## Abstract

We present general guidelines for the design of ultrabroadband, high-energy optical parametric chirped-pulse amplifiers, where maximization of both conversion efficiency and bandwidth and simultaneous suppression of superfluorescence is required. Using a semi-analytical approach together with numerical simulations, we find that the ratio of pump and seed pulse durations is a critical parameter in temporal optimization, and its optimum depends on the amplifier gain. Multi-stage amplifier design thus requires independent optimization of seed chirp at each amplification stage. We find that a small compromise in amplifier bandwidth relative to the full phase-matching bandwidth, through use of the appropriate seed chirp, both maximizes the efficiency-bandwidth product and optimizes the signal-to-noise ratio. On the other hand, maximization of signal bandwidth is found to significantly degrade both the signal-to-noise ratio and the conversion efficiency.

©2009 Optical Society of America

## 1. Introduction

Optical parametric chirped pulse amplification (OPCPA) [1] is nowadays the most promising technology for producing the world’s highest peak power, few-cycle light pulse sources in the visible and near-to-mid infrared range [2, 3, 4, 5, 6, 7, 8]. In OPCPA, a quasi-monochromatic high-energy pump field (such as, for example, a picosecond or nanosecond pulse) is coupled to a chirped, low-energy broadband seed field in a nonlinear crystal. If the seed pulse is sufficiently stretched, good energy extraction from the pump field can be achieved, and subsequent recompression makes it possible to reach very high peak powers. The OPCPA concept has some very important advantages with respect to standard CPA: (i) the parametric amplification process can support gain bandwidths well in excess of those achievable with conventional linear amplifiers, enabling the generation of few-optical-cycle light pulses; (ii) OPCPA has the capability of providing a high gain in a relatively short path length, minimizing the B-integral and allowing a compact, tabletop amplifier setup; (iii) amplification occurs only when there is a pump pulse, so that the amplified spontaneous emission and the consequent pre-pulse pedestal are greatly reduced; (iv) thermal loading effects, apart from parasitic absorption, are completely absent, greatly reducing spatial aberration of the beams. These attributes allow OPCPA to push the limits of high peak power pulse generation at wavelengths at which broadband laser amplification has not been developed, a capability with increasing importance since the advent of Yb- and Nd- based picosecond pump lasers that can deliver hundred-watt (and potentially kilowatt) average powers [9, 10, 11, 12, 13], thus promising simultaneous high peak power and high average power from OPCPA. As a result, today there is much interest in the development of OPCPA as a light source for few-cycle, high-intensity, near-to-mid-infrared pulse-driven applications such as high harmonic generation and attosecond science.

Central to the emergence of OPCPA technology is the issue of maintaining ultrabroadband amplification while pushing the limits of efficient energy conversion from pump to signal. Additionally, the OPCPA may combine low seed energy (≪1 nJ) with high desired gain, and under these conditions the presence of noise at signal wavelengths due to spontaneous parametric generation cannot be ignored. Such amplified noise or ”superfluorescence” may overtake the signal amplification, causing a strong decrease in amplified signal energy and stability, and introduce a temporal pedestal in the compressed pulse which may be detrimental for some applications. Indeed, recent reports of OPCPA in the mid-infrared have found superfluorescence to be a major limiting factor in signal energy scaling [7, 8].

Necessary to the development of OPCPA is a careful investigation into the simultaneous optimization of conversion efficiency, signal bandwidth, and signal-to-noise ratio. Previous studies of OPCPA optimization have treated several aspects of this problem independently [14, 15, 16, 17, 18, 19, 20, 21, 22, 23]. The two earlier works by Ross *et al*. [14, 15] discuss the trade-off between amplifier bandwidth and conversion efficiency that is determined by the signal pulse chirp. Ref. [15] finds that amplifier conversion efficiency and efficiency-bandwidth product are maximized at approximately the same propagation length, and for Gaussian pump and seed pulses finds that a ratio of seed and pump pulse durations, Δ*t _{s}*/Δ

*t*, of 0.57 optimizes the efficiency-bandwidth product. A recent work by Witte

_{p}*et al*. on a terawatt non-collinear OPCPA system focuses on the spectral shaping due to both the phase-matching conditions and the pump intensity profile, and finds a different optimum, Δ

*t*/Δ

_{s}*t*= 0.2-0.3 [16]. This discrepancy, along with others in the literature, highlights the need for a systematic analysis of the optimal seed/pump duration ratio and of its dependence on OPCPA parameters, such as the total gain.

