Abstract

An ultrafast coherent long wavelength MIR source based on difference frequency generation was demonstrated. An average power of 2.5 mW at ∼18 μm was achieved. The angular distribution of the generated MIR source under the condition of tight-focusing limit shows the onset of conical emission of the MIR beam due to on-axis phase mismatching.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

Small coherent sources of radiation in the MIR spectral region of 5 to 20 µm would open up the field of infrared spectroscopy in the “molecular fingerprint region” because almost every molecule has a distinctive signature in its absorption spectra at these wavelengths [1]. Along with applications in environmental monitoring and medical applications, this spectral region is becoming increasingly important for gas sensing, especially the explosive and poisonous gases, which depends on their absorption spectroscopy in this region [2].

To date, there are two types of lasers that can cover the entire MIR region. One is the quantum cascade laser (QCL) [3] that cannot be mode-locked to generate ultrashort pulses for applications such as time-resolved studies and frequency comb spectroscopy. The other is nonlinear frequency down conversion such as difference frequency generation (DFG), optical parametric oscillation (OPO) and optical parametric amplification (OPA), which can generate tunable short pulse MIR radiations [4]. Furthermore, the tunability range of a single MIR QCL is not as abroad as that achieved from nonlinear frequency down conversion processes. Nonlinear frequency down conversion could generate MIR frequency comb sources that combining broad spectral bandwidth, high spectral resolution, and spatial coherence. Gambetta et al. reported on the generation of MIR pulses with a maximum average optical power of 4 mW and wide tunability from 8 to 14 μm DFG in GaSe from an Er: fiber laser oscillator [5]. Beutler et al. realized wider tunability extending from 5 to 17 μm with an average power of 69 mW at 6 μm in the femtosecond regime [6]. Zhou et al. achieved an average power of 5.04 mW MIR pulses centered at 11 μm based on DFG, and it is tunable from 7.4 to 16.8 μm [7]. Steinle et al. achieved a highly stable 350 fs laser system with a gap-free tunability from 1.33 to 2.0 μm and 2.13 to 20 μm by combing OPO, OPA and DFG technology [8]. More recently, intra-pulse difference frequency generation (IDFG) has become a popular way to generate long wave MIR due to its simplicity and passive carrier-envelope-phase stability [9–11], MIR ranging from 6.4 to 16.4 μm with a high average power of 100 mW [9], MIR ranging from 6 to 18 μm with a high average power of 0.5 W [10], and an ultra-broadband coherent radiation coverage ranging from 2 to 17 μm were achieved [11].

At Waterloo, the Ultrafast Laser Group is developing an efficient two-color, short-pulse fiber laser system for the generation of long wavelength MIR femtosecond radiation longer than 16 µm [12–15]. Other MIR sources generated with DFG, tend to fall in power at wavelengths longer than 16 µm. We have measured the power, conversion efficiency and beam profile as a function of focal spot size of the pump down to the tight focusing limit, defined as the pump spot size equal to the generated MIR wavelength. The maximum average MIR power we have achieved to date is 2.5 mW at a peak wavelength of 17.5 µm with a focal spot of 27 µm. The average pump power was 1.6 W giving a photon conversion efficiency of 2.6%. We also noted that when the spot size is reduced to the order of 20 µm, the power decreases and the beam profile is no longer Gaussian. To the best of our knowledge, our results are the first experimental evidence of cone emission in the case of DFG in the tight focusing limit in a crystal.

The experimental system uses a compact, high-average-power, sub-picosecond, two-color fiber-coupled, chirped pulse amplification (CPA) system that has been previously reported by the authors [16]. Two colors are selected from a spectral continuum by using a chirped fiber Bragg grating to filter out a 60 nm spectral region. The two remaining spectral regions are then amplified in a two-stage Yb: fiber amplifier system. The system delivers two-color pulses with the total average power of ~2.3 W at a 65 MHz repetition rate. The power ratio of the short and long color could be changed from 2:1 to 4:1. The total power also slightly changes when the power ratio is varied. MIR power is optimized when the product of the two powers is maximized [14]. We determined the optimum power ratio for MIR generation by rotating the waveplates. We then measured the power, spectrum and pulse duration of the resulting two pump pulses that generated the maximum MIR signal. The resulting short and long wavelength pumps had 1.6 W and 0.8 W average power respectively. The FWHM pulse durations were measured to be 900 and 600 fs, respectively. The spectra of the two pump pulses are shown in Fig. 1.

 

Fig. 1 Input spectrum of the two colors.

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The two output pulses from the fiber amplifier system were sent to the MIR generation setup which is shown in Fig. 2). The nonlinear crystal used to generate a MIR spectrum spanning from 16 to 19 µm (with the FWHM bandwidth between 1.15 and 1.3 μm) by DFG is a 1-mm thick Gallium Selenide (GaSe). GaSe was chosen because of its high nonlinearity in DFG and high transmission in the long wavelength MIR region. According to the calculations, the group velocity mismatch (GVM) between the shorter-wavelength pump and the long-wave MIR radiation was calculated to be 287 fs/mm. Since the pulse duration of the pump is 900 fs, then the pulse splitting length was calculated to be 3.13 mm, which is 3 times our crystal length. Thus we can achieve optimal efficiency with our crystal.

 

Fig. 2 Experimental setup for MIR generation.

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According to Type I and Type II phase matching condition in the DFG process, the short and long color need to be e- and o-polarization in the crystal separately corresponding to p- and s-polarization for the orientation of the nonlinear crystal. Thus, the two pump pulses are set to be orthogonally polarized as follows. The two pulses exit the pulse compressor of the amplifier system vertically separated. A 90-degree periscope was used to both change the polarization of one of the beams and make the two beams the same height. The two beams were then combined using a broadband, thin-film polarizing beam splitter (PBS), which is optimized for s-polarization and so has near 100% reflectivity for s-polarization and only 80% transmission for p-polarization. To have maximum pump power, the short wavelength color was left as the s-polarized beam. Then there was another periscope used to rotate polarization by 90 degrees of the two colors simultaneously one more time before they both entered the crystal. The 1/e2 diameters of two beam spots were both measured to be 3.6 ± 0.5 mm. The two beams were focused into the GaSe crystal by convex lenses with various focal lengths of 10 cm, 15 cm, 20 cm, and 45 cm, and this corresponds to a 1/e2 focused beam radius of 17.8 ± 2.9 µm, 26.7 ± 4.3 µm, 35.6 ± 5.9 µm, and 80.1 ± 13.1 µm respectively in the diffraction limited focusing case. The output MIR power was collected by a 90° off-axis parabolic mirror (diameter = 50.8 mm, reflected focal length = 50.8 mm). Then the generated MIR source, the residual pump, and the signal light passed through a 5-mm thick germanium (Ge) filter, through which the pump and signal light was blocked, only the MIR pulse could pass through. The germanium filter had a measured transmission at 17.5 µm of 25%.

