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Abstract
In this paper, we report that the angular dispersion of the output pulses in a nonlinear process can be efficiently compensated by using a cascaded prism(s) and short hollow-core fiber (HCF) configuration. Here, the prism(s) is used to suppress the angular dispersion and transform it into spatial chirp, while the HCF is used for removing this spatial chirp and the residual angular dispersion, which can also significantly improve the beam quality. The feasibility of this novel method is numerically and experimentally investigated with the ultra-broadband idler pulses centered at 1250 nm wavelength and generated by an LBO crystal based non-collinear optical parametric amplifier. The proof-of-principle experiment shows that the angular dispersion can be effectively removed and ultra-broadband idler pulses with good spectral quality and spatial profile can be obtained. The total transmission efficiency in the experiment is around 67% and the measured $M^2_x$ and $M^2_y$ can reach 1.12 and 1.04, respectively. To the best of our knowledge, this is the first reported ultra-broadband angular dispersion compensation scheme combining prism(s) and HCF, which can remarkably eliminate the angular dispersion while simultaneously possesses high efficiency, good spectral and beam spatial quality.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Angular dispersion widely exists in nonlinear processes, such as optical parametric amplification (OPA) and four-wave mixing in a non-collinear geometry, which results from the broadband phase-matching (i.e. the conservation of momentum and energy) [1–4]. On the one hand, the broadband amplification can be achieved by introducing a suitable angular dispersion in the input pulses [5–10]. On the other hand, the angular dispersion in the output pulses is generally unwanted because it prevents the application of the spatially dispersed pulses. For example, although the newly generated idler pluses in broadband non-collinear OPA (NOPA) feature higher temporal contrast and stable carrier-envelope-phase [11–13], the angular dispersion introduced by non-collinear geometry degrades the spatial-temporal quality and hence affects the direct application of it (amplification or transmission). Another example is the self-diffraction (SD), which has the same contrast enhancement capabilities as the commonly used cross-polarized wave generation [14], but does not require the polarizers [15]. Unfortunately, limited by the serious angular dispersion, the SD technology has not been truly used in the front-end for petawatt class femtosecond laser facility so far [16,17]. In order to remove the angular dispersion of these pulses and then utilize them, an ideal ultra-broadband compensation scheme which can simultaneously possess high efficiency, good spatial and spectral quality is desired. Further, it should be inexpensive and easy to implement.
In previous works, many schemes have been proposed in the terms of compensating the angular dispersion in nonlinear process (especially in OPA) [18–25]. The commonly used angular dispersion compensation schemes consist of an imaging system and angularly dispersive elements, such as prism [21], grating [22–28] or computer-generated hologram [29]. Here, the imaging system is used to image the plane of the nonlinear crystal onto the angularly dispersive elements, which then removes the angular dispersion of pulses. Although these ways show good performance, the residual angular dispersion still exists, especially when there is a mismatch in the imaging system. Another way to keep the idler pulses free from angular dispersion is to use an angularly dispersed signal pulses [19]. However, the structure of such solution is also complex and the amplified signal pulses with angular dispersion are generally unavailable for direct utilization. More recently, a novel method through imaging the idler pulses into a hollow-core photonic crystal fiber (HCPCF) was investigated for the angular dispersion compensation in a broadband degenerate OPCPA with slight non-collinear angle [20]. After the HCPCF, there is no residual angular dispersion and the beam quality is good. Unfortunately, since the limited numerical aperture of HCPCF and the large initial angular dispersion of pulses, the coupling efficiency and broadband compensation cannot be achieved simultaneously and the measured efficiency was only 2.2%. In addition, the schemes using single prism also have been employed in removing the angular dispersion of the signal pulses, which generated from SD [16] or four-wave mixing processes [30]. Nevertheless, the residual angular dispersion still exists and the output beam quality is not good.
In this work, a novel ultra-broadband angular dispersion compensation scheme adopting prism(s) followed by a short hollow-core fiber (HCF) is proposed (see Fig. 1(a)). Here, the prism(s) is used to remove most of the angular dispersion and transform it into spatial chirp firstly, while the HCF is used to eliminate this spatial chirp and the residual small amount of angular dispersion [20]. In addition, using HCF can also improve the beam quality. Compared with the schemes mentioned above, not only the residual angular dispersion is removed, but also a much higher coupling efficiency can be achieved compared to the fiber only scheme [20]. To evaluate the feasibility and superiority of this novel scheme, ultra-broadband idler pulses centered at 1250 nm which generated from an LBO crystal based NOPA are adopted for the numerical and experimental investigation. The proof-of-principle experiment shows that the angular dispersion of the ultra-broadband pulses can be efficiently compensated with a total transmission efficiency around 67%. Moreover, benefiting from the beam quality improvement in HCF, the compensated idler pulses can simultaneously feature good spatial and spectral quality, and the measured $M^2$ factor are $M^2_x = 1.12$ and $M^2_y = 1.04$, respectively. To the best of our knowledge, this is the first reported ultra-broadband pulses angular dispersion compensation scheme by combining prism(s) and HCF, which can simultaneously feature high efficiency, good beam profile and nice spectral quality.
