We have found general expressions relating the high-order pulse front tilt and the high-order angular dispersion in an ultrashort pulse, for the first time to our knowledge. The general formulae based on Fermat’s principle are applicable for any ultrashort pulse with angular dispersion in the limit of geometrical optics. By virtue of these formulae, we can calculate the high-order pulse front tilt in the sub-20-fs UV pulse generated in a novel scheme of sum-frequency mixing in a nonlinear crystal accompanied by angular dispersion. It is also demonstrated how the high-order angular dispersion can be eliminated in the calculation.
©2003 Optical Society of America
Pulse shortening of femtosecond lasers has progressed considerably in recent years, such that a few cycle pulses can now be obtained in the visible~ near-infrared region [1–4]. These results were mainly obtained by the novel technique of spectral broadening in a gaseous nonlinear medium and that of compensation for spectral dispersion. We should, however, remember that, in the background of the success of pulse shortening, spatially dependent delay in the beam profile of an ultrashort laser pulse, called “pulse front tilt,” was intrinsically removed by perfectly aligning the angularly dispersive elements, such as prisms in the pulse compressor [1, 2] and gratings in the phase controller .
It is well known that the angularly dispersive elements induce the pulse front tilt . This effect was expressed by Bor and Rácz in the context of pulse compression and traveling wave excitation of dye lasers  and also analyzed by Martinez using the Kirchhoff-Fresnel integral of a Gaussian beam . The former analysis yielded the result that the pulse front tilt of γ should be related to the angular dispersion of dε/dλ|0 by
with ε, the angle from the angularly dispersive element, and λ, the wavelength as a variable. Subscript 0 denotes the substitution of the fixed wavelength of λ 0 characterizing the ultrashort pulse.
Equation (1) is widely used for evaluating the pulse front tilt in many applications, however, using only the information about the relationship between the first-order derivative of the angle in terms of the wavelength (or the angular frequency) and the group delay, which is also the first-order derivative of the spectral phase, is insufficient for recognizing the change of the pulse shape with the spatial position in the beam profile. While the angular dispersion in the ultrashort pulse can already be experimentally determined up to the second order, as in the recent work done by Varjú et al. [8, 9], we have had no idea how the high-order derivatives of the angle, which we call “high-order angular dispersions,” can affect the high order derivatives of the spectral phase at a certain position in the beam profile relative to that at some other position, which we call “high-order pulse front tilts.” However, we note that Osvay and Ross have shown equations for obtaining the absolute group-delay dispersion (GDD) and third-order dispersion (TOD) induced by angular dispersion at the exit of a misaligned grating pulse compressor .
There had been no need to determine the high order pulse front tilts to date, because the angular dispersion can be eliminated under the ideal condition of angularly dispersive optical elements in the ultrafast laser oscillator/amplifier system and the analysis at the first order is sufficient for the application of the pulse with a tilted pulse front. However, we should remember that the angular dispersion in one of the beams, at least in noncollinear sum-frequency mixing (SFM) or noncollinear optical parametric amplification (OPA), both of which are kinds of three wave mixing in a nonlinear crystal, is intrinsic and nonlinear.
For example, Shirakawa et al. measured the external noncollinear angles of the idler pulse for each wavelength in a noncollinear OPA system and showed that the residual dependence of the angle on the wavelength corrected by the angular-dispersion compensator consisting of a grating and a telescope is nonlinear or somewhat sinusoidal with an amplitude of ~400 µ radians in the modulation, although they were successful in obtaining an almost transform-limited pulse with a duration of 9 fs .
In SFM for the generation of deep-ultraviolet (DUV) or near-vacuum-ultraviolet (NVUV) femtosecond pulses, a noncollinear configuration is required and angular dispersions in the input and the generated pulse are also needed for compensating a large amount of dispersion to the DUV or the NVUV wavelength in a nonlinear crystal, as was shown in ref. 12 and ref. 13. We have analyzed this method in detail as an extension of our novel scheme of SFM reported in ref. 14 and showed that the acceptable power spectrum can be broadened to more than 9 times that obtained in ref. 12 by achieving the GDD-matching condition. In that work, which will be reported elsewhere , the spectral width of the generated DUV (~256 nm) in the experiment was sufficient for forming a sub-20-fs pulse as a Fourier limit, while we had to consider how the residual nonlinear angular dispersion, compensated by an inverse angular disperser, affects the pulse front in the sub-20-fs regime.