The power was measured by fully collecting the MIR after the Ge filter with a large size off-axis mirror and a thermal detector. The power was maximized at each focusing case by optimizing the position of the focal spot, the beam overlap, and incidence angle. The power was maximized at two angles of incidence corresponding to the type I and type II phase matching. As expected, typed II phase matching resulted in a 10% higher power because of lower Fresnel losses. A 50-grooves/mm monochromator fitted with a nitrogen-cooled mercury cadmium tellurium (HgCdTe) detector was used to measure the spectrum of the MIR.

The angular distribution of the generated MIR source in the tight focusing limit was studied using a small detector scanned across the beam in two perpendicular directions. The diameters of the MIR beam are equal within our experimental uncertainty so we assume a circular beam. This circular symmetry agrees with previously published DFG results [9,11] To determine the propagation angle of the generated MIR, we measured the MIR beam profile at three axial positions, where is 37 mm, 46 mm and 55 mm away from the center of the GaSe crystal.

Figure 3 shows the experimental power as a function of focal spot size compared to a numerical simulation of power under two different cases of beam walk-off. By taking the 75% MIR absorption from the filter into consideration, the powers were measured to be 1.06 mW, 2.5 mW, 1.58 mW, and 0.35 mW when the beam radii were 17.8 ± 2.9 µm, 26.7 ± 4.3 µm, 35.6 ± 5.9 µm, and 80.1 ± 13.1 µm in the type 2 phase matching condition, respectively. The simulation is based on J. R. Morris and Y.R. Shen’s model [17]. This theory was worked out for far infrared in the tight focusing limit where the paraxial wave approximation is no longer valid and the generated beam no longer propagates as a Gaussian beam. With tightly focused Gaussian beams, the optimum phase matching does not occur when Δk=0 as in the plane wave approximation. As the focal spot size decreases to the dimensions of the generated wavelengths, the axial phase matching is not optimized and the beam begins to propagates as conical emission.

 

Fig. 3 MIR power in Type 2 phase matching changes with different focal spot size. The solid curves are numerical simulations in different walk-off angle conditions (ζ = 0 and −0.06 rad) and the points are experimental data. The error bars represent the measurement uncertainties.

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According to Morris and Shen model [17], the angular distribution of the output MIR is written as:

dP±(θ,ϕ)dΩ=c2πcos2θ|ET±(kT=ωcsinθ(x^cosϕ+y^sinϕ)|2,
where the P±(θ,ϕ)is the forward and backward mid-infrared power generated by DFG; θ,ϕ are the polar angle and azimuth angle, respectively; ET± is the generated field outside the slab; the subindices “+” and “-” denote forward and backward propagating directions; kT is the transverse vector of the propagating waves;ω is the center frequency of MIR and c is the speed of light; x^ and y^ are unit vectors used in Cartesian coordinates, in which x-direction is parallel to the optical platform, y-direction is vertical to the platform; ET± is derived by solving the wave equations for the MIR generation by DFG where the two input beams were set as Gaussian beams rather than plane-wave beams. In the final expression of Eq. (1), the parameter ET± implicitly contains three important variables which are the position of the input beam waist, the thickness of the crystal, and the focused beam radius. In the simulation, the thickness of the crystal affects the power of the MIR, usually the longer the crystal length, the higher the power. In the simulation, we set the thickness of the crystal to match the 1 mm thickness of the crystal and vary the position of the beam waist in the crystal to maximize the MIR power. The optimum position for the beam waist was the center of the crystal for any beam diameter. We then varied the size of the beam waist for two different beam walk-off angles. In our case, the walk-off angle ζ in the GaSe crystal was calculated to be −0.06 rad, consequently, the simulation curve was done with ζ = 0 and −0.06 rad to show the effect of beam walk-off. As expected, the simulation shows the walk-off angle strongly affects the optimum focused spot size and resulting MIR power, because the interaction length decreases as the walk-off angle increases. For the case of ζ = −0.06 rad, the total average power of the MIR reaches the highest when the beam size decreases to 29.3 μm, which fits well with the experimental results. It should be noted that there is a discrepancy between our and Morris’s model. Morris’s results only show the case when the walk-off angles are small (when ζ = −0.01 or −0.02 rad). At these very small walk-off angles, the optimum focusing size will be shorter than the wavelength. However, their paper does not show the case when the walk-off angle was even larger than −0.02 rad. As we know, the optimum focusing beam size drifts with the walk-off angle. Consequently, when the walk-off angle is a little bit larger, such as the case when ζ = −0.06 rad, the optimum focusing beam size will be longer than the wavelength.

The Morris and Shen model also predicts the propagation angle of the cone emission as a function of spot size. This model was verified by the conical emission observed in THz emission from filaments in the air [18]. The theory shows that conical emission occurs for the condition of the diameter less than the wavelength and the wavelength less than the coherence length given by the phase matching condition. To the best of our knowledge, the results reported here are the first comparison with the theoretical model for MIR wavelengths generated by difference frequency mixing in a nonlinear crystal. We compare the measured beam profiles with the simulated results in Fig. 4. Results in Fig. 4 show that when the focused beam radius is larger than the generated MIR wavelength, the beam shows a Gaussian profile. However, a central dip in the beam profile of the MIR beam appears when the beam radius is 17.8 ± 2.9 µm. This dip indicates that conical emission emerges when the beam size is close to the scale of the MIR wavelength or at the tight-focusing limit. This is the same spot size that results in lower power, indicating that the power is reduced by the imperfect phase matching on-axis where the intensity is highest.

 

Fig. 4 Experimental results and simulations about the Angular Distribution of the Generated MIR with different focused beam radius of (a) 17.8 ± 2.9 µm (b) 26.7 ± 4.3 µm (c) 35.6 ± 5.9 µm (d) 80.1 ± 13.1 µm.

