## Abstract

We report on a novel scheme for generating a broad spectrum in the UV region. This scheme enables us to control the phase of the UV pulse through a frequency-mixing process in a nonlinear crystal. For group velocity matching, it is essential that a monochromatic beam should be sum-frequency mixed with an angularly dispersed beam having a broad spectrum in noncollinear geometry. We found analytically unique solutions for a noncollinear angle, for an angular dispersion of the broadband input beam, and for an angle of the beam from the optical axis in a nonlinear crystal, with the condition that there is no angular dispersion in the output beam. Based on the analysis of this scheme, we obtained UV pulses with a sufficiently broad spectrum for obtaining a sub-20-fs pulsewidths in the experiment. The improvement of conversion efficiency and compensation of chirp are also discussed.

©2003 Optical Society of America

## 1. Introduction

Optical phase control techniques have been widely used for the generation of shaped femtosecond pulses in order to investigate ultrafast coherent interactions in physical and chemical phenomena. Resonant multiphoton transitions can be enhanced with an appropriately shaped femtosecond pulse [1] and the interference between resonant and off-resonant contributions in a coherent transition can be controlled with phase jumps in a chirped femtosecond pulse [2].

Spectra of the phase-controlled femtosecond pulses have been, however, limited to the near infrared (NIR) or visible (VIS) region because instruments for the phase control, such as a liquid-crystal spatial light modulator (LCSLM) [3] or an acousto-optic programmable dispersive filter (AOPDF) [4], are not transparent to other wavelength regions. Thus, arbitrary control of the phase of UV femtosecond pulses has not yet been realized although modulations of amplitude have been demonstrated with frequency doubling of phase-modulated pulses [5] and also with sum-frequency mixing of amplitude-modulated pulses [6].

One possible method for the UV phase control is to follow the same way as NIR region, namely, direct phase control with a transparent material in the UV region or with reflective optics instead of LCSLM. A spatial light modulator made of an array of fused-silica plates [7] or a deformable mirror [8] could manage the UV phase if we could generate an UV pulse with a sufficiently broad spectrum. Another method is indirect phase control though NIR pulses. If the phase of NIR femtosecond pulses can be transferred to UV pulses with frequency mixing in a nonlinear optical medium, we can control the UV phase using the techniques in the NIR range.

Tan et al. demonstrated indirect phase control with a noncollinear optical parametric amplification (NOPA) process in an infrared (IR) region [9]. The principle of the phase transfer as they demonstrated with the NOPA process can be applied to sum-frequency generation because both of the nonlinear optical processes originate on the basis of the same characteristic of a nonlinear optical crystal. As was discussed in Ref. 9, an electric field *E*
_{3}(*t*) generated from the sum-frequency mixing of electric fields *E*
_{1}(*t*) and *E*
_{2}(*t*) is approximately expressed as a product of *E*
_{1}(*t*) and *E*
_{2}(*t*), so that the Fourier component of the generated pulse, *Ẽ*
_{3}(ω), is proportional to the convolution of those of the input pulses, *Ẽ*
_{1}(ω) and *Ẽ*
_{2}(ω):

If one of the input electric fields is quasi monocromatic around the angular frequency ω_{10}, *Ẽ*
_{1}(ω) can be approximated to the delta function,

where *A*
_{1} is a constant with a dimension of the electric field. Hence, the resultant expression of relation (1) is approximately given by,

which shows that *Ẽ*
_{3}(ω) is a replica of the Fourier component of the input electric field *E*˜_{2}(ω) with the frequency shift of ω_{10}. Therefore, we can control the phase of the generated field *E*
_{3}(*t*) through that of the input field *E*
_{2}(*t*). It should be noted that the phase of the input pulse cannot be transferred in the second harmonic generation (SHG) process because the input electricfield is to be convolved with itself [5].

The main problem associated with this scheme is that the pulsewidth of the quasi monochromatic electric field, *E*
_{1}(*t*), should be much longer than that of the femtosecond pulse because of the uncertainty relation between the widths of time and frequency. The temporal mismatch of the input fields can be solved with the stretch of the femtosecond pulse, *E*
_{2}(*t*), by adding a constant chirp, which can be eliminated with a compresser for *E*
_{3}(*t*) after sum-frequency mixing. This scheme is similar to optical parametric chirped pulse amplification (OPCPA) [10] although the wavelength of the chirped pulse is upconverted.

