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In this paper we theoretically study the spatial characteristics of optical parametric amplification (OPA) or chirped pulse OPA (OPCPA) pumped with a phase-aberrated beam. Due to the fact that pump-to-signal phase transfer caused by walk-off effect is highly gain-dependent, in a high gain OPA system the signal beam-quality will be significantly degraded even for a weak walk-off, accompanied with beam tilting and converging or diverging. It is demonstrated that an OPA configuration with walkoff-compensated crystal pair is capable of reducing the phase transfer and hence ensuring signal beam-quality, which may be of importance for designing high-energy OPCPA systems.
©2008 Optical Society of America
1. Introduction
Optical parametric chirped pulse amplification (OPCPA) is a new technique that combines optical parametric amplification (OPA) with chirped pulse amplification (CPA) [1, 2]. By exploiting the second-order nonlinear polarizations, laser systems based on OPA or OPCPA exhibit more flexibility over conventional femtosecond lasers based on broadband solid-state gain media such as Nd:glass and Ti:sapphire [3]. Due to the unique features of efficient conversion, high and broadband gain [4–7], OPCPA technique has been well recognized to be an effective approach to generate high-power femtosecond pulses [8]. Recently, few-cycle pulses with power of ~10 TW [9] and extremely high-power ultrashort pulses approaching petawatt-level [10] have been demonstrated by using OPCPA systems.
For the applications in strong-field physics, laser beam-quality and pulse contrast are of the key issues for CPA laser systems [11]. It is normally regarded that OPCPA technique is capable of ensuring the optical quality of amplified signal field [12, 13]. To date, pulse contrast of OPCPA systems have been extensively studied, and various methods have been proposed to increase the pulse contrast on picosecond time-scale [13–17]. However, less attention has been paid to the beam-quality issue for OPCPAs. It might not be a problem for most low-energy OPCPA systems where the pump laser sources were diffraction-limited or quasi-phase-matching (QPM) crystals were used [18, 19]. The situation, however, will be very different for high-energy OPCPA systems where the high-energy pump lasers are typically far from diffraction-limit [10, 20]. In fact, several papers did consider the beam-quality issue in OPCPA systems [21, 22], which mainly studied the influences of pump beam-profile and dephasing on the OPA gain and signal beam-quality. A theoretical model for the dephasing effects due to angular deviation from ideal phase matching was developed, and the impact of the beam angular content on small signal gain and on conversion efficiency in the strongly depleted regime was evaluated numerically [22]. On the other hand, the phase aberration of a laser beam usually dominates the beam-quality, while intensity profile only has a minor effect. To evaluate the influence of pump beam-quality on OPCPA, therefore, it is necessary to take the pump phase aberration into account since the pump phase will be imprinted partly onto both the idler and signal fields during the OPA process if walk-off or group-velocity mismatch is presented [23, 24]. In this paper we theoretically study the spatial characteristics of optical parametric amplification (OPA) pumped by a phase-aberrated beam. In the spatial domain, the nonlinear process of OPCPA is identical to that of OPA, thus in most parts of our paper we do not make an explicit difference between OPA and OPCPA. Our study was motivated by the recent experimental progress in high-energy OPCPA systems [10, 25]. In these multi-hundred TW OPCPA systems, large-apertured Nd:glass lasers were used as the pumping sources, which are non-diffraction-limited. Our numerical studies may be helpful for understanding the performance of these non-diffraction-limited laser pumped OPCPA systems.
The paper is organized as follows. In section 2, the numerical model is simply discussed. Detailed theoretical studies on OPA with pump phase modulations and walk-off effect are presented in section 3. Comparisons on signal beam-quality, gains, as well as wave-front distortions with different pump beam-quality factors are presented in section 4. In section 5, walkoff-compensated crystal pairs are adopted and studied in order to improve the performance of OPA. Finally, conclusions are given in section 6.