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According to the simulations, the generated power and beam profile are affected mostly by phase mismatching and the input Gaussian spot size but also by the boundary effects of refraction and Fresnel losses on the generated MIR. The boundary effects give relatively equal transmission within 30 degrees of divergence, consequently, the Gaussian beam profile and the phase mismatching are the main factors affecting the final angular distribution in our experimental data. This would change if we had focused yet tighter. We choose not to focus tighter as two-photon absorption (TPA) would start to play a role as the intensity further increased. To determine if TPA was limiting the results, we measured the power of the two colors before and after the GaSe crystal when the beam was in the tight focusing limit. The long color transmission remained constant. However, the higher power short wavelength pump pulse did exhibit a saturated transmission at powers above 0.8 W. Even with this slight TPA, the MIR power increased with increasing pump power.

The MIR spectra were also measured in the different focusing cases to determine if the bandwidth is affected by the loss of phase matching. The resulting spectra are shown in Fig. 5. The bandwidth of the MIR spectra is slightly reduced from 1.3 μm FWHM for the case of the largest focal spot size to 1.15 µm FWHM at the smallest spot size. However, in our case, the imperfect phase matching is not the limit to the measured bandwidth. The MIR bandwidth is just half of what would be expected with Fourier transform limited pulses and perfect phase matching (2.3 µm). For our focusing conditions, our full pump bandwidths were within the phase matching bandwidth. Consequently, the reduced bandwidth is more of a result of the pump pulses not being transform limited.

 

Fig. 5 MIR Spectrum with Different Focal beam radius of (a) 17.8 ± 2.9 µm (b) 80.1 ± 13.1 µm.

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In conclusion, we have reported a DFG-based efficient long wavelength MIR source with an average power of 2.5 mW at ∼18 μm, corresponding to the difference frequency of two colors with the peak wavelengths of 1024 and 1088 nm and average powers of 1.6 and 0.8 W, respectively. The full width at half maximum of the MIR spectra varies between 1.15 and 1.3 μm. The angular distribution of the generated MIR source indicates that a dip emerges in the middle of the MIR beam spot when the focused beam size gets close to tight-focusing limit due to the phase mismatching in the propagating direction. This can be seen as the onset of conical emission of the MIR based on DFG. We demonstrated photon conversion efficiency of over 2% at these long wavelengths in the MIR, when focusing to near the tight focusing limit. In the future, we will increase the peak power of the fiber laser system, by reducing the pulse duration and the repetition rate. We will then be able to reach higher pump intensities without having to reduce the spot size to the tight focusing limit.

Funding

Natural Sciences and Engineering Research Council of Canada; National Natural Science Foundation of China (61527822, 61735005); Natural Science Foundation of Beijing Municipality (4182054).

Acknowledgments

We thank Prof. Yuen-Ron Shen for his help with the guidance of the theoretical work. Xinyang Su also acknowledges the financial support from the China Scholarship Council.

References

1. I. T. Sorokina and K. L. Vodopyanov, eds., Solid-State Mid-Infrared Laser Sources (Springer-Verlag, 2003).

2. C. Bauer, A. K. Sharma, U. Willer, J. Burgmeier, B. Braunschweig, W. Schade, S. Blaser, L. Hvozdara, A. Müller, and G. Holl, “Potentials and limits of mid-infrared laser spectroscopy for the detection of explosives,” Appl. Phys. B 92(3), 327–333 (2008). [CrossRef]  

3. Y. Yao, A. J. Hoffman, and C. F. Gmachl, “Mid-infrared quantum cascade lasers,” Nat. Photonics 6(7), 432–439 (2012). [CrossRef]  

4. V. Petrov, “Frequency down-conversion of solid-state laser sources to the mid-infrared spectral range using non-oxide nonlinear crystals,” Prog. Quantum Electron. 42, 1–106 (2015). [CrossRef]  

5. A. Gambetta, N. Coluccelli, M. Cassinerio, D. Gatti, P. Laporta, G. Galzerano, and M. Marangoni, “Milliwatt-level frequency combs in the 8-14 μm range via difference frequency generation from an Er:fiber oscillator,” Opt. Lett. 38(7), 1155–1157 (2013). [CrossRef]   [PubMed]  

6. M. Beutler, I. Rimke, E. Büttner, P. Farinello, A. Agnesi, V. Badikov, D. Badikov, and V. Petrov, “Difference-frequency generation of ultrashort pulses in the mid-IR using Yb-fiber pump systems and AgGaSe(2),” Opt. Express 23(3), 2730–2736 (2015). [CrossRef]   [PubMed]  

7. G. Zhou, Q. Cao, F. X. Kärtner, and G. Chang, “Energy scalable, offset-free ultrafast mid-infrared source harnessing self-phase-modulation-enabled spectral selection,” Opt. Lett. 43(12), 2953–2956 (2018). [CrossRef]   [PubMed]  

8. T. Steinle, F. Mörz, A. Steinmann, and H. Giessen, “Ultra-stable high average power femtosecond laser system tunable from 1.33 to 20 μm,” Opt. Lett. 41(21), 4863–4866 (2016). [CrossRef]   [PubMed]  

9. I. Pupeza, D. Sánchez, J. Zhang, N. Lilienfein, M. Seidel, N. Karpowicz, V. Pervak, E. Fill, O. Pronin, Z. Wei, F. Krausz, A. Apolonski, and J. Biegert, “High-power sub-two-cycle mid-infrared pulses at 100 MHz repetition rate,” Nat. Photonics 9(11), 721–724 (2015). [CrossRef]  

10. T. P. Butler, D. Gerz, C. Hofer, J. Xu, C. Gaida, T. Heuermann, M. Gebhardt, L. Vamos, W. Schweinberger, J. A. Gessner, T. Siefke, M. Heusinger, U. Zeitner, A. Apolonski, N. Karpowicz, J. Limpert, F. Krausz, and I. Pupeza, “Watt-scale 50-MHz source of single-cycle waveform-stable pulses in the molecular fingerprint region,” Opt. Lett. 44(7), 1730–1733 (2019). [CrossRef]   [PubMed]  

11. Q. Wang, J. Zhang, A. Kessel, N. Nagl, V. Pervak, O. Pronin, and K. F. Mak, “Broadband mid-infrared coverage (2-17 μm) with few-cycle pulses via cascaded parametric processes,” Opt. Lett. 44(10), 2566–2569 (2019). [CrossRef]   [PubMed]  