The group delay walk off in a nonlinear crystal which limits the bandwidth of UV pulses, however, remains as a problem for the generation of pulses shorter than 100 fs in the UV region, while the pioneering work done by Szabó and Bor [11] enabled us to obtain 150-fs pulses of VUV [6]. They demonstrated the advantage of noncollinear angularly dispersed geometry in sum-frequency mixing for the first time.

In this paper, we show a novel scheme for broadband wave-vector matching between a monochromatic visible (VIS) pulse and a NIR (Ti:sapphire laser) pulse of sufficient bandwidth for generating a sub-20-fs UV pulse in a nonlinear crystal with conventional thickness (~1 mm).

The basic formulae for obtaining the wave-vector matching condition in the first order approximation of dispersion is presented in Section 2., and the acceptable bandwidths and other parameters of a nonlinear crystal are calculated in Section 3.. The results of preliminary experiment are shown in Section 4., and finally we discuss issues associated with the scheme for chirped sum-frequency mixing.

## 2. Wave-vector matching in noncollinear geometry

The concept of wave-vector matching or achromatic phase matching in a nonlinear crystal originated in the automatically phase-matched second harmonic generation (SHG) of a dye laser with a few nanoseconds pulsewidth [12]. Following the tuning of the wavelength of the dye laser, the incident angle of the fundamental beam injected to a nonlinear crystal was changed by means of prisms so that the change of the phasematching angle can be compensated for.

Szabó and Bor first applied this principle to broadband frequency doubling in the femtosecond regime [13]. They demonstrated the second harmonic generation from a visible femtosecond laser pulse with a sufficiently broad spectrum in the UV region for obtaining 250-fs pulsewidth by using gratings and a telescope to obtain an appropriate angular dispersion and intensity on a nonlinear crystal. Other experiments on the broadband frequency doubler were reported by Cheville and co-workers [14] and further broadening of the bandwidth was demonstrated by Richman and co-workers [15].

Following this scheme, Nakajima and Miyazaki analyzed the possible arrangements of a broadband frequency tripler which is a sum-frequency mixer with the fundamental and the second harmonic. They concluded that the complete compensation of the first order phase mismatch, which is available to the SHG, was not possible [16]. Osvay and Ross proposed another approach named “chirp-assisted group-velocity matching technique” in sum-frequency generation [17]. This technique, however, may not be suitable for our purpose because one of the input beams must be monochromatic for the indirect phase control while their approach needs broad spectra in both beams for generating off-center sum frequencies.

The schema for broadband phase matching described above essentially use a collinear geometry of input beams, which sets the limit for the broadband frequency tripler [16]. It was Szabó and co-workers who pointed out the importance of a noncollinear geometry [11, 18]. They showed that phase (wave vectors) matching could be achieved only for the wavelengths that are equal to or longer than the central wavelength of the generated beam, as shown in Fig. 6 of Ref. 11, if the central components of the input beams are collinear. This limitation could be eliminated by employing a noncollinear geometry of the central-wavelength component of the input beams, which was applied to the generation of amplitude modulated femtosecond pulses at 200 nm [6]. The beam generated by this method, however, was angularly dispersed, so that other optics such as a prism or a cylindrical lens were needed in order to collimate the beam.

We found that the wave-vector matching condition for sum-frequency mixing of monochromatic beam with spatially dispersed beams by use of a noncollinear geometry, which enable us to obtain a broad spectrum in the UV region without spatial dispersion. The spatial dispersion of the input beam and the angle between the beams at the center wavelength can be uniquely determined on the basis of this condition in the first order approximation of dispersions as described below.

The basic concept of this scheme is similar to that in Ref. 18 and that in NOPA [8] process, as shown in Fig. 1. In NOPA process, wave-vector components in a wide range of optical frequency of a signal beam, which are aligned to a particular direction, can be phase-matched with a wave-vector component of a pump pulse, while an idler beam should be spatially dispersed because the directions of wave-vectors of the idler beam should be different from each other for the compensation of dispersion.

As the NOPA is a process of difference-frequency mixing (DFM), similar argument for wave-vector matching with non-collinear geometry can be made for sum-frequency mixing (SFM) because these two processes both originate in the second order nonlinearity in a crystal. The only difference of non-collinear SFM from NOPA is that the generated beam does not need to be angularly dispersed while the input beam must be, as shown in Fig. 1(b).

