Gain spectra were calculated at critical wavelength degeneracy (CWD) in a collinear phase-matching geometry optical parametric amplification (OPA) process. The frequency bandwidth available through CWD-OPA is broader compared to the gain bandwidth obtained by the non-collinear OPA geometry. A solution for very broad bandwidth chirped pulse amplification based on partially deuterated DKDP (P-DKDP) crystals, pumped by pulsed green lasers, is proposed. 1.38 x 1014 Hz frequency bandwidth and peak intensity gain G ≈62 were calculated in a 5-mm long 58% deuterated DKDP crystal, pumped by 527-nm wavelength at 64-GW/cm2 intensity. Parametric amplification at CWD in few-mm thin P-DKDP crystals, pumped by picosecond pulses of nearly 100-GW/cm2 intensity, possesses a true potential for generating high energy laser pulses compressible to one-cycle duration.
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Optical parametric chirped pulse amplification (OPCPA) is one of the most powerful techniques for generating high-energy femtosecond laser pulses [1,2]. OPCPA combines the traditional chirped pulse amplification (CPA) with optical parametric amplification (OPA) [3–5]. Two pulses with nearly equal duration, a narrow band-width pump pulse and a broad band-width chirped signal pulse, are temporally and spatially overlapped in a nonlinear optical medium. Main advantages of this amplification method include high gain, broad spectral bandwidth, and high contrast of ultra-short pulse intensity related to the background radiation intensity, particularly in case of picosecond pump pulses. Many efforts were devoted to ultra-broad-band amplification supporting sub-10-fs pulses [6–10].
Broad bandwidths (tens up to hundred nm) can be obtained by two phase-matching geometries: collinear OPA at degeneracy, where the signal and idler group velocities are equal ; non-collinear optical parametric process, when the projection of the idler group velocity onto the signal wave-vector is equal to the signal group velocity [11,12]. In both cases, the first order term in the Taylor series expansion of the wave-vectors mismatch around the central phase-matching frequency is canceled (see Section 2).
Broader bandwidth can be obtained in a non-collinear geometry if the second wave-vectors mismatch term becomes zero. For a given pump wavelength, this condition is fulfilled in a non-collinear OPA (NOPA) at a certain central signal wavelength that corresponds to an ultra-broad bandwidth (UBB) signal amplification . Even broader gain bandwidths can be obtained in UBB phase matching regions by use of optimized NOPA geometries [13,14].
In broad gain bandwidth collinear chirp-compensation schemes, the broadband pump and signal waves, propagating collinearly, are chirped in appropriate way to ensure their instantaneous phase matching at all times [15,16].
In this paper, the conditions to get broad frequency bandwidth OPCPA in nonlinear crystals, pumped by narrow-band green lasers, are discussed. Theoretical analysis from Section 2 shows that very broad gain spectra can be achieved at critical wavelength degeneracy (CWD) in a collinear phase-matching geometry. Gain spectra for different type-I OPA geometries in DKDP crystals are calculated in Section 3. New solutions for very broad gain bandwidth chirped pulse amplification at CWD in partially deuterated DKDP (P-DKDP) crystals, pumped by pulsed green lasers, are proposed in Section 4. The broadest gain bandwidth available at CWD seems broad-enough to support the amplification of stretched pulses compressible to one-cycle pulse duration.
2. Theoretical analyses
OPA is a typical three-wave interaction nonlinear optical process, where the conservation of energy and phase matching of wave-vectors, kp,s,i, are required:Fig. 1 ). For a type-I parametric process in a negative uniaxial crystal, the signal and idler waves are ordinary polarized, whereas the pump wave is extraordinary polarized. The θ angle between the pump wave vector and the optical axis is derived from the phase-matching condition and Sellmeier equations for the ordinary and extraordinary refractive indexes of the nonlinear crystal.