_{p}The investigation of signal-to-noise ratio degradation due to the amplification of parametric fluorescence, rather than due to a transfer of pump intensity noise [15, 16, 17, 18, 19, 20, 24], is limited to only a few works [14, 21, 22]. These reports highlight several important concerns in superfluorescence suppression. Ref. [21] illustrates that leading-edge signal-to-noise contrast of the amplified signal pulse may be improved by a slight delay of pump pulse arrival relative to that of the seed pulse, at the expense of conversion efficiency. Ref. [22] studies signal-to-noise contrast as a function of amplifier gain and emphasizes the dependence of the final signal-to-noise ratio on the initial noise intensity. Both papers highlight the “gain quenching” effect of the signal amplification on the noise amplification: as amplification of the signal depletes the available pump energy, both the signal and noise gain are strongly diminished. However, this effect does not prevent a degradation in signal-to-noise ratio observed during saturation of the amplifier gain, where conversion efficiency and bandwidth are maximized [22].

In this paper, we consider the problem of simultaneously maximizing the efficiency and gain bandwidth of OPCPA while avoiding superfluorescence amplification and consequent signal-to-noise ratio degradation. We employ both analytical and numerical analyses to investigate the temporal optimization problem, and where possible, we provide experimental data from a state-of-the-art mid-infrared OPCPA system (Ref. [8]) to support the analysis. We find that not only the efficiency and bandwidth but also the signal-to-noise ratio are each strongly tied to the temporal profiles of the interacting waves and the local phase-mismatch across the interacting pulses, and we explain, through a model of simultaneous signal and noise amplification, how the overall sacrifice in these qualities can be minimized. Furthermore, we demonstrate a sensitivity of the parameters that optimize these qualities to the total gain of the amplifier. As a result, we conclude that careful optimization of seed chirp is necessary at each stage of a multi-stage amplifier in order to prevent significant noise buildup while preserving both gain and bandwidth. We present guidelines for the stage-by-stage optimization, including an empirical formula for optimal signal and pump pulse duration ratio based on numerical simulations and semi-analytical investigation.

The paper is organized as follows.

- In section 2, the conceptual and semi-analytical framework for understanding the optimization is presented. A model for understanding the simultaneous amplification of signal and noise is introduced.
- In section 3, the numerical model is described, and the results of a numerical analysis of a state-of-the-art ultrabroadband amplifier are given and compared against the predictions of section 2. A formula for the optimal ratio of pump and seed durations is derived as a function of gain.
- Section 4 presents a summary of conclusions and practical consequences.

## 2. Underlying principles

The predominant issue in the optimization of efficiency in parametric amplification is the spatio-temporal variation of the small-signal gain due to its dependence on the pump intensity, *g*(*I _{p}*(

*t*,

*x*,

*y*)). This variation prevents transverse spatial and temporal components of the pump wave from becoming fully depleted (by conversion to signal and idler) simultaneously [Fig. 1(a)]; during propagation the most intense coordinates of the pump wave are depleted first. The weaker components are depleted later, only after a reversal of the transfer of energy from pump to signal (i.e., “backconversion”) has already set in at the peak. Begishev

*et al*. identified the existence in theory of a “conformal” signal profile, for every possible pump profile, that would allow simultaneous depletion at all spatio-temporal coordinates of the pump by providing the appropriate requirement for gain at each coordinate [25, 26]. However, in practice, such conformal pump/signal profile pairs are difficult to obtain, even in the simplest case of flattop profiles. Several works have shown that a flattened pump profile (without also a flattened seed profile) can already significantly improve conversion efficiency [20, 23, 27, 28]. Still, except in rare cases where the design of the pump light source allows implementation of temporal and spatial beam shaping without reduction of the available pulse energy, losses due to shaping techniques can outweigh the resulting improvement in conversion efficiency.

In OPCPA the problem of temporal variation of small-signal gain is further complicated by the signal chirp. First of all, the temporal mapping of signal frequencies creates preferential amplification of those frequency components that overlap temporally with the most intense part of the pump pulse. In addition, the mapping of signal frequency to temporal coordinate adds a time dependence of small-signal gain *g*(*I _{p}*(

*t*),Δ

*k*(

*t*)) through the local wavevector mismatch, Δ

*k*(

*t*) =

*k*–

_{p}*k*(

_{s}*t*) –

*k*(

_{i}*t*) [Fig. 1(b)]. The pump and phase-mismatch temporal profiles, therefore, cause spectral narrowing. The full phase-matching bandwidth of the amplifier may be preserved by keeping the seed pulse short enough, relative to the pump pulse, that the pump intensity remains more-or-less constant across the duration of the seed. However, for very short seed pulses only a small fraction of the pump pulse is depleted, and the increased amplifier bandwidth comes with a decrease in conversion efficiency. In this paper, we will investigate the problem of optimization in the time domain.