12. A. M. Al-Kadry and D. Strickland, “Generation of 400 μW at 17.5 μm using a two-color Yb fiber chirped pulse amplifier,” Opt. Lett. 36(7), 1080–1082 (2011). [CrossRef]   [PubMed]  

13. A. Al-Kadry and D. Strickland, “Generation of 400 μW at 17.5 µm using a two-color Yb fiber chirped pulse amplifier: erratum,” Opt. Lett. 43(2), 316 (2018). [CrossRef]   [PubMed]  

14. M. Hajialamdari and D. Strickland, “Tunable mid-infrared source from an ultrafast two-color Yb:fiber chirped-pulse amplifier,” Opt. Lett. 37(17), 3570–3572 (2012). [CrossRef]   [PubMed]  

15. M. Hajialamdari and D. Strickland, “Tunable mid-infrared source from an ultrafast two-color Yb:fiber chirped pulse amplifier: erratum,” Opt. Lett. 43(2), 353 (2018). [CrossRef]   [PubMed]  

16. X. Su, T. Hoang, P. Long, Y. Zheng, and D. Strickland, “A Compact High-Average-Power Femtosecond Fiber-Coupled Two-Color CPA System,” IEEE J. Sel. Top. Quantum Electron. 24(5), 1–5 (2018). [CrossRef]  

17. J. R. Morris and Y. R. Shen, “Theory of far-infrared generation by optical mixing,” Phy. Rev. A 15(3), 1143–1156 (1977). [CrossRef]  

18. Y. S. You, T. I. Oh, and K. Y. Kim, “Off-Axis Phase-Matched Terahertz Emission from Two-Color Laser-Induced Plasma Filaments,” Phys. Rev. Lett. 109(18), 183902 (2012). [CrossRef]   [PubMed]  

References

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  1. I. T. Sorokina and K. L. Vodopyanov, eds., Solid-State Mid-Infrared Laser Sources (Springer-Verlag, 2003).
  2. C. Bauer, A. K. Sharma, U. Willer, J. Burgmeier, B. Braunschweig, W. Schade, S. Blaser, L. Hvozdara, A. Müller, and G. Holl, “Potentials and limits of mid-infrared laser spectroscopy for the detection of explosives,” Appl. Phys. B 92(3), 327–333 (2008).
    [Crossref]
  3. Y. Yao, A. J. Hoffman, and C. F. Gmachl, “Mid-infrared quantum cascade lasers,” Nat. Photonics 6(7), 432–439 (2012).
    [Crossref]
  4. V. Petrov, “Frequency down-conversion of solid-state laser sources to the mid-infrared spectral range using non-oxide nonlinear crystals,” Prog. Quantum Electron. 42, 1–106 (2015).
    [Crossref]
  5. A. Gambetta, N. Coluccelli, M. Cassinerio, D. Gatti, P. Laporta, G. Galzerano, and M. Marangoni, “Milliwatt-level frequency combs in the 8-14 μm range via difference frequency generation from an Er:fiber oscillator,” Opt. Lett. 38(7), 1155–1157 (2013).
    [Crossref] [PubMed]
  6. M. Beutler, I. Rimke, E. Büttner, P. Farinello, A. Agnesi, V. Badikov, D. Badikov, and V. Petrov, “Difference-frequency generation of ultrashort pulses in the mid-IR using Yb-fiber pump systems and AgGaSe(2),” Opt. Express 23(3), 2730–2736 (2015).
    [Crossref] [PubMed]
  7. G. Zhou, Q. Cao, F. X. Kärtner, and G. Chang, “Energy scalable, offset-free ultrafast mid-infrared source harnessing self-phase-modulation-enabled spectral selection,” Opt. Lett. 43(12), 2953–2956 (2018).
    [Crossref] [PubMed]
  8. T. Steinle, F. Mörz, A. Steinmann, and H. Giessen, “Ultra-stable high average power femtosecond laser system tunable from 1.33 to 20 μm,” Opt. Lett. 41(21), 4863–4866 (2016).
    [Crossref] [PubMed]
  9. I. Pupeza, D. Sánchez, J. Zhang, N. Lilienfein, M. Seidel, N. Karpowicz, V. Pervak, E. Fill, O. Pronin, Z. Wei, F. Krausz, A. Apolonski, and J. Biegert, “High-power sub-two-cycle mid-infrared pulses at 100 MHz repetition rate,” Nat. Photonics 9(11), 721–724 (2015).
    [Crossref]
  10. T. P. Butler, D. Gerz, C. Hofer, J. Xu, C. Gaida, T. Heuermann, M. Gebhardt, L. Vamos, W. Schweinberger, J. A. Gessner, T. Siefke, M. Heusinger, U. Zeitner, A. Apolonski, N. Karpowicz, J. Limpert, F. Krausz, and I. Pupeza, “Watt-scale 50-MHz source of single-cycle waveform-stable pulses in the molecular fingerprint region,” Opt. Lett. 44(7), 1730–1733 (2019).
    [Crossref] [PubMed]
  11. Q. Wang, J. Zhang, A. Kessel, N. Nagl, V. Pervak, O. Pronin, and K. F. Mak, “Broadband mid-infrared coverage (2-17 μm) with few-cycle pulses via cascaded parametric processes,” Opt. Lett. 44(10), 2566–2569 (2019).
    [Crossref] [PubMed]
  12. A. M. Al-Kadry and D. Strickland, “Generation of 400 μW at 17.5 μm using a two-color Yb fiber chirped pulse amplifier,” Opt. Lett. 36(7), 1080–1082 (2011).
    [Crossref] [PubMed]
  13. A. Al-Kadry and D. Strickland, “Generation of 400 μW at 17.5 µm using a two-color Yb fiber chirped pulse amplifier: erratum,” Opt. Lett. 43(2), 316 (2018).
    [Crossref] [PubMed]
  14. M. Hajialamdari and D. Strickland, “Tunable mid-infrared source from an ultrafast two-color Yb:fiber chirped-pulse amplifier,” Opt. Lett. 37(17), 3570–3572 (2012).
    [Crossref] [PubMed]
  15. M. Hajialamdari and D. Strickland, “Tunable mid-infrared source from an ultrafast two-color Yb:fiber chirped pulse amplifier: erratum,” Opt. Lett. 43(2), 353 (2018).
    [Crossref] [PubMed]
  16. X. Su, T. Hoang, P. Long, Y. Zheng, and D. Strickland, “A Compact High-Average-Power Femtosecond Fiber-Coupled Two-Color CPA System,” IEEE J. Sel. Top. Quantum Electron. 24(5), 1–5 (2018).
    [Crossref]
  17. J. R. Morris and Y. R. Shen, “Theory of far-infrared generation by optical mixing,” Phy. Rev. A 15(3), 1143–1156 (1977).
    [Crossref]
  18. Y. S. You, T. I. Oh, and K. Y. Kim, “Off-Axis Phase-Matched Terahertz Emission from Two-Color Laser-Induced Plasma Filaments,” Phys. Rev. Lett. 109(18), 183902 (2012).
    [Crossref] [PubMed]