In this paper, we show the relationship between the angular dispersion and the pulse front tilt which is applicable to nonlinear angular dispersion including high-order terms of Taylor expansion for the wavelength or the angular frequency (Section 2). Since the formulae are obtained from only Fermat’s principle, they are general and independent of the content of the individual angularly dispersive optical device, as was pointed out by Bor and Rácz for the first-order pulse front tilt expressed as Eq. (1) . The obtained formulae are the first expressions, to our knowledge, of the high-order pulse front tilt induced by the high-order angular dispersion.
By using the formulae, we calculate the tilted pulse front of the generated DUV pulse in the noncollinear angularly dispersed SFM under the GDD-matching condition in Section 3 and summarize the results in Section 4. We also show the calculated fringe patterns of the interference for the measurement of the angular dispersion  in Section 4.
2. Formulation of the high-order pulse front tilt
We assume that a plane wave of the laser light with an arbitrary angular frequency of ω is diffracted using an optical device such as a prism or a grating, as shown in Fig. 1, where two rays at different positions are named the ‘A’ ray and the ‘B’ ray, respectively. Because the angular dispersion originates from one point at the entrance of the diffractive device for each ray, we can define the fictitious origin, which does not need to lie on the real ray, of the diffracted ray as one point from which rays for any ω component extend in the free space, as is indicated by OA and OB in Fig. 1. Although the point of OB in Fig. 1 is occasionally also the real origin of the diffracted ray, we can proceed with this argument of angular dispersion without losing generality because the fictitious origins of any rays from different entrances inevitably are on the line , which does not need to be perpendicular to the input ‘B’ ray by the definition.
In this configuration of the two rays, we define the diffraction angle as the angle from the normal vector to the line and denote it by θ 0 for the fixed angular frequency of ω 0, which is identified with a near-center angular frequency in an ultrashort pulse. A deviation of the angle from θ 0 induced by a variation of the angular frequency from ω 0, which is Δω in our definition, is denoted by Δθ.
According to Fermat’s principle, the spectral phase at point OA on the ‘A’ ray for the angular frequency ω 0 is equivalent to that at point PB on the ‘B’ ray, which is determined by drawing an intersecting line from OA perpendicularly to the ‘B’ ray, namely,
where ϕA(ω) and ϕB(ω) are the spectral phases of the ‘A’ ray and the ‘B’ ray, respectively, for the angular frequency of ω at the equivalent points on each ray, which are OA(=PA) and PB if ω is equal to ω 0.
We can determine another phase-equivalent point of P′A on the deviated ‘B’ ray for the varying angular frequency of ω 0+Δω by a similar procedure. Point P′A reflects the deviated spectral phase of the ‘A’ ray on the ‘B’ ray, which is denoted by ϕA(ω 0+Δω). On the other hand, the phase-equivalent point on the deviated ‘B’ ray to point PB on the fixed ‘B’ ray is determined by drawing an intersecting line from PB perpendicularly to the deviated ‘B’ ray, shown as P′B in Fig. 1, which reflects the deviated spectral phase of the ‘B’ ray, namely, ϕB(ω 0+Δω). Since the optical path difference of the ‘B’ ray to the ‘A’ ray for the angular frequency of ω 0+Δω is equivalent to , the difference in the spectral phase of the ‘B’ ray from that of the ‘A’ ray is given by
where c is the velocity of light, ℓ is the length of the line , and x 0 is defined as a position on the ‘B’ ray from OA along the axis of , as shown in Fig. 1. Because the variation of Δω is finite, not infinitesimal, this equation gives the exact spectral phase difference of the ray spatially separated by a distance of x 0 from the other ray. We note that the separation must be defined for the rays for the fixed angular frequency of ω 0.
In order to elucidate the relationship between the angular dispersion and the spectral phase, we expand both sides of Eq. (3) in a Taylor series with Δω. By comparing the coefficients in the series at each order of Δω, we can obtain the relation denoted by Eq. (2) at the zeroth order and also obtain the following equations.
These three equations correspond to the coefficients in the series at the first order, the second order, and the third order, respectively.
We can easily see that the left-hand side of Eq. (4) is the difference of the group delay in the ‘B’ ray (τB(ω 0)=dϕB/dω|0) from that in the ‘A’ ray (τA(ω 0)=dϕA/dω|0) and represents the (first order) pulse front tilt, which is clearly seen by converting the variable of the angular frequency into the wavelength such as,
While Eq. (7) expresses the difference in time when the peak of the pulse arrives, corresponding to the separation of x 0, Eq. (8) and Eq. (9) include information on how the pulse shape varies. The left-hand sides of Eq. (8) and Eq. (9) are the GDD difference and the TOD difference, respectively, which make the temporal profile different at each separated position, x 0. Thus they are regarded as high-order effects in the pulse front tilt. We can find that both the GDD difference and the TOD difference are proportional to x 0 and the GDD difference is also proportional to the second order angular dispersion, while the right-hand side of Eq. (9) contributes all the angular dispersions, up to the third order, to the TOD difference.