To gauge the maximum possible conversion efficiency of the amplifier, the concept of the temporal region of significant gain of the pump pulse is useful. Consider parametric amplification of a seed pulse with spectrum centered at signal frequency *ω _{s}* by a pump pulse with center frequency

*ω*such that

_{p}*ω*=

_{p}*ω*+

_{s}*ω*. If the idler is unseeded, prior to significant pump depletion the gain obeys the relation

_{i}*G*=

*I*(

_{s}*L*)/

*I*(0) = 1 + (Γ

_{s}^{2}/γ

^{2})sinh

^{2}(

*γL*), where $\gamma =\sqrt{{\mathrm{\Gamma}}^{2}-{(\mathrm{\Delta k}/2)}^{2}}$ and Γ

^{2}= 2

*ω*

_{s}ω_{i}*d*

_{eff}

^{2}

*I*/

_{p}*n*

_{s}n_{i}n_{p}*ε*

_{0}

*c*

^{3}(definitions provided later) [29]. If the gain is reasonably large,

*i.e*., Γ

*L*≫ 1,

which, in the case of perfect phase matching, can be recast in the form

In this case, since $\mathrm{\Gamma}~\sqrt{{I}_{p}},$ the gain is time-dependent and follows the pump pulse intensity profile: $G\left(t\right)~\mathrm{exp}\left[\sqrt{{I}_{p}\left(t\right)}\right].$
*G*(*t*) is plotted in Fig. 2(a) alongside *I _{p}*(

*t*) for a Gaussian pump pulse and unchirped, phase-matched seed. The peak gain is 100 and the plotted gain profile is normalized to one. Significant gain will only be possible for a seed pulse overlapped with the pump pulse within a central region about

*t*= 0. The shaded interval of Fig. 2(a) corresponds to the region

*t*< |

*t*| where

_{g}*G*(

*t*) ≥

*e*

^{−1}

*G*

_{0}, with peak gain

*G*

_{0}=

*G*(

*t*= 0). Let us calculate the centroid bounds, ±

*t*, corresponding to a gain

_{g}*G*(

*t*) =

_{g}*e*

^{-1}

*G*

_{0}, in the case of perfect phase matching. If we define the small-signal gain

*g*(

*t*) = 2Γ(

*t*)

*L*and

*g*

_{0}=

*g*(0), we obtain

For a Gaussian pump pulse, described by *I _{p}*(

*t*) =

*I*

_{0}· exp(-(

*t*/

*τ*

_{0})

^{2}) with full-width-at-half-maximum (FWHM) duration ${\mathrm{\Delta}t}_{p}=2{\tau}_{0}\xb7\sqrt{\mathrm{In}2},$ we may rearrange Eq. (3) to find

Since pump depletion will typically occur only where there is significant signal gain, *t _{g}* thus gives a measure of the maximum possible energy extraction from the pump. Note that

*t*is a function of the peak gain

_{g}*G*

_{0}, and that a power amplification stage with low

*G*

_{0}will have a larger gain centroid width than a pre-amplification stage with high

*G*

_{0}. Thus, the power amplier can extract more energy from the pump pulse, resulting in a higher maximum conversion efficiency. Figure 2(c) plots the temporal gain profile of an amplifier with unchirped signal pulse for several values of

*G*

_{0}, each curve normalized to 1, and Table 1 tabulates the pump centroid width. The difference in

*t*for pre- and power amplifiers implies that a different seed pulse chirp will optimize amplification at each stage. For example, a power amplifier stage with

_{g}*G*

_{0}= 10

^{2}has 1.5-times wider a region of significant gain than a preamplifier stage with

*G*

_{0}= 10

^{5}.

For a chirped-pulse parametric amplifier, we need to include also the temporally-varying wavevector mismatch, Δ*k*(*t*). According to Eq. (1), wavevector mismatch leads to a reduction in the gain at the wings of the signal pulse if phase-matched at the center of the pulse. In Fig. 2(b), the red (dashed) curve represents the gain profile of the unchirped signal pulse of Fig. 2(a), while the green (dot-dashed) curve includes the effects of the temporally-varying wavevector mismatch and the resulting decrease in the small-signal gain. The signal pulse is linearly chirped such that the edges of the phase-matching bandwidth, *ω* – *ω _{s}* = ±

*δ*

*ω*, where

_{b}*G*(

*ω*±

_{s}*δω*) =

_{b}*e*

^{−1}

*G*(

*ω*), are mapped to coordinates

_{s}*t*= ±

*t*. The new temporal region of significant gain, –

_{g}*t′*<

_{g}*t*<

*t′*where

_{g}*G*(

*t′*) =

_{g}*e*

^{−1}

*G*

_{0}, is narrower.

