signal beam at the plane *z* = *z*
_{0} + *L* in situation II. Thus, when it propagates to the exist
plane *z* = -*z*
_{0} + *L* , the DF wave generated at *z* = -*z*
_{0} in situation I is the phase-conjugated
wave of that generates at *z* = *z*
_{0} + *L* in situation II, because in situation II the locally
generated DF wave is also the phase-conjugated wave of the signal beam. In a similar way, when they propagate to the exist plane, all the DF waves generated at other planes in situation I should have their corresponding phase-conjugated waves in the locally generated DF waves in situation II. Consequently, the overall DF beam at the exit plane in situation I, which is the coherent combination of the DF waves generated at different positions in the crystal, is the phase-conjugated wave of that in situation II.

#### 3.4. Division between the subluminal and the superluminal region

Figure 1 and Eq. (5) indicate that, in the region |*z*|< *z _{R}* of a focused Gaussian beam, the
phase-velocity-induced

*δk*decreases with the transverse coordinate

*r*and is equal to zero at a certain radial position. This property results in an interesting characteristic shown in Fig. 5 where the DFG beam profiles for different values of phase-mismatch Δ

*k*are plotted. The beam profiles for + Δ

*k*and - Δ

_{n}*k*(where Δ

_{n}*k*presents a certain phase mismatch value) intersect each other, and the transverse positions of intersections are identical for all pairs of the phase-mismatch +Δ

_{n}*k*and -Δ

_{n}*k*. This property reflects the fact that, at this transverse position, the intensity of the DF wave in the case of perfect PM (Δ

_{n}*k*= 0) is the largest compared to those in the cases of phase-mismatch, and the decrease of the DF intensity due to phase-mismatch Δ

*k*= +Δ

*k*is equal to that in the case of Δ

_{n}*k*= -Δ

*k*.

_{n}According to Fig. 4(b), when the entrance plane is at the beam waist (*z* =0), the locally generated DF wave in every plane inside the crystal is a Gaussian beam, and the beam radii of DF waves vary from those of the signal beam between *z* =0 and *z* =*L* planes. Therefore during the DFG process both the signal and the locally generated DF wave are Gaussian (although the overall DF beam at the exit plane deviates from the standard Gaussian beam). And as a result, as shown in Fig. 5, the transverse position where the intersections locate at is
in a good agreement with the radial position (*r* =1.075*w*
_{0}) where the phase velocity of the
Gaussian beam at *z* = *L*/2 is equal to the standard light velocity *c* . This property will be valuable in identifying the subluminal region of the Gaussian beam, which is important in many applications such as electron acceleration.

It is important to discuss the obtained results in the application to electron acceleration where the laser field is usually an ultrashort one. With femtosecond pulses, the nonlinear coupled-wave equations should generally take the effect of GVM into consideration [22]. In fact, as shown in previous work the effect of GVM would influence the conversion efficiency as well as the optimal focusing ratio in frequency mixing using “thick” nonlinear crystal [22]. In our simulations, however, “thin” crystals with typical length 0.1~0.2 mm are used, which is much shorter than the GVM length for pulses ~100 fs. Thus the effect of GVM is very weak, and the results obtained in this paper may be adequately applied to the case with femtosecond pulses.

## 4. Conclusion

In summary, the DFG between a planar pump wave and a focused signal beam has been investigated. The numerical results have shown that the DF waves present interesting beam patterns related to the nonuniform phase-velocity distribution of the focused signal beam. The resulted beam patterns can conversely reflect the relationship among the phase velocities of the interacting waves and might be useful in detecting the slightly nonuniform distribution of the phase-velocity across focused beams, which is important in some applications such as the CAS.

## Acknowledgments

This work was partially supported by the Natural Science Foundation of China (grant Nos. 10335030, 60538010 and 10576009), and the Science and Technology Commission of Shanghai (grant Nos. 05JC14005 and 05SG02).

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