We note that the derivation of the phase in Eq. (3) or the pulse front tilts in Eq. (7)~Eq. (9) are independent of the diffractive device. The only assumptions are that (i) the diffracted rays for each angular frequency component has an origin (as one point), and (ii) the origins of spatially separated rays are aligned, thus, the results are general and applicable to all the angularly dispersed pulses that satisfy these assumptions in the limit of geometrical optics.
In spite of the generality, we should note that we can get no information on the spectral phase or dispersion itself from Eq. (7)~Eq. (9) which only give the relative difference of dispersion between the two rays. The spectral dispersions at x 0 depend on how the pulse passes thorough the diffractive element and also on the dispersion suffered due to the other optical devices before entering and after leaving the diffractive element.
3. Examples: deep UV pulse generated with GDD-matched SFM
As was briefly discussed in Section 1, our aim of deriving the formula for the high-order pulse front tilt is to see how the residual angular dispersion affects the DUV pulse generated with the GDD-matched scheme. We give a brief explanation of this scheme in this section while the detailed analysis will be reported elsewhere .
For the purpose of broadening the width of the acceptable power spectrum in SFM, the three kinds of wave vectors interacting in a nonlinear crystal should be noncollinear and angularly dispersed, as shown in Fig. 2. With an appropriate angle of αab(ωb) between the quasi-monochromatic input pulse A and the broadband input pulse B, which is assumed to be a sub-20-fs Ti:sapphire laser pulse with an angular frequency of ωb, the output angle from the optical axis in the generated pulse, θc, varies, following the change of the angular frequency of ωc such as
where is the fixed polar angle of the wave vector in the quasi-monochromatic input pulse and αac(ω c) is the noncollinear angle between the input pulse A and the generated pulse C obtained from
which originates from the wave-vector matching condition. Assuming type I SFM in a BBO crystal, the wave numbers of the two input pulses, and kb(ωb) are independent of polar angles to the optical axis. Thus Eq. (11) explicitly describes the dependence of the output angle on the generated pulse with the photon-energy conservation of .
In our analysis and experiments of the SFM using the sub-20-fs Ti:sapphire laser  with the wavelength centered at ~800 nm and the narrow band second harmonic of the Ti:sapphire laser at 376.5 nm, the width of the acceptable power spectrum, shown as the red hatched area in Fig. 3, exceeds 15 nm in the DUV region (~256 nm) and the spectrum obtained in the experiment, shown as the light blue hatched area in Fig. 3, agrees well with that of the input Ti:sapphire laser. The broadband nature of this scheme is due to the extension of the wave-vector matching condition from the first order (group delay, GD), which was reported in ref. 14, to the second order (GDD).
Because the output DUV pulse has a large angular dispersion shown as the blue solid curve in Fig. 3, which is calculated from Eq. (11) and from the refraction on the output surface of the BBO crystal, we must compensate it using an inverse angular disperser consisting of a telescope and a diffractive device. A fused silica prism with an apex angle of 73° combined with a telescope with a magnification factor of unity, which was adopted in the preliminary experiment , can provide almost the same angular dispersion as the inverse of that to the generated pulse up to the second order. This condition of the incident angle (83.59°) to the prism and the magnification factor of the telescope could be found by adjusting the incident angle so that the ratio of the first-order angular dispersion to the second-order one as a result of passing through the prism should be matched with that in the generated DUV pulse. Residual output angle and the first- and second-order angular dispersions are shown as dotted curves in Fig. 4. The cubic dependence of the angle on the wavelength, shown as the dotted light blue curve, reflects the residual third-order angular dispersion, although the first- and second-order angular dispersions, shown as a dotted dark green curve and a dotted dark brown curve, respectively, cross the zero line near the center wavelength.
We can see how this nonlinear angle difference affects the pulse front by using Eq. (3) and assuming that the central part of the spatial profile of the generated DUV pulse can be the transform limit with an appropriate dispersion compensator, namely, ϕA=const. in Eq. (3).