2. Numerical model
The coupled-wave equations for OPA in spatial domain, with type-I phase matching, collinear configuration, can be treated in analogy with that in time domain [8]. The equations that govern the evolution of the three parametrically interacting waves are
Where E j(z, x) is the field envelope normalized to the input pump field E 0, and j=s, i, p refers to the signal, idler, and pump waves, respectively. For the sake of simplicity, a computational less demanding one transverse dimensional model is used in our simulations. Gaussian pump beam is assumed throughout the paper, though different forms of phase modulation might be involved. The space variable x is normalized to the radius of pump beam waist w, and Δk is the wave-vector mismatch among the three waves. We define the nonlinear length as LNL=nλp/(πχ (2) E 0), which is a measure of pump intensity. The signal and idler beam walk-off refer to the pump beam, so walk-off terms only for these two beams appear in the equations; signal walk-off length (L sp) equals that of the idler (L ip) in type- I phase matching collinear configuration, and is defined as Lsp=Lip=w/ρ, where ρ is the walk-off angle, and the ratio of L sp (L ip) to crystal length L indicates the practical walk-off magnitude of the OPA process in the nonlinear crystal. The diffraction lengths (L 2j) are defined as L 2j=2πw2/λj. Diffraction involved in the case of large beam size has a minor effect. For example, a beam with waist radius of 2 mm and wavelength 1 μm will have a diffraction length of ~24 m which is much longer than the typical crystal length used in existing systems. Thus the diffraction effects are ignored in this paper. In the calculations, we assume that the initial signal beam has a Gaussian shape with uniform phase, and Δk is set to be zero since its effect is mainly reducing signal gain and conversion efficiency, which is fully discussed [22], and also the investigation of the effect of walk-off separately can explicitly clarify its impact on output signal beam. The standard split-step method is used to solve the nonlinear equations numerically.
3. The rule of phase transfer from pump to signal
Firstly the impact of walk-off on signal phase with phase-aberrated beam pumping is addressed. For simplicity, we assume a form of the pump beam with sinusoidal phase modulation as E(x,0)=E 0 exp(-x 2+ia sin2(nπx)), where the parameters a and n correspond to the modulation amplitude and spatial frequency, respectively. For simplicity, we set n to be different integers in the following calculations, but it is not necessary to do so in general. More general form of pump phase aberration can be characterized by beam-quality M 2 p, and the OPA with such a non-diffraction-limited pump beam will be discussed in section 4.
Figure 1 shows the phase distributions of output signal and idler in a single OPA stage. It can been seen that under the condition of no walk-off, both the period and amplitude of phase modulation on the idler wave are identical with that of the pump beam, while the phase of the signal retains its initial uniform shape, indicating that the OPA just amplifies the signal energy and will not affect signal beam-quality, which is highly required in laser systems employing OPA and OPCPA. However, if walk-off exists, the originally consistent phases of the pump and idler will slip spatially, thus their phase difference (i.e. the phase of the signal) becomes disordered too, and the output signal bears part of the pump aberrations. Since part of the pump beam aberration transfers to the signal beam, the modulations imposed upon the idler decreases.
Fig. 1. Calculated output signal (a) and idler phase (b) in the cases with (red and blue curves) and without (black curve) walk-off. The red and blue curve correspond to two different nonlinear lengths of LNL=0.23L and LNL=0.15L respectively. Other parameters used in the simulations: Lsp=5L, L=10 mm, a=0.5, n=2, and Es(0)=10-9.
The amplitude of the phase modulations imposed onto the signal depends on the OPA gain. If the signal amplification is very small, such as the situation that the input signal intensity is comparable with that of the pump beam, i.e. difference frequency generation process, the signal phase will not be affected. If the signal is highly amplified, the signal phase will be modulated significantly, and larger amplitude of phase modulation on the signal beam may be resulted with larger pump intensity, as shown in Fig. 1. Meanwhile, as the pump beam has a Gaussian shape, the amplitude of phase modulation at the central part of the signal is larger.
Fig. 2. (a). The transverse distributions of signal phase after a single OPA stage with different walk-off lengths, and (b) the phase distributions of pump beam (black curve) and output signal further amplified by a second OPA. The red and blue curves correspond to Lsp/L of 5 and 100, respectively. Each OPA with the same nonlinear length LNL=0.23L, L=10 mm, a=0.5, n=2. Input signal of the first OPA is: Es(0)=10-9.
Furthermore, this signal phase degradation depends on the walk-off magnitude. Figure 2(a) shows the transverse phase distributions of signal beam for different values of Lsp/L by varying the walk-off angle (i.e., the width of the pump beam is fixed). We can see that as Lsp/L gets smaller, the amplitude of the induced signal phase increases. When Lsp/L=5, the phase of the signal is strongly modulated and is comparable to that of the pump; even for very small walk-off magnitude such as Lsp/L=100, the situation corresponding to a practical walk-off magnitude in KDP crystal with length of 10 mm and beam diameter of 30 mm, the modulation is still noticeable, which suggests that the influence of walk-off may not be ignored in high-energy OPCPA systems. Since the pump beam and signal beam will slip away farther with larger walk-off magnitude, the phase modulations of the signal beam rise steadily. In a multi-stage OPA system, signal gain can be very high, and the phase modulation of the output signal at the last stage may be greater than that of pump at moderate walk-off, as shown in Fig. 2(b). Since the signal phase modulations will be greatly intensified in further OPAs, the originally negligible effect of phase transfer in a single OPA at small walk-off might be considered in a multi-stage OPA system. Inasmuch as the output signal will be greatly disturbed when pump phase is modulated, pump phase modulations should be restrained for those OPAs with walk-off.