To identify the specific features of this wave-vector matching scheme, we return to a basic equation of the wave-vector mismatch function Δκ :

where we define κ
_{a}
, κ
_{b}
, κ
_{c}
as the wave vectors of an input monochromatic beam, an input broadband beam and a generated broadband beam with the dependences on optical angular frequencies of ω
_{a}
, ω
_{b}
and ω
_{c}
respectively, which should satisfy the photon energy conservation law;

Although the optical angular frequencies are arbitrary variables in any spectrum range, ω
_{a}
is supposed to be visible, ω
_{b}
to be near infrared and ω
_{c}
to be in the UV region. With small variations of these termes ω
_{j}
→ω
_{j}
+Δω
_{j}
(
_{j}
=*a*, *b*, *c*), Δκ can be expanded in the first order of deviation;

$$\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}+\left\{\frac{d{\mathit{k}}_{c}}{d{\omega}_{c}}{|}_{0}-\frac{d{\mathit{k}}_{b}}{d{\omega}_{b}}{|}_{0}\right\}\Delta \omega +\mathit{O}\left(\Delta {\omega}^{2}\right).$$

In Eq. (6), we replace three variables of angular frequencies to three constants of the center angular frequencies, ω
_{a0}
, ω
_{b0}
, and ω
_{c0}
, after differentiation of the wave vectors. The symbol “|_{0}” denotes this substitution to the constant. Because we assume that beam A is monochromatic, Δω
_{a}
is equal to 0, and therefore we changed the notation Δω
_{c}
=Δω
_{b}
to Δω regarding the photon energy conservation of Eq. (5).

To achieve the wave-vector matching condition (WVMC) in any order of Δω, we first apply Δω to zero and also the wave-vector mismatch (Δκ=0) in Eq. (6).

which is an usual wave-vector (phase) matching condition at the center angular frequencies. Substituting Eq. (7) and Δκ=0 into Eq. (6) and dividing this equation by Δω, we obtain the equation;

in the limit of Δω→0. This vector representation of equation means that the inverse of the group velocities of beam B and C must be the same in each component of the wave vectors. Equation (8) is a generalized expression of the group velocity matching condition (GVMC) which was shown in Ref. 19 and Ref. 20 for the analysis of NOPA. The equation in Ref. 19 and Eq. (15) in Ref. 20 are special cases of this expression in which no angular dispersion was imposed on one of the broadband beams. We will discuss this condition in further details later.

Although we can obtain the second and higher order conditions for Δω with the vector representation of dispersions by following the procedure described above, we do not calculate them because it is very difficult to control such highly nonlinear angular dispersion in the experiment.

We note that unique solutions may be obtained from these two vector equations of WVMC and GVMC at the center angular frequencies if there exist four unknown variables, because these equations do not include six but four independent equations to be solved. This is due to a restraint that the wave vector κ
_{c}
should be on the plane sustained with κ
_{a}
and κ
_{b}
. Thus, we set two orthogonal unit bases of the vector; one is parallel to the direction of κ_{a0} defined as *e*
_{a0} and the other, *e*
_{b0}, is perpendicular to κ_{a0} and κ_{a0}× κ_{b0} as shown in Fig. 2. The subscript “0” denotes values at the center angular frequencies.

By using these unit vectors, we can express κ_{j}(*j* = *a, b, c*) as linear summations of projections to the unit vectors;

where we define the absolute values of wave vectors as κ_{a}(ω_{a})≡|κ_{a}(ω_{a})|, κ_{b}(ω_{b})≡κ_{b}(ω_{b})|, and κ_{c}(ω_{c})≡κ_{c}(ω_{c})|. Noncollinear angles of α_{ab} (ω_{b}) and α_{ac} (ω_{c}), which are schematically shown in Fig. 2, are defined according to inner products: cosα_{ab}≡κ_{b}·*e*
_{a0}/κ_{b} and cos α_{ac}≡κ_{c}·*e*
_{a0}/κ_{c}. Dependences of ω_{b} and ω_{c} on the noncollinear angles of α_{ab} and α_{ac} are essential for satisfying the GVMC.

Substituting Eq.’s (9)~(11) into the WVMC of Eq. (7), and using orthogonality between the bases, we obtained two scaler equations;

where subscript “0” denotes the values at the center angular frequencies.

These equations can also be obtained straightforwardly from a geometrical relation at the center angular frequencies as shown in Fig. 2 [20] and it seems that we have no need for the strict calculations described above. The GVMC, however, is not evident from the geometrical relation because it relates to angular dispersions in both beam B and beam C.