2019 (2)

2018 (4)

2016 (1)

2015 (3)

I. Pupeza, D. Sánchez, J. Zhang, N. Lilienfein, M. Seidel, N. Karpowicz, V. Pervak, E. Fill, O. Pronin, Z. Wei, F. Krausz, A. Apolonski, and J. Biegert, “High-power sub-two-cycle mid-infrared pulses at 100 MHz repetition rate,” Nat. Photonics 9(11), 721–724 (2015).
[Crossref]

V. Petrov, “Frequency down-conversion of solid-state laser sources to the mid-infrared spectral range using non-oxide nonlinear crystals,” Prog. Quantum Electron. 42, 1–106 (2015).
[Crossref]

M. Beutler, I. Rimke, E. Büttner, P. Farinello, A. Agnesi, V. Badikov, D. Badikov, and V. Petrov, “Difference-frequency generation of ultrashort pulses in the mid-IR using Yb-fiber pump systems and AgGaSe(2),” Opt. Express 23(3), 2730–2736 (2015).
[Crossref] [PubMed]

2013 (1)

2012 (3)

Y. Yao, A. J. Hoffman, and C. F. Gmachl, “Mid-infrared quantum cascade lasers,” Nat. Photonics 6(7), 432–439 (2012).
[Crossref]

Y. S. You, T. I. Oh, and K. Y. Kim, “Off-Axis Phase-Matched Terahertz Emission from Two-Color Laser-Induced Plasma Filaments,” Phys. Rev. Lett. 109(18), 183902 (2012).
[Crossref] [PubMed]

M. Hajialamdari and D. Strickland, “Tunable mid-infrared source from an ultrafast two-color Yb:fiber chirped-pulse amplifier,” Opt. Lett. 37(17), 3570–3572 (2012).
[Crossref] [PubMed]

2011 (1)

2008 (1)

C. Bauer, A. K. Sharma, U. Willer, J. Burgmeier, B. Braunschweig, W. Schade, S. Blaser, L. Hvozdara, A. Müller, and G. Holl, “Potentials and limits of mid-infrared laser spectroscopy for the detection of explosives,” Appl. Phys. B 92(3), 327–333 (2008).
[Crossref]

1977 (1)

J. R. Morris and Y. R. Shen, “Theory of far-infrared generation by optical mixing,” Phy. Rev. A 15(3), 1143–1156 (1977).
[Crossref]

Agnesi, A.

Al-Kadry, A.

Al-Kadry, A. M.

Apolonski, A.

T. P. Butler, D. Gerz, C. Hofer, J. Xu, C. Gaida, T. Heuermann, M. Gebhardt, L. Vamos, W. Schweinberger, J. A. Gessner, T. Siefke, M. Heusinger, U. Zeitner, A. Apolonski, N. Karpowicz, J. Limpert, F. Krausz, and I. Pupeza, “Watt-scale 50-MHz source of single-cycle waveform-stable pulses in the molecular fingerprint region,” Opt. Lett. 44(7), 1730–1733 (2019).
[Crossref] [PubMed]

I. Pupeza, D. Sánchez, J. Zhang, N. Lilienfein, M. Seidel, N. Karpowicz, V. Pervak, E. Fill, O. Pronin, Z. Wei, F. Krausz, A. Apolonski, and J. Biegert, “High-power sub-two-cycle mid-infrared pulses at 100 MHz repetition rate,” Nat. Photonics 9(11), 721–724 (2015).
[Crossref]

Badikov, D.

Badikov, V.

Bauer, C.

C. Bauer, A. K. Sharma, U. Willer, J. Burgmeier, B. Braunschweig, W. Schade, S. Blaser, L. Hvozdara, A. Müller, and G. Holl, “Potentials and limits of mid-infrared laser spectroscopy for the detection of explosives,” Appl. Phys. B 92(3), 327–333 (2008).
[Crossref]

Beutler, M.

Biegert, J.

I. Pupeza, D. Sánchez, J. Zhang, N. Lilienfein, M. Seidel, N. Karpowicz, V. Pervak, E. Fill, O. Pronin, Z. Wei, F. Krausz, A. Apolonski, and J. Biegert, “High-power sub-two-cycle mid-infrared pulses at 100 MHz repetition rate,” Nat. Photonics 9(11), 721–724 (2015).
[Crossref]

Blaser, S.

C. Bauer, A. K. Sharma, U. Willer, J. Burgmeier, B. Braunschweig, W. Schade, S. Blaser, L. Hvozdara, A. Müller, and G. Holl, “Potentials and limits of mid-infrared laser spectroscopy for the detection of explosives,” Appl. Phys. B 92(3), 327–333 (2008).
[Crossref]

Braunschweig, B.

C. Bauer, A. K. Sharma, U. Willer, J. Burgmeier, B. Braunschweig, W. Schade, S. Blaser, L. Hvozdara, A. Müller, and G. Holl, “Potentials and limits of mid-infrared laser spectroscopy for the detection of explosives,” Appl. Phys. B 92(3), 327–333 (2008).
[Crossref]

Burgmeier, J.

C. Bauer, A. K. Sharma, U. Willer, J. Burgmeier, B. Braunschweig, W. Schade, S. Blaser, L. Hvozdara, A. Müller, and G. Holl, “Potentials and limits of mid-infrared laser spectroscopy for the detection of explosives,” Appl. Phys. B 92(3), 327–333 (2008).
[Crossref]

Butler, T. P.

Büttner, E.

Cao, Q.

Cassinerio, M.

Chang, G.

Coluccelli, N.

Farinello, P.

Fill, E.

I. Pupeza, D. Sánchez, J. Zhang, N. Lilienfein, M. Seidel, N. Karpowicz, V. Pervak, E. Fill, O. Pronin, Z. Wei, F. Krausz, A. Apolonski, and J. Biegert, “High-power sub-two-cycle mid-infrared pulses at 100 MHz repetition rate,” Nat. Photonics 9(11), 721–724 (2015).
[Crossref]

Gaida, C.