Fig. 3. (a). Calculated output signal phase distributions with different pump phase modulation amplitudes, a=0.1, 0.5, and 2.0 correspond to the black, red and blue curves, respectively. n=2. (b) signal phase distributions with different modulation frequencies of the pump beam, n=2, 3, 5, 9 refers to the black, red, blue and green curves, respectively. a=0.5. Other parameters used in the simulations: LNL=0.23, Lsp=5L, L=10 mm, and Es(0)=10-9.
The dependence of signal phase degradation on the modulation amplitude and spatial frequency of the pump beam phase are shown in Fig. 3. The amplitude of the induced signal phase increases with that of the pump beam, which is trivial and can be anticipated. The spatial frequency-dependence, however, shows that the amplitude of signal phase modulation vibrates with increasing spatial frequency.
4. The effect of walk-off and pump beam-quality in phase transfer
As validated in section 3, walk-off may lead to pump-to-signal phase transfer, which will surely affect the OPA gain and signal beam-quality. However, we should point out that phase transfer may also occur in the absence of birefringent walk-off if a non-collinear phase matching configuration is adopted.
To discuss the OPA performance affected by pump beam-quality, it is of practical significance to construct a non-diffraction-limited pump beam. Let us consider a one-dimensional Gaussian beam with a slowly varying phase aberration: Ep(0,x)=E 0 exp[-x 2+iΦ(x)], where Φ(x)=α exp[-(x/2.5)2]+β exp[-(x/2.5)4], α and β are constant factors of the phase aberration. Such a form of aberration may well describe, in the sense of beam-quality factor, the typical laser beams obtained from high-power laser facilities with spatial filtrations [26]. For comparison, three sets of the parameters with α=0, β=0; α=94.0, β=94.7 and α=177.0, β=174.7 corresponding to beam-quality factors (M 2 p) of 1, 10 and 20, respectively, are chosen in the calculations. Thereafter we use the term of M 2 p in the text as well as graph labeling for the sake of simplicity, but one should note that it corresponds to the above specific form of phase aberration and parameter sets of α and β.
Fig. 4. Numerical results of the output signal gain (a) and beam-quality M 2 s (b) in a single stage OPA with various walk-off in the regime of pump non-depletion (Es(0)=10-9); while graphs (c) and (d) corresponding to a strongly pump-depleted regime (Es(0)=10-2). Other parameters used in the simulations: LNL=0.15L; L=10 mm.
The simulation results about output signal beam-quality and OPA gain or conversion efficiency are summarized in Fig. 4. As expected, the gain (efficiency) of OPA pumped with an ideal Gaussian beam (M 2 p=1) is the highest compared to all the cases of non-diffraction limited pump beams. Since pump-to-signal phase transfer occurs in OPA with walk-off, both the larger pump beam-quality factor M 2 p and stronger walk-off will result in worse signal beam-quality and lower OPA gain (efficiency) in both the pump non-depletion and depletion regimes. The effects of pump beam-quality on OPA performance are very significant at moderate or strong walk-off (Lsp/L<5). For very weak walk-off (Lsp/L>10), pump beam-quality affects little on signal beam-quality and OPA gain in a single OPA stage. Eventually in the absence of walk-off as those in OPAs with QPM, ideal Gaussian pump beam and non-diffraction limited pump beams will give identical OPA performances, and OPA gain or efficiency depends only on pump intensity.
Fig. 5. Calculated output signal phases for two different walk-off lengths Lsp/L. Parameters used in the simulations: M 2 p=10; LNL=0.15L; L=10 mm; Es(0)=10-9.
Fig. 6. Calculated output signal phases for two different spatial frequencies with pump phase modulation in the form of (a) Φ(x)=(nx)2 and (b) Φ(x)=(nx)3. n=1, 2 refer to the black and red curve, respectively. Other parameters used in the simulations: LNL=0.15L; L=10 mm; Es(0)=10-9.