The most important point is that α_{ac0} defined as a value of α_{ac} at the center angular frequency ω_{c0} depends on α_{ab0} defined as a value of α_{ab} at the center angular frequency ω_{b0}. In other words, α_{ac0} becomes a function of α_{ab0} by solving the Eq.’s (12) and (13) in terms of α_{ac0}. The angle of κ_{a0} from an optical axis in a nonlinear crystal defined as θ_{a0} in Fig. 2 is also determined by these equations depending on α_{ab0} because some of the wave numbers (κ_{a0}, κ_{b0}, κ_{c0}) should be functions of θ_{a0}.

On the other hand, the angular dispersions of *d*α_{ab}/*d*ω_{b|0} and *d*α_{ac}/*d*ω_{c|0} are related to the inverses of the group velocities of *d*κ_{b}/*d*ω_{b|0} and *d*κ_{c}/*d*ω_{c|0} through the GVMC as follows;

$$\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}-\left\{\frac{d{k}_{c}}{d{\omega}_{c}}{|}_{0}\mathrm{cos}\left({\alpha}_{{\mathit{ac}}_{0}}\right)-{k}_{{c}_{0}}\mathrm{sin}\left({\alpha}_{{\mathit{ac}}_{0}}\right)\frac{d{\alpha}_{\mathit{ac}}}{d{\omega}_{c}}{|}_{0}\right\}=0,$$

$$\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}-\left\{\frac{d{k}_{c}}{d{\omega}_{c}}{|}_{0}\mathrm{sin}\left({\alpha}_{{\mathit{ac}}_{0}}\right)+{k}_{{c}_{0}}\mathrm{cos}\left({\alpha}_{{\mathit{ac}}_{0}}\right)\frac{d{\alpha}_{\mathit{ac}}}{d{\omega}_{c}}{|}_{0}\right\}=0,$$

which are obtained from Eq.’s (8), (9), (10), and (11). Again, the subscript “0” denotes values at the center angular frequencies.

Remembering that α_{ac0} and θ_{a0} are functions of α_{ab0} by using WVMC and that the inverses of the group velocities are also the functions of α_{ab0}, the unknown variables remaining in Eq.’s (14) and (15) are the noncollinear angle, α_{ac0}, the angular dispersion of beam B, *d*α_{ab}/*d*ω_{b|0}, and the angular dispersion of beam C, *d*α_{ac}/*d*ω_{c|0}. Thus, we may treat one of the variables as a free parameter for various purposes. If any of the angular dispersions of the input (beam B) and generated beams (beam C) can be permitted in our experiment, we can find an appropriate GVMC for a given noncollinear angle, while the noncollinear angle and the angular dispersion of the input beam (beam B) are uniquely determined if we specify *d*α_{ac}/*d*ω_{c|0}. This analysis leads to the conclusion that Eq. (8) or Eq.’s (14)~(15) are generalizations of the group velocity matching condition shown in Ref. 20.

For simplicity, we restrict the condition that there is no angular dispersion of beam C in the first order approximation;

which is similar to a non-angular dispersive signal beam in the NOPA process.

We can easily find an equation to determine the noncollinear angle from Eq.’s (14), (15), and (16) by removing the angular dispersion of the input beam;

where the group velocities of the input and the output beam at the center angular frequencies, ${\upsilon}_{g0}^{b}$ and ${\upsilon}_{g0}^{c}$, are given by

Equation (17) indicates that the projection of the group velocity of the input beam onto κ_{c0} should be the same as that of the output beam and this equation is comparable to the group velocity matching condition in NOPA [19].

It is not clear whether we can find a solution for Eq. (17) in terms of the noncollinear angle α_{ab0}, because (1) α_{ac0} is a function of α_{ab0}, and (2) ${\upsilon}_{g0}^{b}$ and ${\upsilon}_{g0}^{c}$ may depend on α_{ab0} according to polarizations of the input and output beams. When we can find α_{ab0}=${\mathrm{\alpha}}_{{\mathit{\text{ab}}}_{0}}^{\text{gvm}}$ such that Eq. (17) is satisfied, the angular dispersion of the input beam needed for GVMC, *d*α_{ab}/*d*${{\mathrm{\omega}}_{b}|}_{0}^{\text{gvm}}$, is expressed as

by eliminating *d*κ_{c}/*d*ω_{c|0} from Eq.’s (14) and (15) with Eq. (16), and substituting ${\mathrm{\alpha}}_{{\mathit{\text{ab}}}_{0}}^{\text{gvm}}$ into α_{ab0}. This equation can be interpreted as the condition that the end of the wave-vector of the input beam B should be aligned, such that wave-vector components of the generated beam should be parallel.