Fig. 1. (a) Schematic setup of the novel angular dispersion compensation configuration. (b) Path of a ray through a prism. (c) The simulated influence of spatial chirp introduced by prism.
2. Design of the angular dispersion compensation scheme
As demonstrated in Fig. 1(a), in principle, the novel ultra-broadband angular dispersion compensation configuration should consist of a prism, a pair of cylindrical mirrors for down-collimation, a focusing mirror, a short HCF and a collimated mirror. First, most angular dispersion of the initial pulses is compensated and transformed into spatial chirp through passing a prism with appropriate apex angle and incident angle. Next, the cylindrical concave and convex mirrors are used to reduce the beam size along the direction with spatial chirp, which can make the beam ellipticity close to 1 and hence enhance the coupling efficiency in the HCF. Then, the quasi-collimated pulses are coupled into a short HCF via the focusing mirror (lens or concave mirror or off-axis parabolic mirror), which can eliminate both the spatial chirp and residual angular dispersion [20], and improve the beam quality [31]. After the HCF, the collimated mirror (lens or concave mirror or off-axis parabolic mirror) suppresses the beam divergence and outputs a quasi-collimated laser beam.
Compared with the schemes consisting of an imaging system and angularly dispersive elements [21–25], the residual angular dispersion is removed in our scheme. Moreover, since the spatial filtering effect in HCF, the beam quality can be improved in our configuration. Since the most angular dispersion is compensated by the prism before coupling into the HCF, a higher coupling efficiency and a better output spectral profile can be simultaneously achieved compared to the fiber only geometry [20].
There are two key points in this compensation scheme: (i) One is to choose a AR coted prism with appropriate apex angle and incident angle to ensure that most angular dispersion can be compensated at the full wavelength range. Fig. 1(b) demonstrate the dispersion characteristics of a prism. For a given apex angle $\theta$, the relationship between emergent angle $\theta _2$ and incident angle $\theta _1$ is given by:
3. Performance of the angular dispersion compensation scheme
In order to evaluate the feasibility and performance of our compensation scheme, a NOPA working at 925 nm central wavelength and pumped by a 532 nm Nd:YAG laser is adopted in the proof-of-principle experiment to provide the ultra-broadband idler pulses with angular dispersion [39]. As shown in Fig. 2(a), the signal pulses are generated by a nonlinear temporal filter combining mid-infrared OPA, noble gas-filled HCF and femtosecond SHG [40]. After a double-grating Öffner stretcher, the signal pulses with 210 nm spectral width are stretched to $\sim$3 ns pulse duration. The injected energy and beam diameter of the signal for the NOPA is about 10 $\mathrm {\mu J}$ and 1.2 mm, respectively. The pump laser with an energy of 70 mJ and a pulse duration of 4 ns is down-collimated to 1.5 mm diameter and imaged to the LBO crystal, which is 10 mm (W)$\times$10 mm (L)$\times$30 mm (T) in size and coated with anti-reflection films on both sides. The LBO crystal works in the XOY plane. The phase-matching angle and the inner non-collinear angle are ($\theta = 90^{\circ }$, $\phi = 13.78^{\circ }$) and 1.37 $^{\circ }$, respectively. As shown in Fig. 2(d), the spectral width of amplified signal pulses with an energy of 4.47 mJ is close to 200 nm. Correspondingly, the newly generated idler pulses with an energy of 2.59 mJ should have a spectral width close to 400 nm. However, limited by the relatively low gain of idler pulses and the insufficient dynamic range of spectrometer, the measured bandwidth of idler pulses is just close to 300 nm as demonstrated in Fig. 4(a).
Fig. 2. (a) The NOPA for providing the 1250 nm centered ultra-broadband idler pulses with angular dispersion. (b) Layout of the proof-of-principle experiment. (c) The non-collinear phase-matching geometry. (d) The measured spectrum of amplified signal pluses.