Since the beam-quality factor is mainly determined by the phase of a beam, we shall discuss the phase distribution of the output signal when walk-off exists. As presented in Fig. 5, the output signal phase shows an interesting feature of phase tilting, and the larger the walk-off magnitude (smaller Lsp/L), the larger the amplitude of the signal phase as well as its slope. The situation in the pump depleted regime also shows similar feature (not plotted). In these demonstrating calculations, different magnitudes of walk-off are obtained by assuming different walk-off angles while keeping the beam width fixed. Such a phase tilting may also be introduced by other forms of pump beam aberration. For example, Fig. 6(a) presents the phase distribution of the signal beam obtained by assuming Φ(x)=(nx)2 on the pump beam, where phase tilting is obvious. As a comparison, Fig. 6(b) gives the corresponding results obtained with Φ(x)=(nx)3, which doesn’t show phase tilting. The results given by Fig. 6 imply that the occurrence of signal phase aberration depends on the form of phase aberration on the pump beam. This phase aberration of the signal beam is the combined effect of spatial walk-off and non-diffraction-limited beam pumping, which causes the phase differences of the pump and idler beam, as is clarified in section 3.
To characterize the wave-front distortions of the signal beam completely, both the curvature and tilting of the output signal wave are calculated. The cylindrical curvature of the wave fronts was characterized by use of the method described by Siegman [27]. The tilting angle β is the first moment in spatial-frequency domain, defined by
Where the symbol s is the spatial frequency and
The evolutions of the signal tilting angle along the crystal for two different values of Lsp are presented in Fig. 7. It shows that larger walk-off angle leads to larger tilting angle with the same nonlinear interaction length; the tilting angle increases monotonically and begins to saturate at the corresponding walk-off length after which the interacting waves separate considerably. This is consistent with the phase transfer analysis.
Fig. 7. Calculated tilt angle of the signal with increasing interaction length in the crystal. The black and red curve correspond to Lsp=2 cm and 1.5 cm, respectively. Other parameters used in the simulations: M 2 p=10; LNL=0.15L; Es(0)=10-9.
Figure 8 compares the output signal tilt angles and curvatures for pump beam-quality factor of M 2 p=1 with those of M 2 p=10 and M 2 p=20. From these figures we can see that the overall rules are: (a) when the pump laser is an ideal Gaussian beam (M 2 p=1), both the signal beam direction and curvature will retain their initial incident values, even if walk-off exists; (b) however, the signal beam direction and curvature will be affected with a non-diffraction-limited pump laser, and the larger the M 2 p, the worse the signal beam wave-front distortion. As an example for moderate walk-off (Lsp/L=5) and pump beam-quality (M 2 p=10), the output signal will be quite divergent with a radius of curvature ~3×102 m if we take a crystal length L=10 mm typically. This suggests that the signal beam should be adjusted for collimation before its further amplification in later OPAs.
Fig. 8. Calculated signal beam tilt and curvature versus walk-off length at three different values of pump beam-quality. Parameters used in the simulations: LNL=0.15L; L=10 mm; Es(0)=10-9.
Wave-front distortion was previously studied in the process of second-harmonic generation (SHG) by Smith, et al., [28]. It should be pointed out that the wave-front distortion discussed in this paper is different from that of the previous research. Whereas the work of Smith emphasizes that a non-zero phase-velocity mismatch ch Δk may lead to intensity-dependent phase shifts of the generated second-harmonic wave if there exists walk-off, or linear absorption, etc., in this paper we discuss the process of OPA and set the condition of phase-matching (Δk=0). The mechanism of signal wave-front distortion discussed here is due to the effect of pump-to-signal phase transfer, which depends on not only the walk-off but also the phase aberration of pump beam.
The pump intensity or OPA gain can also affect the phase transfer hence the signal beam quality in the OPA process, as shown in Fig. 9. It shows that signal beam-quality M 2 s gets very large at large walk-off magnitude, and higher OPA gain may also lead to worse signal beam-quality with the same walk-off magnitude. Since OPA crystals with walk-off and high overall gain are involved, signal beam-quality would be an unavoidable problem in multi-stage large-aperture OPA systems due to the phase transfer effect. To improve the signal beam-quality, in the next section we will study an OPA configuration using the walkoff-compensated OPA crystal pair that was previously proposed to increase the angular and frequency acceptance bandwidths of frequency conversions [29, 30].
Fig. 9. Calculated signal beam-quality versus walk-off length at two different pump beam intensities. Star symbol: LNL=0.15L; square symbol: LNL=0.1L. Other parameters used in the simulations: M 2 p=10; L=10 mm; Es(0)=10-9.