In principle, as the signal chirp is increased, the region -*t _{g}* <

*t*<

*t*will on average contain signal frequencies that are closer to the phase-matched frequency; Δ

_{g}*k*remains small across the bounds and

*t′*tends to

_{g}*t*, increasing the maximum possible conversion efficiency. However, the larger the chirp, the smaller the portion of the signal bandwidth that will fit within the region –

_{g}*t′*<

_{g}*t*<

*t′*and thus the smaller the effective amplifier bandwidth. In the other limiting case of vanishingly small chirp, all the seed colors see approximately the peak pump intensity; in this case,

_{g}*t′*is determined solely by phase matching, and

_{g}*t′*≪

_{g}*t*. The bandwidth approaches the full phase-matching bandwidth, but the energy extraction is limited since only a small temporal window of the pump can be depleted.

_{g}The trade-off between amplifier conversion efficiency and bandwidth makes the optimization of the seed chirp subject to the user’s desired characteristics of the pulse source. The maximization of peak power is a typical goal and will inform our optimization analysis. In this case, maximization of the efficiency-bandwidth product is desired. The optimal seed chirp can be estimated by analysis of Eq. (1). Figure 3 plots the efficiency-bandwidth product calculated from Eq. (1) for given values of the peak gain *G*
_{0}, varying the seed chirp and thus influencing the corresponding mismatch function Δ*k*(*t*). For later comparison, we consider the amplifier materials and geometry that will be discussed in section 3, and operate at degeneracy (*i.e*., *ω _{s}* =

*ω*=

_{i}*ω*/2). We approximate the wave-vector mismatch for the broadband amplifier to second-order (with first-order term vanishing due to the matched signal and idler group velocities) by Δ

_{p}*k*(

*ω*) = −(

*ω*–

*ω*)

_{s}^{2}

*β*

_{2}(

*ω*), where

_{s}*β*

_{2}(

*ω*) =

_{s}*∂*

^{2}

*k*/

*∂ω*

^{2}|

_{ωs}is the material group-velocity dispersion evaluated at central signal frequency

*ω*. The chirp is linear and each component

_{s}*ω*is mapped at

*t*= (

*ω*–

*ω*)/GDD (group delay dispersion). Since significant energy extraction from the pump occurs only where there is significant signal gain,

_{s}*efficiency*was calculated as the fraction of pump energy included in the bounds –

*t′*<

_{g}*t*<

*t′*and the

_{g}*bandwidth*corresponds to 2

*δω*/2

_{g}*π*, where we define

*ω*±

_{s}*δω*as the frequencies mapped to

_{g}*t*= ±

*t′*Figure 3 demonstrates a clear optimum chirp for each value of

_{g}*G*

_{0}, as expected, since efficiency increases and bandwidth decreases with increasing chirp. In order to compare the gain width of the unchirped, phase-matched amplifier to that of the chirped-pulse amplifier optimized for maximum efficiency-bandwidth product, the values of

*t′*corresponding to the optimum chirp at each

_{g}*G*

_{0}are tabulated in the second line of Table 1. The parameters 2

*t*/Δ

_{g}*t*and 2

_{p}*t′*/Δ

_{g}*t*exhibit the same behavior as

_{p}*G*

_{0}is varied: larger peak gains result in narrower regions of significant gain. Consistently,

*t′*<

_{g}*t*.

_{g}Note, for this analysis and in all theoretical examples of the paper we consider the case of a parametric amplifier with ultrabroad phase-matching bandwidth achieved through matching of signal and idler group velocities, with second-order approximation Δ*k*(*ω*) ~ (*ω* – *ω _{s}*)

^{2}. The analysis, however, can also be applied to the unmatched group velocity case, where Δ

*k*(

*ω*) ~ (

*ω*–

*ω*). In this case there will be different quantitative results, but similar trends.

_{s}We now turn to the matter of degradation of signal-to-noise ratio during amplification, when the amplifier is seeded by both signal and quantum noise. An important difference between signal gain and noise gain can be understood by means of the same conceptual picture used above [Fig. 2(a),(b)]. Initial quantum noise is stationary, *i.e*., seed fluctuations of all frequencies are present at all times [30]. Thus, phase-matched quantum noise is available at all times. The noise gain profile, therefore, is like the signal gain profile of an unchirped, phase-matched parametric amplifier, with Δ*k* = 0 at all *t*. As a result, in OPCPA, at each temporal coordinate of the interacting pulses the local gain of signal and noise photons is different: the noise temporal gain profile is determined solely by the local pump intensity, while the signal temporal gain profile is is determined by both the local pump intensity and the instantaneous wavevector mismatch. Simultaneous amplification of signal and noise is thus depicted by Fig. 2(b), where the unchirped-pulse amplifier gain profile (red, dashed) represents the noise, and the chirped-pulse amplifier gain profile (green, dot-dashed) represents the signal. Apart from *t* = 0, where both signal and noise photons experience Δ*k* = 0, there is higher gain for noise than for signal. In the example of Fig. 2(b), at *t* = ±*t _{g}*, the signal photons experience a gain <

*e*

^{−1}lower than that of coincident noise photons that are perfectly phase-matched.