Angles from an optical axis in the nonlinear crystal for each wave vector at the center angular frequencies (θ_{a0}, θ_{b0}, θ_{c0}) can also be determined from Eq. (12), Eq. (13) and Eq. (17). We do not have sufficient space to show every formula for giving these angles in various configurations of polarizations, thus, we show only the angle of the output beam at the center angular frequency, θ_{c0} in Type I WVMC of a uniaxial nonlinear crystal;

where *n*
_{o}(ω) and *n*
_{e}(ω) are refractive indices for ordinary ray and extraordinary ray at the frequency ω, respectively. The “averaged” refractive index, *n̄*
_{ab0} (α_{ab0}), is defined as

$$\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\equiv {\left\{{\left[{\omega}_{\mathit{bc}0}{n}_{o}\left({\omega}_{{b}_{0}}\right)\mathrm{sin}{\alpha}_{{\mathit{ab}}_{0}}\right]}^{2}+{\left[{\omega}_{{\mathit{ac}}_{0}}{n}_{o}\left({\omega}_{{a}_{0}}\right)+{\omega}_{\mathit{bc}0}{n}_{o}\left({\omega}_{{b}_{0}}\right)\mathrm{cos}{\alpha}_{{\mathit{ab}}_{0}}\right]}^{2}\right\}}^{\frac{1}{2}},$$

where the weight parameters, *w*
_{ac0} and *w*
_{bc0} are ω_{a0}/ω_{c0} and ω_{b0}/ω_{c0}, respectively.

Equation (21) is valid for an arbitrary noncollinear angle of α_{ab0}. By substituting ${\mathrm{\alpha}}_{{\mathit{\text{ab}}}_{0}}^{\text{gvm}}$ into α_{ab0} in Eq. (21), we can obtain θ_{c0}=${\mathrm{\theta}}_{c0}^{\text{gvm}}$ which satisfies GVMC.

We have now obtained the general formulae giving the noncollinear angle (Eq.(17)), the angular dispersion of the input beam (Eq. (20)), and the angle from the optical axis for the output beam (Eq. (21)), in order to satisfy the 0th order and the first order WVMC in sum-frequency generation.

## 3. Acceptance function of a nonlinear crystal

The bandwidth of the mixed beam is limited by the wave-vector mismatch Δκ defined in Eq. (4), even though WVMC and GVMC are fulfilled at the center angular frequencies (or the center wavelengths) of beam A, beam Band beam C, because of high-order wave-vector mismatch in terms of Δω. In the approximation of the small depletion limit of the input beam, the electric field amplitude of mixed beam in the frequency domain is proportional to a convolution of the electric field amplitudes of the input beams and the integral of the phase mismatch factor e^{iΔκ‖ζ} parallel to the direction of κ_{c}. Thus, we define a complex function of η(ω_{a0}, ω_{b}) as

$$\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}={e}^{i\Delta {k}_{\parallel}\left({\omega}_{{a}_{0}},{\omega}_{b}\right)\frac{L}{2}}\mathrm{sinc}\left\{\Delta {k}_{\parallel}\right({\omega}_{{a}_{0}},{\omega}_{b}\left)\frac{L}{2}\right\},$$

where Δκ_{‖} is the projection of Δκ onto κ_{c}, ζ is a scalar coordinate parallel to κ_{c}, and *L* is the length of a nonlinear crystal (the factor *L*
^{-1} is multiplied in front of the integration in order to normalize the function). This function determines an additional phase shift of Δκ_{‖} (ω_{a0}, ω_{b}) *L*/2 and the acceptable bandwidth represented by a sinc factor in a SFM process. We call η(ω_{a0}, ω_{b}) the acceptance function in accordance with Ref. 6. We calculate |η(ω_{a0}, ω_{b})|^{2} as an acceptable power spectrum with respect to the angular frequency of the input beam ω_{b} (or with respect to the wavelength λ_{b}) of a beta-barium borate (BBO) crystal.

Regarding the experimental setup of the sub-20-fs CPA system of the Ti:sapphire laser in our laboratory [21], the center wavelength of the broadband input beam, λ_{b0} =2π*c*/ω_{b0}, is set to be 800 nm in the calculations. The wavelength of the input of a monochromatic beam, λ_{a0}=2π*c*/ω_{a0}, is fixed at 532 nm which is the wavelength of the second harmonic of a Nd:YAG laser.