Galzerano, G.

Gambetta, A.

Gatti, D.

Gebhardt, M.

Gerz, D.

Gessner, J. A.

Giessen, H.

Gmachl, C. F.

Y. Yao, A. J. Hoffman, and C. F. Gmachl, “Mid-infrared quantum cascade lasers,” Nat. Photonics 6(7), 432–439 (2012).
[Crossref]

Hajialamdari, M.

Heuermann, T.

Heusinger, M.

Hoang, T.

X. Su, T. Hoang, P. Long, Y. Zheng, and D. Strickland, “A Compact High-Average-Power Femtosecond Fiber-Coupled Two-Color CPA System,” IEEE J. Sel. Top. Quantum Electron. 24(5), 1–5 (2018).
[Crossref]

Hofer, C.

Hoffman, A. J.

Y. Yao, A. J. Hoffman, and C. F. Gmachl, “Mid-infrared quantum cascade lasers,” Nat. Photonics 6(7), 432–439 (2012).
[Crossref]

Holl, G.

C. Bauer, A. K. Sharma, U. Willer, J. Burgmeier, B. Braunschweig, W. Schade, S. Blaser, L. Hvozdara, A. Müller, and G. Holl, “Potentials and limits of mid-infrared laser spectroscopy for the detection of explosives,” Appl. Phys. B 92(3), 327–333 (2008).
[Crossref]

Hvozdara, L.

C. Bauer, A. K. Sharma, U. Willer, J. Burgmeier, B. Braunschweig, W. Schade, S. Blaser, L. Hvozdara, A. Müller, and G. Holl, “Potentials and limits of mid-infrared laser spectroscopy for the detection of explosives,” Appl. Phys. B 92(3), 327–333 (2008).
[Crossref]

Karpowicz, N.

T. P. Butler, D. Gerz, C. Hofer, J. Xu, C. Gaida, T. Heuermann, M. Gebhardt, L. Vamos, W. Schweinberger, J. A. Gessner, T. Siefke, M. Heusinger, U. Zeitner, A. Apolonski, N. Karpowicz, J. Limpert, F. Krausz, and I. Pupeza, “Watt-scale 50-MHz source of single-cycle waveform-stable pulses in the molecular fingerprint region,” Opt. Lett. 44(7), 1730–1733 (2019).
[Crossref] [PubMed]

I. Pupeza, D. Sánchez, J. Zhang, N. Lilienfein, M. Seidel, N. Karpowicz, V. Pervak, E. Fill, O. Pronin, Z. Wei, F. Krausz, A. Apolonski, and J. Biegert, “High-power sub-two-cycle mid-infrared pulses at 100 MHz repetition rate,” Nat. Photonics 9(11), 721–724 (2015).
[Crossref]

Kärtner, F. X.

Kessel, A.

Kim, K. Y.

Y. S. You, T. I. Oh, and K. Y. Kim, “Off-Axis Phase-Matched Terahertz Emission from Two-Color Laser-Induced Plasma Filaments,” Phys. Rev. Lett. 109(18), 183902 (2012).
[Crossref] [PubMed]

Krausz, F.

T. P. Butler, D. Gerz, C. Hofer, J. Xu, C. Gaida, T. Heuermann, M. Gebhardt, L. Vamos, W. Schweinberger, J. A. Gessner, T. Siefke, M. Heusinger, U. Zeitner, A. Apolonski, N. Karpowicz, J. Limpert, F. Krausz, and I. Pupeza, “Watt-scale 50-MHz source of single-cycle waveform-stable pulses in the molecular fingerprint region,” Opt. Lett. 44(7), 1730–1733 (2019).
[Crossref] [PubMed]

I. Pupeza, D. Sánchez, J. Zhang, N. Lilienfein, M. Seidel, N. Karpowicz, V. Pervak, E. Fill, O. Pronin, Z. Wei, F. Krausz, A. Apolonski, and J. Biegert, “High-power sub-two-cycle mid-infrared pulses at 100 MHz repetition rate,” Nat. Photonics 9(11), 721–724 (2015).
[Crossref]

Laporta, P.

Lilienfein, N.

I. Pupeza, D. Sánchez, J. Zhang, N. Lilienfein, M. Seidel, N. Karpowicz, V. Pervak, E. Fill, O. Pronin, Z. Wei, F. Krausz, A. Apolonski, and J. Biegert, “High-power sub-two-cycle mid-infrared pulses at 100 MHz repetition rate,” Nat. Photonics 9(11), 721–724 (2015).
[Crossref]

Limpert, J.

Long, P.

X. Su, T. Hoang, P. Long, Y. Zheng, and D. Strickland, “A Compact High-Average-Power Femtosecond Fiber-Coupled Two-Color CPA System,” IEEE J. Sel. Top. Quantum Electron. 24(5), 1–5 (2018).
[Crossref]

Mak, K. F.

Marangoni, M.

Morris, J. R.

J. R. Morris and Y. R. Shen, “Theory of far-infrared generation by optical mixing,” Phy. Rev. A 15(3), 1143–1156 (1977).
[Crossref]

Mörz, F.

Müller, A.

C. Bauer, A. K. Sharma, U. Willer, J. Burgmeier, B. Braunschweig, W. Schade, S. Blaser, L. Hvozdara, A. Müller, and G. Holl, “Potentials and limits of mid-infrared laser spectroscopy for the detection of explosives,” Appl. Phys. B 92(3), 327–333 (2008).
[Crossref]

Nagl, N.

Oh, T. I.

Y. S. You, T. I. Oh, and K. Y. Kim, “Off-Axis Phase-Matched Terahertz Emission from Two-Color Laser-Induced Plasma Filaments,” Phys. Rev. Lett. 109(18), 183902 (2012).
[Crossref] [PubMed]

Pervak, V.

Q. Wang, J. Zhang, A. Kessel, N. Nagl, V. Pervak, O. Pronin, and K. F. Mak, “Broadband mid-infrared coverage (2-17 μm) with few-cycle pulses via cascaded parametric processes,” Opt. Lett. 44(10), 2566–2569 (2019).
[Crossref] [PubMed]

I. Pupeza, D. Sánchez, J. Zhang, N. Lilienfein, M. Seidel, N. Karpowicz, V. Pervak, E. Fill, O. Pronin, Z. Wei, F. Krausz, A. Apolonski, and J. Biegert, “High-power sub-two-cycle mid-infrared pulses at 100 MHz repetition rate,” Nat. Photonics 9(11), 721–724 (2015).
[Crossref]

Petrov, V.