The intensity gain of the amplified signal can be obtained by solving the coupled wave equations that describe the nonlinear amplification process [11,17]. Under the slowly varying envelope approximation, assuming no significant pump depletion and no initial idler beam, the intensity gain G is given by:
The full width half maximum (FWHM) phase-matching bandwidth (PMB), within the large-gain approximation (ΓL>>1), is determined by
The phase matching condition is fulfilled for ωs0 and ωi0 frequencies that satisfy the Eqs. (1). If the signal frequency changes to ωs0 + Δω, by energy conservation the idler frequency changes to ωi0 -Δω:
Broad phase-matching bandwidths are obtained when the wave-vectors mismatch Δk changes as slowly as possible with the change of the signal frequency. The wave-vectors mismatch around the phase-matching frequency is given by the Taylor series expansion:
In general, the first order term Δk(1) determines the phase matching frequency bandwidth, which can be calculated as:
Large band-width phase matching is obtained if the first order term Δk(1) can be canceled. In a collinear phase-matching geometry, it corresponds to the degeneracy (ωs = ωi), where the group velocity of the signal and idler are equal. In this case, the phase mismatch Δk must be expanded to the second order, giving
Out of degeneracy, this requirement if fulfilled for a non-collinear geometry [Fig. 1(b)] if the phase-matching conditions and the cancellation of the first order mismatch term are simultaneously satisfied. For type–I OPA, the refractive indexes of the signal and idler waves are a function of the wavelength through the Sellmeier equation for ordinary polarization, whereas the pump wave refractive index is determined by Sellmeier equatons for ordinary and extraordinary polarizations and phase matching angle θ to the optical axis. The internal crystal angle between signal and pump wave-vectors α, the angle between signal and idler wave-vectors β, and θ angle are determined by Fig. 1(b)].
A significantly increase of the gain bandwidth can be obtained if the second order mismatch Δk(2) term becomes zero too. The ultra broad band (UBB) NOPA conditions were derived from Eqs. (9) by equalizing to zero the second derivatives with ω of the phase-matching relations.
For a certain pump wavelength, if Eqs. (9) and (10) are simultaneously satisfied, θ, α, and β angles and a pair of signal and idler wavelengths corresponding to an ultra-broad band gain bandwidth can be calculated .
A broader gain bandwidth could exist if the third-order mismatch term in the Taylor series would be equal to zero simultaneously with first and second order terms. I derived this additional condition by equalizing to zero and by combining the third order derivatives with ω of the two phase-matching equations from (9)
Considering the number of six variables (ωp, ωs, ωi,, θ, α, β) involved in a parametric amplification process, the solution of the six-equations system composed by (9), (10), and (11) represents the conditions for obtaining the broadest gain signal bandwidth in a certain nonlinear crystal. The solution of this equations system consists in a collinear phase matching geometry (α = 0, β = 0) and degeneracy (ωs = ωi = ωp/2) at the critical wavelength, λc, where the group velocity dispersion of the signal/idler is
For a type-I OPA, the signal central wavelength for the broadest gain bandwidth is given byEqs. (9) and Sellmeier equations.
At critical wavelength degeneracy (CWD), the FWHM phase-matching bandwidth, Δν(4), is determined by the fourth order mismatch term, Δk(4), of the Taylor series expansion (6)
3. Gain bandwidths calculation
I considered a type-I parametric process in DKDP crystals. Calculations were performed under the approximation of plane waves, high parametric gain, negligible pump depletion, and quasi-monochromatic pump wave. I used Sellmeier equations for refractive indexes of DKDP crystals . For certain pump and signal wavelengths, in case of a non-collinear process, phase matching angles (θ, α, and β) for broad spectral bandwidth were calculated with Eqs. (9). For a certain pump wavelength, in case of UBB NOPA, the phase matching angles and the suitable signal/idler wavelengths are derived from Eqs. (9) and (10). In case of a parametric amplification process at CWD, the signal central wavelength was calculated using Eq. (13), whereas the θ angle is derived from the phase-matching condition for collinear OPA. The gain spectra were calculated by use of (1), (2), (3), and (9) equations.
The intensity gain spectra versus signal wavelength for an 80-mm length DKDP crystal (100% level of deuteration), at 527-nm pump wavelength and 1-GW/cm2 pump intensity, are shown in Fig. 2(a) . Curve A represents the gain spectrum for a collinear interaction at 0.95 μm central signal wavelength. The gain spectra B, C, and D are derived for a collinear degenerated parametric process (λs = λi = 1.054 μm), NOPA (at λs = 0.95 μm central signal wavelength), and UBB-NOPA (λs = 0.90 μm), respectively. Figure 2(b) shows the intensity gain for OPA at CWD, λp = 0.561 μm, λs = λi = 1.122 μm. Calculations are summarized in the Table 1 .