The degradation in signal-to-noise ratio to be expected during amplification, arising from the difference between the total noise gain and total signal gain, can be estimated, therefore, by the area between the two gain curves of Fig. 2(b), and depends on the signal pulse chirp. Figure 3 (dashed line) plots the difference between noise and signal gain as a function of seed chirp for the case *G*
_{0} = 10^{6}. This difference approaches zero at a chirp slightly higher than that which maximizes the efficiency-bandwidth product. Optimum peak power, therefore, may be obtained with reasonably low degradation of signal-to-noise ratio.

A full summary of the amplifier properties determined by the signal pulse chirp is presented in Fig. 4. Here we consider Gaussian pump and seed profiles measured by their full width at half maxima, Δ*t _{p}*,Δ

*t*, respectively. With Δ

_{s}*t*= Δ

_{s}*t*/3, the signal pulse fits within a largely unvarying portion of the pump intensity profile [Fig. 4(a)]. As a result, there is little clipping of the signal pulse at the wings due to gain narrowing, and the effective amplifier bandwidth is nearly the full phase-matching bandwidth [Fig. 4(e)]. However, since the signal carrier frequency sweeps quickly in time, so does Δ

_{p}*k*(

*t*), and as a result the temporal gain profile of the signal is much narrower than that of the noise. The narrow signal gain profile means the conversion efficiency will be small, as only a fraction of the pump pulse will be depleted, and the large area between signal and noise gain profiles means the signal-to-noise ratio will strongly degrade after amplification. As the signal chirp increases [Fig. 4(b)-(d)], the signal gain profile widens relative to the noise gain profile, resulting in larger conversion efficiency and higher signal-to-noise ratio. On the other hand, since there is more clipping of the signal pulse during amplification, there is a narrower effective amplifier bandwidth [Fig. 4(f)-(h)]. The extreme is shown in Fig. 4(d), where Δ

*t*= 2Δ

_{s}*t*. Here, the very slow variation in signal frequency, resulting in a slow variation in Δ

_{p}*k*(

*t*), makes the noise and signal gain profiles nearly identical. However, since only a small portion of the signal bandwidth fits within the central part of the pump pulse, there is severe spectral clipping, and the effective amplifier bandwidth is much smaller than the phase-matching bandwidth [Fig. 4(h)]. The nearly equal signal and noise gain seen in Fig. 4(d) can be understood from another point of view: since the effective amplifier bandwidth covers only the flat, phase-matched central region of the phase-matching bandwidth, Δ

*k*is essentially zero over the full significant gain region of the pump pulse. Therefore, there is little preferential amplification of the noise.

## 3. Numerical simulations and results

The analytical and conceptual models introduced in the previous section allowed us to establish trade-offs between conversion efficiency, gain bandwidth and signal-to-noise ratio optimization in OPCPA. In order to test these predictions in the saturation regime of amplification, where significant pump depletion occurs and both efficiency and bandwidth are maximized, we numerically integrated the nonlinear coupled equations describing the parametric amplification process. Our simulations exclude the spatial transverse dimensions, allowing isolation of the temporal behavior. We consider an ultrabroadband OPCPA system that is known to be sensitive to superfluorescence [8]. The amplifier is pumped by a 9-ps FWHM Gaussian pulse at 1.047 *μ*m and seeded by a broadband pulse at 2.094 *μ*m, for operation around degeneracy. The interaction was calculated in a 3-mm long periodically-poled stoichiometric lithium tantalate (PPSLT) crystal. The seed had a supergaussian spectrum (*I _{s}*(

*ω*) ~ exp [−(

*ω*–

*ω*)

_{s}^{8}]) with FWHM bandwidth of 69 THz, which well matches the 15-fs (~2-optical-cycle) phase-matching bandwidth of the amplifier. These input parameters are close to the experimental conditions of Ref. [8], chosen for ease of comparison between simulations and experiments. We simulate the second-order nonlinear interaction occurring in the parametric amplifier between signal, idler, and pump fields, labeled

*m*=s, i, p, respectively, with carrier angular frequency

*ω*and wavenumber

_{m}*k*. We represent the total electric field as [31]:

_{m}$$\phantom{\rule{3.2em}{0ex}}{A}_{i}\left(z,t\right)\xb7{e}^{j\left({\omega}_{i}t-{k}_{i}z\right)}+$$

$${\phantom{\rule{3.2em}{0ex}}A}_{p}\left(z,t\right)\xb7{e}^{j\left({\omega}_{p}t-{k}_{p}z\right)}+\mathrm{c}\mathrm{.}\mathrm{c}\},$$

where *A _{m}*(

*z*,

*t*) denotes the complex field amplitude. The coupled equations describing the nonlinear interaction are derived from the nonlinear propagation equation [32],

applied on the total field *E*(*z*,*t*). Here, *D*(*z*,*t*) = *ε*
_{0}∫*ε _{r}*(

*τ*)

*E*(

*z*,

*t*–

*τ*)d

*τ*is the linear electric induction field accounting for the dispersion of the medium [33], and

*P*= 2

_{NL}*ε*

_{0}

*d*

_{eff}*E*(

*z*,

*t*)

^{2}is the nonlinear polarization, where

*d*is the effective second-order nonlinear coefficient. Equation 6 was solved by the split-step method [34] in the frequency domain. Periodic poling was accounted for by changing the sign of

_{eff}*d*along the propagation coordinate

_{eff}*z*. Numerical errors were reduced suitably by setting the propagation step size Δ

*z*= 0.1

*μ*m, much smaller than the used poling period, ∧ = 31.2

*μ*m. It is important to point out that the representation of fields given in Eq. 5 assumes that pump, signal and idler are three separate fields. Justifying this treatment, the amplifier we model employs a small non-collinear angle between signal and idler, used both to allow their separation after amplification (since they have opposite temporal chirp) and to avoid signal-idler interference for preservation of carrier-envelope phase of the signal.

In order to test the dependence of the optimization problem on the desired peak gain, the simulations were run for 3 configurations: *E _{p}*/

*E*= 10

_{s}^{2},

*E*/

_{p}*E*= 10

_{s}^{4}and

*E*/

_{p}*E*= 10

_{s}^{6}, where

*E*/

_{p}*E*is the initial pump to signal energy ratio. The parameter

_{s}*E*/

_{p}*E*is closely related to the maximum gain,

_{s}*G*

_{0},

*i.e.*, the gain that in principle could be obtained from the amplifier if all the pump energy could be transferred to the signal and idler at the peak of the pulse, by the approximate equivalence

*G*

_{0}≃

*E*/

_{p}*E*/2, since

_{s}*E*/

_{p}*E*≃

_{s}*I*(0)/

_{p}*I*(0). For each of the configurations we performed the following analysis:

_{s}- We changed the duration of the chirped seed pulse from 1 to 12.5 ps by introducing a group delay dispersion variable from 2500 to 30000 fs
^{2}, in steps of 2500 fs^{2}; - For each given seed duration, the amplification process was evaluated for a range of pump peak intensities,
*I*, experimentally corresponding to changing equally the pump and seed beam diameters at the input surface of the crystal. This action, increasing the pump intensity at a fixed propagation length, corresponds to increasing the gain, and increases the degree of amplifier saturation;_{p} - We deduced the pump intensity giving the highest conversion efficiency
*η*, where the conversion efficiency,_{max}*η*, is defined as the sum of amplified signal and idler energies divided by the initial pump energy, and at this intensity we evaluated the corresponding efficiency-bandwidth product (*η*· Δ_{max}*v*).

We note, since *η*(*I _{p}*) peaks and Δ

*v*(

*I*) slowly rises,

_{p}*η*and (

_{max}*η*· Δ

*v*)

*occur at approximately the same value of*

_{max}*I*, and

_{p}*η*· Δ

_{max}*v*is a close approximation to (

*η*· Δ

*v*)

*for each seed duration. Since we use a plane-wave model,*

_{max}*η*represents the fractional conversion of pump to signal and idler in the case of flattop signal and pump beam profiles with matched beam widths. A typical set of simulations is reported in Fig. 5, which shows the efficiency and bandwidth of the amplifier for various seed durations and pump peak intensities, calculated for the case

*E*/

_{p}*E*= 10

_{s}^{4}. Figure 5 confirms the behavior predicted in section 2 and presents additional behavior pertaining to the pump-depletion regime: (i) the amplified bandwidth decreases with increasing seed pulse duration, due to the progressively lower gain experienced by the wings of the spectrum; (ii) for each seed pulse duration there is an optimum peak intensity that guarantees the highest efficiency [squares in Fig. 5(a)] (higher intensities induce back-conversion at the peak of the pump pulse that exceeds additional conversion at the wings); (iii) as seed duration is increased, the maximum possible conversion efficiency increases; (iv) for a given seed duration, as the amplifier reaches maximum conversion the bandwidth increases with intensity due to saturation of gain at the center of the pulse and preferential amplification at the wings.