Pronin, O.

Q. Wang, J. Zhang, A. Kessel, N. Nagl, V. Pervak, O. Pronin, and K. F. Mak, “Broadband mid-infrared coverage (2-17 μm) with few-cycle pulses via cascaded parametric processes,” Opt. Lett. 44(10), 2566–2569 (2019).
[Crossref] [PubMed]

I. Pupeza, D. Sánchez, J. Zhang, N. Lilienfein, M. Seidel, N. Karpowicz, V. Pervak, E. Fill, O. Pronin, Z. Wei, F. Krausz, A. Apolonski, and J. Biegert, “High-power sub-two-cycle mid-infrared pulses at 100 MHz repetition rate,” Nat. Photonics 9(11), 721–724 (2015).
[Crossref]

Pupeza, I.

T. P. Butler, D. Gerz, C. Hofer, J. Xu, C. Gaida, T. Heuermann, M. Gebhardt, L. Vamos, W. Schweinberger, J. A. Gessner, T. Siefke, M. Heusinger, U. Zeitner, A. Apolonski, N. Karpowicz, J. Limpert, F. Krausz, and I. Pupeza, “Watt-scale 50-MHz source of single-cycle waveform-stable pulses in the molecular fingerprint region,” Opt. Lett. 44(7), 1730–1733 (2019).
[Crossref] [PubMed]

I. Pupeza, D. Sánchez, J. Zhang, N. Lilienfein, M. Seidel, N. Karpowicz, V. Pervak, E. Fill, O. Pronin, Z. Wei, F. Krausz, A. Apolonski, and J. Biegert, “High-power sub-two-cycle mid-infrared pulses at 100 MHz repetition rate,” Nat. Photonics 9(11), 721–724 (2015).
[Crossref]

Rimke, I.

Sánchez, D.

I. Pupeza, D. Sánchez, J. Zhang, N. Lilienfein, M. Seidel, N. Karpowicz, V. Pervak, E. Fill, O. Pronin, Z. Wei, F. Krausz, A. Apolonski, and J. Biegert, “High-power sub-two-cycle mid-infrared pulses at 100 MHz repetition rate,” Nat. Photonics 9(11), 721–724 (2015).
[Crossref]

Schade, W.

C. Bauer, A. K. Sharma, U. Willer, J. Burgmeier, B. Braunschweig, W. Schade, S. Blaser, L. Hvozdara, A. Müller, and G. Holl, “Potentials and limits of mid-infrared laser spectroscopy for the detection of explosives,” Appl. Phys. B 92(3), 327–333 (2008).
[Crossref]

Schweinberger, W.

Seidel, M.

I. Pupeza, D. Sánchez, J. Zhang, N. Lilienfein, M. Seidel, N. Karpowicz, V. Pervak, E. Fill, O. Pronin, Z. Wei, F. Krausz, A. Apolonski, and J. Biegert, “High-power sub-two-cycle mid-infrared pulses at 100 MHz repetition rate,” Nat. Photonics 9(11), 721–724 (2015).
[Crossref]

Sharma, A. K.

C. Bauer, A. K. Sharma, U. Willer, J. Burgmeier, B. Braunschweig, W. Schade, S. Blaser, L. Hvozdara, A. Müller, and G. Holl, “Potentials and limits of mid-infrared laser spectroscopy for the detection of explosives,” Appl. Phys. B 92(3), 327–333 (2008).
[Crossref]

Shen, Y. R.

J. R. Morris and Y. R. Shen, “Theory of far-infrared generation by optical mixing,” Phy. Rev. A 15(3), 1143–1156 (1977).
[Crossref]

Siefke, T.

Steinle, T.

Steinmann, A.

Strickland, D.

Su, X.

X. Su, T. Hoang, P. Long, Y. Zheng, and D. Strickland, “A Compact High-Average-Power Femtosecond Fiber-Coupled Two-Color CPA System,” IEEE J. Sel. Top. Quantum Electron. 24(5), 1–5 (2018).
[Crossref]

Vamos, L.

Wang, Q.

Wei, Z.

I. Pupeza, D. Sánchez, J. Zhang, N. Lilienfein, M. Seidel, N. Karpowicz, V. Pervak, E. Fill, O. Pronin, Z. Wei, F. Krausz, A. Apolonski, and J. Biegert, “High-power sub-two-cycle mid-infrared pulses at 100 MHz repetition rate,” Nat. Photonics 9(11), 721–724 (2015).
[Crossref]

Willer, U.

C. Bauer, A. K. Sharma, U. Willer, J. Burgmeier, B. Braunschweig, W. Schade, S. Blaser, L. Hvozdara, A. Müller, and G. Holl, “Potentials and limits of mid-infrared laser spectroscopy for the detection of explosives,” Appl. Phys. B 92(3), 327–333 (2008).
[Crossref]

Xu, J.

Yao, Y.

Y. Yao, A. J. Hoffman, and C. F. Gmachl, “Mid-infrared quantum cascade lasers,” Nat. Photonics 6(7), 432–439 (2012).
[Crossref]

You, Y. S.

Y. S. You, T. I. Oh, and K. Y. Kim, “Off-Axis Phase-Matched Terahertz Emission from Two-Color Laser-Induced Plasma Filaments,” Phys. Rev. Lett. 109(18), 183902 (2012).
[Crossref] [PubMed]

Zeitner, U.

Zhang, J.

Q. Wang, J. Zhang, A. Kessel, N. Nagl, V. Pervak, O. Pronin, and K. F. Mak, “Broadband mid-infrared coverage (2-17 μm) with few-cycle pulses via cascaded parametric processes,” Opt. Lett. 44(10), 2566–2569 (2019).
[Crossref] [PubMed]

I. Pupeza, D. Sánchez, J. Zhang, N. Lilienfein, M. Seidel, N. Karpowicz, V. Pervak, E. Fill, O. Pronin, Z. Wei, F. Krausz, A. Apolonski, and J. Biegert, “High-power sub-two-cycle mid-infrared pulses at 100 MHz repetition rate,” Nat. Photonics 9(11), 721–724 (2015).
[Crossref]

Zheng, Y.