For a given crystal and a certain pump wavelength, in order to enlarge the signal gain bandwidth, higher pump intensity and shorter crystals are necessary. In case of UBB-NOPA, as can be derived from Eqs. (4) and (6), the gain bandwidth scales as
A FWHM gain bandwidth as broad as 4.89 x 1013 Hz was calculated for UBB-NOPA at 900 nm signal central wavelength. For generating 5-fs laser pulses, a gain bandwidth at least 2-times broader would be necessary. Pulse intensities of two order of magnitude higher (nearly 100 GW/cm2), crystals one order of magnitude shorter (few-mm length), and optimized NOPA geometries are necessary to reach the required bandwidth. Taking into account the DKDP damage threshold intensity, pump pulses of about 1 ps duration may be used. The peak intensity gain G2 significantly decreases by a factor
The FWHM-PMB of 7.71 x 1013 Hz, obtained for collinear OPA at CWD, is larger by a factor of 1.6 compared to the gain bandwidth for UBB-NOPA. Very broad gain bandwidths could be available simultaneously with a high parametric gain in relatively long nonlinear crystals.
The required pump wavelength of 561 nm, non-available for usual pulsed pump lasers, and relatively long central wavelength (1122 nm) of the seed pulse make the CWD-OPA process in DKDP crystals of limited practical interest. For parametric amplification at CWD, nonlinear crystals with the critical wavelength fitted to the emission wavelength of usual green pump lasers (e.g., second harmonic of Nd:YAG at 532 nm, Nd:glass at 527 nm, Yb:YAG at 515 nm) are necessary.
4. P-DKDP crystals application
Large aperture KDP and DKDP crystals are considered like most appropriate media for OPCPA for generating ultra-short laser pulses at PW power level [2,13]. Critical wavelengths of 986 nm and 1122 nm, for KDP and DKDP crystals, respectively, were calculated from Eqs. (13). In order to get the gain bandwidth at CWD-OPA, nonlinear crystals with a critical wavelength of 1030 nm would be necessary for Yb:YAG green lasers pumping (515 nm wavelength), whereas critical wavelengths of 1054 nm and 1064 nm are required in case of second harmonic Nd:glass and Nd:YAG pump lasers. Ultra-broad band seed pulses (700-1400 nm) with around 1-μm central wavelength can be obtained by spectral broadening techniques (e.g., in gas filled hollow crystal fibers ).
Refractive indexes of partially deuterated DKDP (P-DKDP) crystals are given by 18], with Eqs. (13) and (17), the deuteration level necessary to get the broadest gain bandwidth by CWD-OPA has been calculated. Deuteration levels of 58% and 40% were derived for P-DKDP crystals pumped by 527-nm and 515-nm waves, respectively. The corresponding θ phase-matching angles are 38.5 and 39.8 degrees.
Gain spectra for a 58% P-DKDP crystal pumped by a 527-nm wavelength Nd:glass laser at 1 GW/cm2, 4 GW/cm2, 25 GW/cm2, and 64 GW/cm2 pump intensities are presented in Fig. 3(a) . From intensity damage threshold considerations, these pump intensities can be used when pump laser pulses have durations in the range of nanosecond, hundred of picoseconds, ten picoseconds, and few picoseconds, respectively. The crystal lengths were chosen to get the same peak gain factor in all cases: 40 mm, 20 mm, 8 mm, and 5 mm. The calculated FWHM-PMB were 0.77 x 1014 Hz, 0.95 x 1014 Hz, 1.22 x 1014 Hz, and 1.38 x 1014 Hz, respectively. Similar results were obtained for a 40% P-DKDP crystal pumped by a 515-nm Yb:YAG [Fig. 3(b)].