Figure 6 demonstrates the behavior of *η _{max}* · Δ

*v*. For example, the optimal seed duration when

*E*/

_{p}*E*= 10

_{s}^{4}(filled squares) is 8.5 ps, corresponding to Δ

*t*/Δ

_{s}*t*= 0.94 and contrasting with the cases

_{p}*E*/

_{p}*E*= 10

_{s}^{2}(open circles) and

*E*/E

_{p}^{s}= 10

^{6}(open squares), also calculated. This behavior in the depleted-pump regime of amplification matches the behavior predicted in section 2 for the non-depleted-pump regime: higher initial pump-to-seed energy ratios (

*i.e*., peak gains in section 2) call for shorter seed pulses. The trend with increasing

*E*/

_{p}*E*is in very good agreement with Table 1 for increasing

_{s}*G*

_{0}. Fitting the results of Fig. 6 with the analytical formula for the scaling of 2

*t*/Δ

_{g}*t*with peak gain, Eq. (4), using a variable correction coefficient,

_{p}*a*, and

*G*

_{0}=

*E*/

_{p}*E*/2, we find excellent agreement (Fig. 7). We find

_{s}where *a* = 2.1, and therefore (Δ*t _{s}*)opt ≃ 1.7(2

*t*). The extra factor of 1.7 can be attributed to the increase in temporal gain profile width due to amplifier saturation: gain at the peak of the pump pulse saturates before gain at the wings does, which pushes out the wings of the gain profile. Even with the effects of saturation, however, the simple non-pump-depletion-regime analysis of section 2 still recovers the scaling of (Δ

_{g}*t*/Δ

_{s}*t*)opt with

_{p}*E*/

_{p}*E*. The factor

_{s}*a*also accounts implicitly for the particular spectral intensity profile of the signal,

*I*(

_{s}*ω*),a characteristic of the amplifier not included in the analytical model. Eqs. (4) and (7), therefore, can be used to scale the optimal chirp from one peak gain to another.

To confirm also the behavior of signal-to-noise ratio as a function of the seed chirp as predicted by section 2, we redid the simulations corresponding to the data points of Fig. 6 for the *E _{p}*/

*E*= 10

_{s}^{6}case (open squares), adding initial noise distributions. To capture the features of superfluorescence noise, to the signal and idler fields we add initial noise fields modeled as complex stochastic random variables for each frequency component. These noise fields model, in a semiclassical picture, the incoming vacuum fluctuations that are transformed into superfluorescence by the amplifier. The frequency-domain real and imaginary parts of these complex stochastic variables are independent and Gaussian-distributed with zero-mean and variance that scales with the quantum energy, σ

^{2}∝

*h*̄

*ω*[30]. In this example, the amount of initial noise energy (signal plus idler) in frequencies falling within the signal and idler phase-matching bandwidth amounts to 3.8% of the initial signal pulse energy. Figure 6 plots the calculated degradation of signal-to-noise ratio as a function of seed chirp (triangles). As predicted, the signal-to-noise ratio performance improves significantly as the seed chirp increases, leveling off at close to the same value that maximizes the efficiency-bandwidth product. This completes the confirmation of a general conclusion of section 2 regarding OPCPA optimization: a small sacrifice in amplifier bandwidth relative to the full phase-matching bandwidth of the amplifier simultaneously allows optimal efficiency-bandwidth product and good robustness of signal-to-noise ratio.

Finally, as a proof that the numerical model we used for the optimization process is consistent with a real OPCPA system, we compared simulations with measured spectra and efficiencies from an amplifier with the same parameters and seeded by pulses with bandwidths corresponding to 32 fs [8]. Figure 8 illustrates the very good qualitative and quantitative agreement between numerical (a) and experimental (b) gain and bandwidth trends. In the same figure, we compare calculated (c) and measured (d) amplified spectra corresponding to the best total efficiency at the given seed chirp. This match provides confidence in the use of the trends and scaling rules found above to inform the design of an OPCPA system.

## 4. Conclusions and practical consequences

We have investigated the optimization of OPCPA when ultrabroad bandwidth, high gain, and suppression of superfluorescence noise are each important. Noteworthy conclusions of the analysis are listed below.

- In OPCPA, the simultaneous amplification of signal and background superfluorescence noise can be treated as simultaneous chirped-pulse and non-chirped-pulse amplification, respectively. Instantaneous signal amplification is sensitive both to the local pump intensity and instantaneous wavevector mismatch, whereas the instantaneous noise amplification is sensitive to only the local pump intensity.
- Like the conversion efficiency and bandwidth of the amplifier, the robustness of signal-to-noise ratio is tied to the ratio of initial signal and pump pulse durations, Δ
*t*/Δ_{s}*t*. Considering all three qualities simultaneously, as seed chirp is increased, the maximum conversion efficiency increases, the amplifier bandwidth decreases, and the signal-to-noise ratio increases. A small sacrifice in effective amplifier bandwidth relative to the full phase-matching bandwidth of the amplifier can significantly improve the signal-to-noise ratio._{p} - The optimal ratio of pump and seed durations also depends on the gain. The scaling of the optimum Δ
*t*/Δ_{s}*t*with the peak gain of the amplifier is well captured by an analytic formula, Eq. (7). For example, the ratio of optimum Δ_{p}*t*/Δ_{s}*t*values for_{p}*G*_{0}= 10^{2}and 10^{5}is 1.5.