The conventional SFM scheme of a collinear geometry in the 100-femtosecond regime typically requires that the thickness of a nonlinear crystal be less than 1 mm for sufficient bandwidth [18, 22–24], thus the thickness is fixed at 1 mm in the calculation below.

Based on the formulae in section (2.), and Sellmeier’s equations for refractive indices reported in Ref. 25, we obtained the angle from the optical axis for the output beam, θ_{c0}, the noncollinear angle between the two input beams at the center wavelength, α_{ab0}, and the angular dispersion with respect to the wavelength (instead of the angular frequency given by Eq. (20)), *d*α_{ab}/*d*λ_{b|0} in the type I configuration as θ_{c0}=54.81°, α_{ab0}=±26.15°, and *d*α_{ab}/*d*λ_{b|0}=±3.5884×10^{-4} rad/nm, respectively, where a plus sign is indicates the same direction of θ_{a0} and the plus and minus signs of α_{ab0} and *d*α_{ab}/*d*λ_{b|0} must be in order. These two sets of solutions with plus and minus signs give the same result of wave-vector mismatch because the refractive indices of the input beams do not depend on θ_{b0} and θ_{c0}.

The phase mismatch, Δκ_{‖}
*L*, with the crystal length of 1 mm, is shown as a function of the wavelength of Ti:sapphire laser in Fig. 3(a). A brown, dotted curve is the phase mismatch in a collinear geometry and a red, dot-dashed curve is that in a noncollinear angularly dispersed geometry (NADG) with GVMC. It is noted that not only is the phase mismatch zero, but also tangent of the curve is zero at the center wavelength reflecting GVMC.

An acceptable power spectrum, |η(λ_{b})|^{2}, is also shown below the phase mismatches in the same figure. If we take the acceptable spectral width as the full-width at half maximum (FWHM) of the acceptable power spectrum, it is 4.8 nm for the brown, dotted curve of the collinear geometry, while it is 82 nm for the red, dot-dashed curve. Thus, the NADG with GVMC makes it possible to generate a spectrum 17 times broader than that for the collinear geometry.

The NADG does not need a complete GVMC at the center wavelength. Because the phase mismatch parabolically changes around the center wavelength, we can broaden the acceptable bandwidth to >100nm. A blue curve in Fig. 3(a) shows the phase mismatch with a slight tilt of BBO from the complete GVMC. The curve crosses the zero line at two points and the acceptable power spectrum, which is shown in Fig. 3(a) as a blue curve with hatched area, has two peaks corresponding the crossing points of phase mismatch. Although the central dip is 20% lower than the peaks, this spectrum is sufficient for generating sub-20-fs pulses.

The type II configuration of polarizations, in which the polarization of the monochromatic beam is parallel to that of the output beam, may be a suitable alternative to the NADG of type I wave-vector matching. We show an acceptable power spectrum of type II wave-vector matching with collinear geometry in 1-mm-thick BBO as a dotted, brown curve in Fig. 3(b). The acceptable spectral width is 14 nm, which is still much narrower than that of the NADG of type I configuration. The NADG of type II configuration, however, broadened the spectrum significantly.

We found the parameters of NADG for GVMC; the angle from the optical axis for the output beam, θ_{c0}, is 72.83°, the noncollinear angle between the two input beams at the center frequeny, α_{ab0}, is 13.9°, and the angular dispersion, *d*α_{ab}/*d*λ_{b|0}, is 1.8168×10^{-4} rad/nm. The acceptable spectral width under these conditions is 113nm (refer to the red, dot-dashed curve in Fig. 3(b)), and we can achieve further broadening of the spectrum with a slight tilt of the crystal (blue, solid curve with hatching in Fig. 3(b)).

As compared with the type I configuration, type II seems more suitable for broadband wave-vector matching for the generation of ultrashort pulses. However, the nonlinearities of the crystal are important for an efficient SFM as well as for the broad bandwidth. The effective nonlinear coefficient at the center wavelength in type I configuration, ${d}_{\text{eff}}^{\mathrm{I}}$, is given by

$$\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\simeq {d}_{11}\mathrm{cos}\left(3\phi \right)\mathrm{cos}{\theta}_{{c}_{0}},$$

while that in the type II configuration, ${d}_{\text{eff}}^{\text{II}}$, is

$$\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\simeq -{d}_{11}\mathrm{cos}{\theta}_{{a}_{0}}\mathrm{cos}{\theta}_{{c}_{0}}\mathrm{sin}\left(3\phi \right),$$