Considering the moderate angular dispersion of the idler pulses and also for the sake of simplicity, the cylindrical concave and convex mirrors are omitted in the proof-of-principle experiment. In the experimental setup (see Fig. 2(b)), uncoated fused silica prism with an apex angle of $\theta = 70^{\circ }$ is introduced to transform the angular dispersion into spatial chirp. A concave mirror with focal length of 250 mm is used to couple the laser beam into the HCF which placed in air. The length and the core diameter of the HCF are 25 cm and $\mathrm {300\ \mu m}$, respectively. After the HCF, a silica lens with focal length of 100 mm is utilized to collimate the laser beam without considering the chromatic aberration in the proof-of-principle experiment.
3.1 Analysis of the angular dispersion
The phase-matching geometry of NOPA is illustrated in Fig. 2(c), where the angle $\alpha$, $\beta$ and $\gamma$ refer to the non-collinear angle (internal, between signal and pump), the internal idler angle (between idler and pump) and the external idler angle (between idler and surface normal in air), respectively. Taking into account the conservation of momentum in NOPA process and the refraction on crystal-air surface, the external idler angle $\gamma$ can be written in the form:
Fig. 3. (a) The calculated external angle (left, red solid line) and angular dispersion (right, blue dashed line) of initial idler pulses. (b) The calculated emission angle difference between the central wavelength and the other wavelengths (left, red line) and residual angular dispersion (right, blue line) of compensated idler pulses.
To compensate this angular dispersion of idler pulses by prism(s) with an apex angle of $\theta = 70^{\circ }$, different compensation schemes are calculated according to Eq. (1). As demonstrated in Fig. 3(b), most of the angular dispersion can be compensated using a single prism with incident angle of $\theta _1=85^{\circ }$ (for 1250 nm). However, the quite large incident angle $\theta _1$ will result in a low transmission since the Brewster angle for the uncoated fused silica prism is around $55.4^{\circ }$ at 1250 nm. Therefore, two-prisms scheme ($\theta _1=64^{\circ }$ for 1250 nm) is designed for higher transmission efficiency. As can be seen from Fig. 3(b), there are no obvious differences in residual angular dispersion between these two schemes. Specifically, the calculated residual emission angle difference between the central wavelength and the other wavelengths can significantly reduce to below $0.04^{\circ }$, while the corresponding angular dispersion can decrease to -2$\sim$4 $\mathrm {\mu {}rad/nm}$. Hence, two-prisms geometry is adopted in the proof-of-principle experiment under all above considerations. In addition to the numerical analysis, the measured spectrum of initial idler pulses at different positions along angular dispersion direction (see the triangle markers from 1 to 5 in the right insert of Fig. 2(b)) are also demonstrated in Fig. 4(a), which can represent the entire bandwidth of initial idler pulses. It can be clearly seen that there is an obvious angular dispersion in the initial idler pulses. The NF profile of initial idler pulses at the output surface of LBO crystal is also illustrated in Fig. 4(b).
Fig. 4. (a) The measured spectrum of initial idler pulses at different position along angular dispersion direction. (b) The NF profile of initial idler pulses at the output surface of LBO crystal.
3.2 Proof-of-principle experiment
According to above analysis, the proof-of-principle experiment is demonstrated using a two-prisms geometry as shown in Fig. 2(b). Here, the most angular dispersion of idler pulses is firstly compensated by passing two consecutive prisms at $\sim 64^{\circ }$ incident angle under the consideration of both transmission efficiency and residual angular dispersion. Then, a silver-coated concave mirror (f=250 mm) placed at a small incident angle is used to couple the quasi-collimated idler pulses into a 250 mm long HCF with $\mathrm {300\ \mu m}$ core diameter, which can remove the residual angular dispersion and spatial chirp. Finally, the output idler pulses without both angular dispersion and spatial chirp are collimated via a lens with 100 mm focal length. Not far behind the collimated lens (CL), an iris is used to block the components which cannot be effectively coupled. To realize a fine adjustment, the prisms are mounted on a tip, tilt, and rotation stage, while the HCF is installed on two home-made 3-axis holders. On the one hand, the prism is finely rotated around $64^{\circ }$ incident angle to minimize the residual angular dispersion while maintaining high efficiency. On the other hand, the diameter of beam focus and the position of HCF input end are finely optimized through the beam profile measurement with a CCD. According to section 2., around $\mathrm {192\ \mu m}$ (in theory) $1/e^2$ long-axis diameter of focal spot is adopted at the input of the HCF. Moreover, the measured beam ellipticity at the input end of HCF is around 1.15 which has finite influence on coupling efficiency. In order to determine the performance of this novel angular dispersion compensation scheme, some key characteristics of idler pulses are examined by energy meter, NIR spectrometer and CCD.