5. The improvement of signal beam-quality
The OPA configuration using walkoff-compensated crystal pair is schematically shown in Fig. 10. The optical axis of a crystal pair are set to be antiparallel, such that the walk-off direction is reversed in the second crystal [29], and the incident pump and signal beams are centered at the same position at the entrance of the first crystal. Originally the walkoff-compensated crystal pair was studied for the purpose to increase the interacting length hence the conversion efficiency, and to increase the acceptance angle compared with a single crystal of the same overall length [29], while in this paper we suggest this OPA configuration to reduce the transfer of aberrations and hence to improve the signal beam-quality. In principle, reducing the phase transfer is very closely related to increasing the pump acceptance angle [30].
The two crystals in walkoff-compensated OPA configuration may be considered as a single OPA stage, in which both the crystal length and pump intensity for the two crystals should be the same for optimally reducing the pump-to-signal phase transfer. As shown in Fig. 11, we presented the results of output signal phases in walkoff-compensated configuration and ordinary two-stage OPAs (i.e., identical orientation of the crystal axis). As discussed in section 4, the signal phase output from the first OPA crystal experiences a distortion showing tilting due to walk-off and pump phase aberration. The signal phase tilting will be enhanced in the second OPA crystal if its optical axis is parallel to that of the first OPA crystal. The situation will be very different in the walkoff-compensated OPA configuration, however, the signal phase tilting induced from the first OPA crystal can be compensated in the second antiparallel OPA crystal, and the phase of the final amplified signal nearly resumes its initial uniform shape.
Fig. 11. Calculated output signal phases in a two-stage OPA system. Red curve: walkoff-compensated configuration; blue curve: ordinary two OPAs. The signal phase of the first-stage OPA (black curve) is also given. Parameters used in the simulations: M 2 p=10; LNL=0.23L; Lsp=5L; L=10 mm; Es(0)=10-9.
Fig. 12. Calculated total OPA gain, signal beam-quality, beam tilt and curvature versus walk-off length in a two-stage OPA system with and without walkoff-compensation. Parameters used in the simulations are the same as those in Fig.11.
Figure 12 summarizes the detailed simulation results about the walkoff-compensated configuration and ordinary two-stage OPAs. Fig. 12 (a) shows that the overall OPA gain in the walkoff-compensated configuration is higher than that of ordinary two-stage OPAs. Due to further degradation of the signal phase, the phase tilting, beam-quality and curvature will be enlarged in the second OPA crystal if its optical axis is parallel to that of the first OPA crystal. In the walkoff-compensated configuration, the final signal tilt angle and curvature may resume their initial values, while the beam-quality factor M 2 s can be controlled to a satisfactory level by setting a proper value of Lsp/L. The deviation of M 2 s from M 2 s=1 may still occur in such a walkoff-compensated configuration, as shown in Fig. 12(b), if the walk-off is sufficiently large (i.e., smaller Lsp/L). This can be attributed to the distortion on the intensity profile of the signal. In the case of large walk-off, the performance may be enhanced by displacing the incident signal relative to the pump.
6. Conclusion
We have studied the detailed spatial behaviors of the OPA process including the effects of pump phase aberrations and walk-off. By numerical simulations in the spatial domain, we have shown that: (1) an OPA process is sensitive to the overall phase of the interacting waves, in which the transfer of phase distortions on the pump beam to the signal is unavoidable when walk-off exists. The effect of pump-to-signal phase transfer should be taken into account even at very weak walk-off; (2) Due to the combined effect of walk-off and non-diffraction-limited beam pumping, the amplified signal wave-front will be aberrated. Meanwhile, OPA gain, conversion efficiency, and signal beam-quality will also be affected; (3) The phase degradation of the signal beam in a conventional multi-stage OPA system could be more severe, and sometimes may be even worse than that of the pump beam, thus the walk-off compensation scheme is preferable; and (4) the walkoff-compensated crystal pair has been studied in order to improve the performance of OPA, which is capable of reducing the phase transfer. The signal beam-quality can be controlled to a satisfactory value in a walkoff-compensated OPA crystal pair. We suggest walkoff-compensated OPA configuration in highenergy large-aperture OPCPA system, in which the walkoff-compensated crystal pair should be designed as a single OPA stage and may be regarded as a simple replacement of QPM structures.
Acknowledgments
This work was partially supported by the Natural Science Foundation of China (grant Nos. 60538010, 60725418 and 10776005), and the National Basic Research Program of China (973 Program) (Grant No. 2007CB815104).
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