These features of OPCPA have a number of practical consequences for the design of an amplifier. A high-gain parametric amplifier is often split into two or more stages, including pre-amplification, with high gain, and power amplification, with relatively low gain (for example, see Fig. 9.) The benefit of this practice can be understood by means of Figs. 2, 3, and 6: as peak gain decreases, both the maximum achievable conversion efficiency and maximum achievable efficiency-bandwidth product increase. Therefore, by placing most of the gain in a pre-amplifier stage, and only 10^{2} gain or lower in the final stage, the final peak power of the amplifier can be maximized.

Several realistic scenarios in multi-stage OPCPA are subject to problems in design if separate optimization of the seed chirp in each amplification stage is disregarded, and superfluorescence noise suppression is a concern. For example:

- In the schematic of Fig. 9(a), if the signal stretcher is designed to optimize chirp for maximum efficiency-bandwidth product in the power amplification stage, Δ
*t*/Δ_{s}*t*will be larger than optimal in the pre-amplifcation stage, causing unwanted spectral clipping that would result in a signal bandwidth too narrow to properly seed the power amplifier. To recover the lost bandwidth, the operator of this system has no choice but to drive the pre-amplification stage into heavy saturation [see Fig. 5(b)]. The resulting loss in efficiency in the pre-amplifier may not be problematic, but the heavy saturation would result in strong degradation of signal-to-noise ratio, since high gain would take place at the wings of the pulse, where there would be a large discrepancy between noise and signal gain._{p} - If, alternatively, the system is designed to maximize amplifier bandwidth, then optimization of the pre-amplifier stage is crucial. Since the gain centroid of the pre-amplifier is considerably narrower than the that of the power amplifier, the pre-amplifier will set a maximum seed chirp. Once the pre-amplifier is optimized for suitably-large bandwidth, Δ
*t*/Δ_{s}*t*will be too small in the power amplification stage, resulting in both a low efficiency and a poor signal-to-noise ratio._{p}

These problems can be avoided by placement of a third dispersive element between pre- and power amplification stages, to allow independent optimization of each stage. A good design strategy is illustrated by Fig. 9(b). First, the stretcher dispersion is chosen to optimize the signal chirp in the pre-amplification stage, GDD = *α*. Second, the compressor dispersion is chosen to optimize the signal chirp in the power amplification stage, GDD = *β*. Last, a third dispersive element, placed between amplification stages, is chosen to compensate for the dispersion mismatch between stretcher and compressor, GDD = *α* − *β*. An acousto-optic programmable dispersive filter (AOPDF) or other variable dispersive device is particularly suitable for this element, since it allows both experimental testing of the optimal chirp at the power amplifier and the ability to correct higher order dispersion terms.

We note, the empirical formula for the scaling of Δ*t _{s}*/Δ

*t*with

_{p}*E*/

_{p}*E*for the OPCPA system we modeled [Eq. (7), with

_{s}*a*= 2.1] depended on the particular characteristics of the amplifier, including the initial pump intensity profile, the initial seed spectrum, and Δ

*k*(

*ω*). While it is possible to reproduce the simulations and determination of a of section 3 for any system, alternatively, one may experimentally determine the appropriate seed chirp for either stage of a multi-stage amplifier and then use Eq. (7) to determine the optimum chirp at the other stages.

Finally, for when robustness of signal-to-noise ratio is crucial, we mention the importance of at least a small sacrifice in effective amplifier bandwidth relative to the full phase-matching bandwidth of the amplifier by keeping the ratio Δ*t _{s}*/Δ

*t*suitably large. This both allows the elimination of a significant discrepancy between signal and noise gain profiles and ensures that all temporal coordinates of the region of significant gain are seeded (an important consideration when the amplifier is driven into saturation). Since the final signal-to-noise ratio of a multistage amplifier depends on the suppression of noise amplification at each stage, it is important to correctly optimize the signal chirp at each stage.

_{p}This work was supported by the U.S. Air Force Office of Scientific Research (AFOSR) (FA9550-06-1-0468 and FA9550-07-1-0014) through the Defense Advanced Research Projects Agency (DARPA) Hyperspectral Radiography Sources program and the Progetto Rocca.

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