Gain spectra for a CWD-OPA process in a 5-mm long 58% P-DKDP crystal in comparison with an UBB-NOPA process in a 100% deuterated DKDP crystal, both pumped by 527-nm green lasers at 64-GW/cm2, are shown in Fig. 4(a) . The crystal lengths were chosen to get the same peak gain. The calculated FWHM-PMB of 1.38 x 1014 Hz for the 58% P-DKDP crystal is about 1.23 times broader than 1.12x 1014 Hz in case of UBB-NOPA in the DKDP crystal. As derived from Eqs. (14) and (15), due to the difference between power scaling factors (1/8 and 1/6), for the same peak gain, the gain bandwidth in case of CWD-OPA changes slower with pump intensity and crystal length compared to an UBB-NOPA process. As a result, when pump intensity decreases, the ratio between the available gain bandwidths in the above mentioned OPA processes will be higher. The calculated gain frequency bandwidths were 1.33 and 1.27 times broader in case of CWD-OPA in P-DKDP crystals compared to UBB-NOPA in DKDP crystals at 4 GW/cm2 and 25 GW/cm2 pump intensity, respectively. At 1-GW/cm2 pump intensity, the FWHM-PMB of 7.7 x 1013 Hz calculated for a 40-mm long 58% P-DKDP is 1.4 times broader than 5.5 x 1013 Hz in case of UBB-NOPA in a DKDP crystal [Fig. 4(b)]. In conditions of similar peak gain and pump intensity, broader frequency bandwidth is available in case of CWD-OPA in P-DKDP crystals. Under conditions of equal peak gain (G ≈62), for CWD-OPA in a 10.7 mm long P-DKDP crystal, pumped by 14-GW/cm2 intensity, a FWHM-PMB of 1.12 x 1014 Hz was calculated, as broad as the one obtained for UBB-NOPA in a 5.4 mm long DKDP crystal at 64-GW/cm2 pump intensity [Fig. 4(a)].
When pumping with very high laser intensity of 1-2 ps pulse duration, 5-fs pulses up to 1-PW level could be generated by temporal compression of chirped pulses amplified by optimized UBB-NOPA processes in thin DKDP crystals .
Gain bandwidths available in amplifier stages based on OPCPA at CWD in P-DKDP crystals, pumped by 10-ps duration and 10-GW/cm2 intensity pump pulses, allow the generation of 5-fs laser pulses. Optical synchronization of pump and seed pulses becomes less critical than in case of 1-ps pulses overlapped in UBB-NOPA processes. When pumping with nanosecond pulses at ~1-GW/cm2 intensity, the available frequency bandwidth at CWD is broad enough to amplify chirped laser pulses compressible to sub-10 fs duration. Considering a pump intensity of 100 GW/cm2, and a 2.5-mm thin P-DKDP crystal, a peak gain about 4 and a frequency bandwidth broader than 1.5 x 1014 Hz were calculated. This gain bandwidth seems broad enough to amplify supercontinuum ps-stretched pulses, compressible down to one-cycle pulse duration.
Very broad bandwidth parametric amplification at CWD could be considered as an alternative to the optimized UBB NOPA. If I compare OPA at CWD with UBB NOPA, an advantage of parametric amplification at CWD could be the possibility to obtain similar gain bandwidths in amplifier stages based on longer nonlinear crystals, with a higher gain factor. Under conditions of similar parametric gains, a similar gain bandwidth could be obtained at CWD by pumping with lower laser intensity. On the other hand, in case of a purely collinear parametric amplifier architecture and type I near degeneracy nonlinear process, a difficult problem is the separation of signal and idler waves that have the same polarization. A quasi-collinear geometry with a very small angle between the pump and signal beams could be used. In case of high power ultrafast laser systems with large aperture signal beams (beam diameter of 10 cm or more), a drawback of this solution is the long distance propagation necessary for the signal and idler beams separation. Further experimental investigations are necessary to prove the validity of OPCPA at CWD in ultrashort pulsed laser amplifiers, particularly for PW-class femtosecond laser systems.
Very broad gain bandwidth for optical parametric amplification in non-linear crystals can be obtained at critical wavelength degeneracy in a collinear phase-matching geometry. By choosing the appropriate deuteration level of P-DKDP crystals, the critical wavelength can be adjusted to satisfy the conditions of parametric amplification at CWD when pumping with available pulsed green lasers (e.g., second harmonic of Nd:YAG, Yb:YAG, and Nd:glass). Frequency bandwidths achieved at CWD in P-DKDP crystals at 10-GW/cm2 intensity and 10-ps duration of pump pulses are broad-enough to amplify chirped laser pulses compressible down to 5 fs pulsewidth. Sub 10-fs duration laser pulses could be generated by ultrafast laser systems based on OPCPA at CWD in P-DKDP crystals pumped by nanosecond green laser beams at ~1 GW/cm2 pump intensity. By pumping few-mm long P-DKDP crystals with 1-ps green laser pulses at nearly 100-GW/cm2 pump intensity, the available gain bandwidth at CWD seems broad-enough to amplify stretched laser pulses compressible down to one-cycle pulse duration. OPCPA at CWD could be considered as a solution for ultrashort pulsed laser amplifier systems.
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