Figure 5 shows the variation of the tilted pulse front with the change of the incident angle to the fused silica prism in the inverse angular disperser. The temporal profile of the pulse is calculated from the measured spectrum shown in Fig. 3. The color bar at the top left indicates the incident angle relative to that under the condition of the dotted curves in Fig. 4. Temporal modulation in the pulse shape, mainly due to the TOD, cannot be removed even in the limited spatial range of ~±1mm, which corresponds to the beam radius in the experiment, at any incident angle to the prism. At near 0°-incidence, the main peak of the pulse front accompanied by substantial number of subpeaks is bent around the spatial center of the beam. These modulations and bending of the pulse front could not be visualized until we obtain the general formula of the pulse front tilt in Eq. (3).
While Fig. 5 shows an interesting example of high-order effects in a pulse front, this result is disappointing in terms of the compensation of the high-order angular dispersion in the generated pulse from the GDD-matched scheme for sub-20-fs DUV pulses. Thus, we add a grating with a groove density of 1200 ℓ/mm and replace the telescope with that having a transverse magnification factor of 2.5/1.4, which means that the angular dispersion in the generated DUV pulse is reduced to 1.4/2.5. This hybrid configuration with the grating and the prism enables us to control the ratios of both the second- and third-order angular dispersion to the first-order one, which results in the compensation of almost all angular dispersions up to the third order with the incident angles of 66.00° to the grating and 79.06° to the prism, as shown by solid curves in Fig. 4.
By using the hybrid compensation technique, the tilted pulse front exhibits almost no modulation or bend within a range of the spatial profile of ±6 mm in this calculation, even if the incident angle to the prism is slightly changed, as shown in Fig. 6. We have only to pay attention to the first-order angular dispersion in order to obtain the spatially correct pulse without a tilted pulse front in this case.
4. Summary and discussion
We have derived the general formulae for calculating the tilted pulse front including the high-order angular dispersions in the ultrashort pulse, for the first time to our knowledge. The formulae enable us to evaluate not only the gradient but also the distortion of the pulse front, which has been demonstrated in the novel scheme of SFM for the generation of sub-20-fs DUV pulses under the GDD-matching condition.
We should, however, note that the formulae cannot always yield the correct solution of the tilted pulse front because the analysis is only valid under the assumption that the geometrical optics can be applied to the plane wave and that the origins of the rays are aligned. Thus, we have no idea of the high-order pulse front tilts for a Gaussian beam propagating with diffraction, for which Martinez gave the solution of the first-order pulse front tilt . Neither can we predict the high-order distortion of the pulse front as a result of passing through a spherical lens, for which Bor and Horváth analyzed the effect of group delay .
Fortunately, the spatial profile of the DUV pulse in our experiment seems to have a nearly flat top and the beam diameter (~2 mm) is much larger than the wavelength. Hence, we can expect the analysis to be valid for this DUV pulse that resembles a plane wave.
In order to confirm the high-order angular dispersion in the experiment, a method with interference, proposed by Varjú et al., is very useful . The angularly dispersed beam is divided into two by the beam splitter of a Mach-Zehnder interferometer in which one of the beams is spatially reversed in such a way that the angular dispersion has the opposite sign. Then the two beams are spatiotemporally overlapped on another beam splitter of the interferometer. Spectrally resolved fringes in the interferogram reflect the propagating angle for each spectral component, thus they can be used to determine the angular dispersion.
We demonstrate in calculation how the spectrograms can be seen by this method with changes of the angular dispersion corresponding to the changes of the pulse front tilt in Fig. 5 and Fig. 6, and show those in Fig. 7 and Fig. 8, respectively. The angle between the two beams in the interferometer is assumed to be 600 µ radians in these calculations.
Nonlinearity in the residual angular dispersion compensated with the prism-only inverse angular disperser is clearly perceived from the anomalous bend of the fringe at ~253 nm and ~258 nm, which correspond to the local minimum and the local maximum of the angle plotted as the light-blue dotted curve in Fig. 4, when the relative incident angle to the prism is set near 0°. On the other hand, the straight lines along the wavelength in the fringe in Fig. 8 exhibit almost no angular dispersion in the compensated beam with the hybrid inverse angular disperser at the 0° relative incidence to the prism. These images of the fringes in numerical calculations will help us to finely adjust the inverse angular disperser in the experiment.
We acknowledge financial support through a Grant-in-Aid for Scientific Research on Priority Areas (No. 14077222) of the Japanese Ministry of Education, Culture, Sports, Science, and Technology, and from the Research Foundation for Opto-Science and Technology in Hamamatsu.
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