X. Su, T. Hoang, P. Long, Y. Zheng, and D. Strickland, “A Compact High-Average-Power Femtosecond Fiber-Coupled Two-Color CPA System,” IEEE J. Sel. Top. Quantum Electron. 24(5), 1–5 (2018).
[Crossref]

Zhou, G.

Appl. Phys. B (1)

C. Bauer, A. K. Sharma, U. Willer, J. Burgmeier, B. Braunschweig, W. Schade, S. Blaser, L. Hvozdara, A. Müller, and G. Holl, “Potentials and limits of mid-infrared laser spectroscopy for the detection of explosives,” Appl. Phys. B 92(3), 327–333 (2008).
[Crossref]

IEEE J. Sel. Top. Quantum Electron. (1)

X. Su, T. Hoang, P. Long, Y. Zheng, and D. Strickland, “A Compact High-Average-Power Femtosecond Fiber-Coupled Two-Color CPA System,” IEEE J. Sel. Top. Quantum Electron. 24(5), 1–5 (2018).
[Crossref]

Nat. Photonics (2)

Y. Yao, A. J. Hoffman, and C. F. Gmachl, “Mid-infrared quantum cascade lasers,” Nat. Photonics 6(7), 432–439 (2012).
[Crossref]

I. Pupeza, D. Sánchez, J. Zhang, N. Lilienfein, M. Seidel, N. Karpowicz, V. Pervak, E. Fill, O. Pronin, Z. Wei, F. Krausz, A. Apolonski, and J. Biegert, “High-power sub-two-cycle mid-infrared pulses at 100 MHz repetition rate,” Nat. Photonics 9(11), 721–724 (2015).
[Crossref]

Opt. Express (1)

Opt. Lett. (9)

G. Zhou, Q. Cao, F. X. Kärtner, and G. Chang, “Energy scalable, offset-free ultrafast mid-infrared source harnessing self-phase-modulation-enabled spectral selection,” Opt. Lett. 43(12), 2953–2956 (2018).
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T. Steinle, F. Mörz, A. Steinmann, and H. Giessen, “Ultra-stable high average power femtosecond laser system tunable from 1.33 to 20 μm,” Opt. Lett. 41(21), 4863–4866 (2016).
[Crossref] [PubMed]

T. P. Butler, D. Gerz, C. Hofer, J. Xu, C. Gaida, T. Heuermann, M. Gebhardt, L. Vamos, W. Schweinberger, J. A. Gessner, T. Siefke, M. Heusinger, U. Zeitner, A. Apolonski, N. Karpowicz, J. Limpert, F. Krausz, and I. Pupeza, “Watt-scale 50-MHz source of single-cycle waveform-stable pulses in the molecular fingerprint region,” Opt. Lett. 44(7), 1730–1733 (2019).
[Crossref] [PubMed]

Q. Wang, J. Zhang, A. Kessel, N. Nagl, V. Pervak, O. Pronin, and K. F. Mak, “Broadband mid-infrared coverage (2-17 μm) with few-cycle pulses via cascaded parametric processes,” Opt. Lett. 44(10), 2566–2569 (2019).
[Crossref] [PubMed]

A. M. Al-Kadry and D. Strickland, “Generation of 400 μW at 17.5 μm using a two-color Yb fiber chirped pulse amplifier,” Opt. Lett. 36(7), 1080–1082 (2011).
[Crossref] [PubMed]

A. Al-Kadry and D. Strickland, “Generation of 400 μW at 17.5 µm using a two-color Yb fiber chirped pulse amplifier: erratum,” Opt. Lett. 43(2), 316 (2018).
[Crossref] [PubMed]

M. Hajialamdari and D. Strickland, “Tunable mid-infrared source from an ultrafast two-color Yb:fiber chirped-pulse amplifier,” Opt. Lett. 37(17), 3570–3572 (2012).
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M. Hajialamdari and D. Strickland, “Tunable mid-infrared source from an ultrafast two-color Yb:fiber chirped pulse amplifier: erratum,” Opt. Lett. 43(2), 353 (2018).
[Crossref] [PubMed]

A. Gambetta, N. Coluccelli, M. Cassinerio, D. Gatti, P. Laporta, G. Galzerano, and M. Marangoni, “Milliwatt-level frequency combs in the 8-14 μm range via difference frequency generation from an Er:fiber oscillator,” Opt. Lett. 38(7), 1155–1157 (2013).
[Crossref] [PubMed]

Phy. Rev. A (1)

J. R. Morris and Y. R. Shen, “Theory of far-infrared generation by optical mixing,” Phy. Rev. A 15(3), 1143–1156 (1977).
[Crossref]

Phys. Rev. Lett. (1)

Y. S. You, T. I. Oh, and K. Y. Kim, “Off-Axis Phase-Matched Terahertz Emission from Two-Color Laser-Induced Plasma Filaments,” Phys. Rev. Lett. 109(18), 183902 (2012).
[Crossref] [PubMed]

Prog. Quantum Electron. (1)

V. Petrov, “Frequency down-conversion of solid-state laser sources to the mid-infrared spectral range using non-oxide nonlinear crystals,” Prog. Quantum Electron. 42, 1–106 (2015).
[Crossref]

Other (1)

I. T. Sorokina and K. L. Vodopyanov, eds., Solid-State Mid-Infrared Laser Sources (Springer-Verlag, 2003).

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Figures (5)

Fig. 1
Fig. 1 Input spectrum of the two colors.
Fig. 2
Fig. 2 Experimental setup for MIR generation.
Fig. 3
Fig. 3 MIR power in Type 2 phase matching changes with different focal spot size. The solid curves are numerical simulations in different walk-off angle conditions (ζ = 0 and −0.06 rad) and the points are experimental data. The error bars represent the measurement uncertainties.
Fig. 4
Fig. 4 Experimental results and simulations about the Angular Distribution of the Generated MIR with different focused beam radius of (a) 17.8 ± 2.9 µm (b) 26.7 ± 4.3 µm (c) 35.6 ± 5.9 µm (d) 80.1 ± 13.1 µm.
Fig. 5
Fig. 5 MIR Spectrum with Different Focal beam radius of (a) 17.8 ± 2.9 µm (b) 80.1 ± 13.1 µm.

Equations (1)

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d P ± (θ,ϕ) dΩ = c 2π cos 2 θ | E T± ( k T = ω c sinθ( x ^ cosϕ+ y ^ sinϕ) | 2 ,

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