where φ is the common polar angle of the three beams. Substituting numerical parameters into these equations, we conclude that the effective nonlinear coefficient in the type II configuration is less than one over ten of that in type I; ${d}_{\text{eff}}^{\mathrm{I}}$ =2.5${d}_{36}^{\text{KDP}}$ and ${d}_{\text{eff}}^{\text{II}}$ =0.22${d}_{36}^{\text{KDP}}$, where ${d}_{36}^{\text{KDP}}$ is a nonlinear coefficient of KDP and we use the relative nonlinearity (*d*
_{11}=4.1${d}_{36}^{\text{KDP}}$) reported in Ref. 26. Giving priority to the nonlinearity rather than the bandwidth, we adopted the type I configuration in the experiment.

## 4. Experiment

In order to confirm the feasibility of NADG for GVMC, we performed an experiment with a sub-20-fs Ti:sapphire chirped pulse amplification (CPA) laser system [21]. A monochromatic beam mixed with this broadband laser was supplied by the frequencydoubled output of a Q-switched Nd:YAG laser with energy of 25 mJ (Minilase II, New Wave Research) and a pulse duration of 4.5 ns. The trigger signal of the Q-switch was supplied from a delay generator (DG535, Stanford Research Inc.) synchronized with the signal driving a Pockels cell in a regenerative amplifier in the Ti:sapphire CPA system.

The configuration of these two beams for NADG is schematically shown in Fig. 4. The amplified pulses, which bypass the compressor in the CPA system, with an energy of 28 mJ and a pulse duration of ~500 ps, were led to a spatial disperser consisting of a Brewster prism of s-LAL10 glass (OHARA) and a 1/8 telescope of two concave mirrors, with which the image of the beam on the surface of the prism was relayed in a 1-mm thick BBO crystal with an appropriate angular dispersion. We note that the beam diameter of the Ti:sapphire laser on the prism was set to be ~12 mm, which is smaller than that in the original CPA system [21] (~20 mm), in order to obtain the appropriate beam diameter for a spatial overlap with the Green beam in the BBO crystal. The polarization of Ti:sapphire laser was rotated by 90 degrees at the output of the prism with a quartz rotator although it maintained a horizontal direction until the beam passed through the prism.

Because of the refraction on the surface of the BBO crystal, the noncollinear angle between the two input beams at the center wavelength should be set at 44.38° and the angular dispersion outside the crystal should be adjusted to 6.3369×10^{-4} rad/nm. The 1/8 telescope magnifies the angular dispersion eight times, hence the angular dispersion induced by means of the prism required for the appropriate GVMC should be 7.9211×10^{-5} rad/nm, which can be obtained from the adjustment of the incident angle of the Brewster prism at around 53.8°. The spatial chirp (it means that the different parts of the beam belongs to different wavelengths) caused by defocusing of the image within the 1-mm thick crystal is estimated to be ~40µm for 110-nm spectral width, which is ~1/40 of the beam diameter (~1.5mm).

The monochromatic green beam was sent via a mirror on a translation stage and was imaged onto the surface of the crystal with a diameter of approximately 2 mm.

The measured spectra of the input (Ti:sapphire) and the output UV pulses to the wavelengths (λ_{b} and λ_{c}) are shown in Fig. 5(a) and Fig. 5(b), respectively. These spectra were recorded with single-shot measurements. The spectral width of the input pulse is approximately 100 nm and ensures the pulse width of 16 fs if we adopt an appropriate procedure for the compensation of dispersion as reported in Ref. 21. The UV spectrum of is centered at ~320 nm with a width of ~17 nm.

By converting the wavelengths to optical frequencies (ν_{TiS}=*c*/ω_{b}, ν_{UV}=*c*/ω_{c}), we can see that the bandwidths of these power spectra are almost the same (refer to Fig. 6). The bottom axis of the optical frequency of the generated UV pulse in Fig. 6 is shifted by an offset of optical frequency of the mixed monochromatic beam (ν_{a0}=*c*/λ_{a0}), which is equal to 563.52 THz, from the top axis of the Ti:sapphire laser frequency.

A central dip in the UV spectrum may be due to the filtering of the acceptance function. We adjusted the angle of the BBO crystal and the direction of the second harmonic of the Nd:YAG laser so that the spectral width could be maximized, and hence, the central dip of the acceptable power spectrum in the experiment might be deeper than that in the calculation. We calculated the spectrum of Ti:sapphire filtered by the acceptable power spectrum with a slight change of the noncollinear angle from the ideal condition. It is shown as a brown, dot-dashed curve in Fig.6, which is comparable to the generated UV spectrum.