The measured HCF total coupling efficiency (the ratio of energy at HCF output and HCF input) is 79%, while the effective coupling efficiency (the ratio of energy after iris and at HCF input) is 70%. Considering the transmittance of two-prisms is 98% and the reflectance loss of mirrors (silver-coated), the measured total transmission efficiency (the ratio of energy after iris and at prism input) is around 67%. Moreover, all spectral components can be efficiently coupled into the HCF, which can be demonstrated by comparing Fig. 4(a) and Fig. 5(a). Benefiting from the effective angular dispersion compensation before coupling into the HCF, more than one orders of magnitude improvement in efficiency is achieved compared to the fiber only scheme (only $\sim$2.2% for broadband case) [20].
Fig. 5. The performance of the cascaded prism(s) and HCF configuration. The spectrum measured at five different positions (a) and $M^2$ factor measurement (b) of the idler pulses after the angular dispersion compensation. The measured NF profiles at 25 mm (c) after HCF output, 100 mm (d) and 500 mm (e) after CL, and the FF profile (f).
In order to check the residual angular dispersion and spatial chirp, the collimated idler pulses are propagated more than 500 mm and then the spectrum at five different positions (see the circle markers in the right insert of Fig. 2(b)) are measured and demonstrated in Fig. 5(a). It can be seen that the idler pulses after compensation almost have no residual angular dispersion and spatial chirp, and possess a good spectral profile. Besides the good spectral quality, high efficiency and excellent angular dispersion compensation, the output beam profile is another important aspect. Thanks to the spatial filtering effect in HCF, which is insensitive to the input beam profile [31], a significant improvement on the beam quality is realized as compared to the initial beam profile (see Fig. 4(b)). The measured $M^2$ factor of the collimated idler pulses is $M^2_x = 1.12$ and $M^2_y = 1.04$ as illustrated in Fig. 5(b), while the FF profile is plotted in Fig. 5(f). Moreover, the measured NF profile at 25 mm after HCF output, 100 mm and 500 mm after CL are also illustrated in Fig. 5(c), Fig. 5(d) and Fig. 5(e), respectively. By comparing Fig. 5(d) and Fig. 5(e), there is no obvious degradation in NF quality when the transmission distance increases. All these NF and FF profile can clearly show that the beam quality of idler pulses after angular dispersion compensation is quite good.
All experiment results show that the prism(s) and short HCF based ultra-broadband angular dispersion compensation configuration can simultaneously feature excellent compensation capability, high transmission efficiency, good spectral quality and nice beam profile. We believe that more excellent comprehensive performance should be achieved when the single prism with suitable apex angle and AR coating, cylindrical concave and convex mirrors pair, as well as off-axis parabolic mirror is adopted. Moreover, this prism(s) and HCF based scheme can be easily generalized to other ultra-broadband nonlinear process with large angular dispersion and pulses energy up to 100 mJ level [41]. In the near future, we will investigate the amplification and compression of the compensated idler pulses. Corresponding feasibility and potential have been revealed in published works [42,43].
4. Conclusion
In this work, a novel ultra-broadband angular dispersion compensation scheme which can simultaneously achieve high efficiency, good spectral and beam quality is proposed by adopting a cascaded prism(s) and short HCF configuration. Most of the angular dispersion is firstly compensated and transform to spatial chirp by prism(s), and the residual part as well as spatial chirp are then removed by HCF. Moreover, the beam quality can be improved due to the spatial filtering effect in HCF. The feasibility of this novel scheme is demonstrated in a proof-of-principle experiment using ultra-broadband idler pulses with a central wavelength of 1250 nm which generated by an LBO based NOPA. The experiment results show that our scheme can not only compensate the angular dispersion over the entire spectrum with high efficiency, but also feature both good spectral quality and excellent beam profile. Specifically, the measured total transmission efficiency is around 67% while the measured $M^2_x$ and $M^2_y$ can reach 1.12 and 1.04, respectively. This prism(s) and HCF based configuration provide a reliable solution for ultra-broadband angular dispersion compensation and can be easily generalized to other nonlinear processes with large angular dispersion.
Funding
National Key Research and Development Program of China (2017YFE0123700); The Strategic Priority Research Program of the Chinese Academy of Sciences (XDB1603); National Natural Science Foundation of China (61925507); Program of Shanghai Academic Research Leader (18XD1404200); Shanghai Municipal Science and Technology Major Project (2017SHZDZX02); Shanghai Sailing Program (19YF1453100); Natural Science Foundation of Shanghai (20ZR1464600); Youth Innovation Promotion Association of the Chinese Academy of Sciences; International Partnership Program of Chinese Academy of Sciences (181231KYSB20200040).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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