We can see that there is some kind of modulation in the spectrum of the UV pulse. It may be due to the instability of the temporal profile of the Nd:YAG laser induced by mode-beating, although we could not verify this because of the limited temporal resolution of the detection system.

Energies of the UV pulses were measured by an energy meter (Rm-3700, Laser Probe Inc.). Solid circles in Fig. 7 are the energies of the UV pulse. They are plotted along with the relative delays of the triggering of the Q-switch of Nd:YAG laser and compared to the pulse shape of the second harmonic of the Nd:YAG laser measured with a PIN photodiode and a digital oscilloscope of which the bandwidth is 300 MHz. Considering that the pulsewidth of the Ti:sapphire laser is much shorter than that of the second harmonic of the Nd:YAG laser, it is appropriate that the plots of solid circles agree well with the pulse shape. The energy of the UV pulse is only 1.3 micro joule at most. This low conversion efficiency (<10^{-4}) may be due to the low intensities of the input beams of 2GW/cm^{2}.

The output beam was diverged with diffraction. However, the ratio of the beam diameter in the direction of angular dispersion to that in the perpendicular direction kept almost unity. Thus, we conclude that there is almost no angular dispersion, although this result should be quantitatively confirmed by using a method for measuring angular dispersion [27].

## 5. Summary and discussion

We have proposed and analyzed a novel scheme for broadband sum-frequency mixing for the indirect phase control of sub-20-fs pulses in the UV region. The key points of this method are as follows: (1) one of the input beams should be (quasi-) monochromatic, and (2) the broadband input beam must be angularly dispersed and non-collinearly mixed with the other monochromatic beam. We found that we could uniquely determine the direction of the beams to the optical axis, the noncollinear angle, and the angular dispersion of the input beam by use of the analytical expressions of the wave-vector matching and the group-velocity matching conditions, with the restriction of there being no angular dispersion of the output beam at the center angular frequencies. The bandwidth of the acceptable power spectrum was sufficiently broad for the generation of sub-20-fs pulses around the wavelength of 320 nm with 1-mm-thick BBO, which is not possible in a conventional collinear SFM scheme.

The feasibility of broadband wave-vector matching in this new scheme was verified in the experiment by using a CPA system with a sub-20-fs Ti:sapphire laser and the second harmonic of the Q-switched Nd:YAG laser. The bandwidth of the UV beam was sufficiently broad to obtain sub-20-fs pulses and this result indicates that the indirect phase control of UV pulses in the sub-20 fs regime is possible.

The energy obtained in the experiment, however, is too low to be applied to nonlinear optical experiments such as high-order harmonic generation. The beam diameter of the YAG laser is larger than that of the Ti:sapphire laser in order to cover the spatial walk-off estimated to be ~500µm. The mismatch of the beam diameter may reduce the efficiency, however, the main reason for the low conversion efficiency is the low intensities of the input beams [24]. Thus, we are now planning a new arrangement involving the use of compressed pulses with a slight chirp from the CPA system in order to reduce the pulse width to the sub-picosecond regime. In the SFM of these pulses with their band-pass-filtered SH pulses, the intensities of the two input beams would be ~10^{2} times higher than those of the beams described in the previous section. The acceptable power spectrum in a 0.8-mm-thick BBO with NADG ensures ultrashort pulses in the sub-20-fs regime with a wavelength of around 275 nm in this arrangement.

Compensation of the chirp in the generated beam will still be a problem even if we could obtain a broad spectrum and sufficient energy. In our estimation, a pair of Brewster prisms made of fused silica or sapphire with an appropriate separation and insertion can eliminate group delay (the second order), dispersion (GDD), and the third order dispersion (TOD) simultaneously when we give a negative chirp to a 16-fs Ti:sapphire laser pulse by increasing the separation between gratings consisting of the compressor in the CPA system, in order to broaden the pulse width up to 0.6 ps. Indirect phase control with the LCSLM in the CPA system will enable us to realize the further compensation of high order dispersions in order to obtain both a transform limited and a shaped pulse in the sub-20-fs regime. We are currently preparing the experiment incorporating the new arrangement.

This work was supported by Grant-in-Aid for Scientific Research on Priority Areas No. 14077222 and Grant from Research Foundation for Opto-Science and Technology.

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