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Abstract
Femtosecond long-wave-infrared (LWIR) pulses have found applications in several fields, but their generation is limited to ${{\rm CO}_2}$ lasers and solid-state frequency converters. Waveguide-based Raman red shifting provides another promising solution to efficiently generate LWIR pulses. Here, we numerically study LWIR pulse generation in a hydrogen-filled hollow-core fiber. Several excitation schemes are considered, involving one or two pulses at either the same or different wavelengths. The analysis reveals that a waveguide structure enables tailoring of the Raman gain, which is required to produce pulses at LWIR wavelengths. With ${\sim}5 {\text -} {\rm mJ}$ and 50-fs input pulses, clean 400-µJ and 88-fs pulses at 12 µm are theoretically generated with 41% total quantum efficiency. The simulations also provide insight into the nonlinear dynamics of the Raman gain, where the concept of a phonon amplifier underlies the optimal performance that can be achieved. Only the two-pulse scheme with a two-color source creates a good phonon amplifier for efficient LWIR generation.
© 2023 Optica Publishing Group
1. INTRODUCTION
Ultrafast fiber-based sources have drawn considerable interest in both the scientific and industrial communities due to their robustness, compactness, and alignment-free operation. However, conventional silica-based solid-core fibers are limited to the 350- to 2500-nm range. Chalcogenide glasses have extended the transmission window up to 12 µm, but their fabrication remains challenging [1]. Hollow-core fibers, on the other hand, have enabled a wide range of studies in gas-based nonlinear optics in various spectral ranges [2–19]. By selecting appropriate gas species, pulses from ultraviolet to mid-wave infrared (3–5 µm) have been generated via supercontinuum [2,3], Raman scattering [4–18], or optical parametric amplification (OPA) [19]. In addition, gas-filled hollow-core fibers allow working with much higher pulse energy than solid-core fibers.
Ultrashort long-wave-infrared (LWIR, 8–15 µm) pulses have found wide applications in fields such as ultrafast molecular spectroscopy [20–23] and strong-field light–matter physics [24–30]. In spectroscopy, characteristic vibrational absorption bands in the fingerprint region ($400 {-} 1800\;{{\rm cm}^{- 1}}$) are useful to identify complex molecules [31]. The high peak power of ultrafast pulses can also induce nonlinear multiphoton interactions unique to the LWIR regime [32]. In strong-field physics, longer wavelengths can efficiently drive electron tunneling ionization and, through high harmonic generation, produce energetic attosecond pulses in the extreme ultraviolet range due to the inverse-quadratic frequency scaling of the ponderomotive energy [33–36]. Such long-wavelength scaling also underlies several physical phenomena leading, for example, to efficient laser-wakefield acceleration [37–39], generation of intense gamma rays through inverse Compton backscattering [40], and nonlinear self-guiding propagation in the outdoor atmosphere [41]. Fiber sources would be attractive, but the generation of short LWIR pulses from either solid-core or hollow-core fibers has not been reported.
${{\rm CO}_2}$ lasers and solid-state frequency converters are the most common LWIR sources. ${{\rm CO}_2}$ lasers are mature and widespread [42]. Chirped-pulse amplification in a high-pressure mixed-isotope ${{\rm CO}_2}$ amplifier has been used to generate 2-ps pulses at 9.2 µm [43]. Pulse compression produces pulses as short as 50 fs [44]. Although the performance of ${{\rm CO}_2}$ lasers is remarkable, solid-state sources are more compact and easier to maintain. OPA or difference frequency generation (DFG) has produced impressive results: for example, 180-fs pulses at 7 µm [45], 142-fs pulses at 9 µm [46], and 101-fs pulses at 10 µm [47]. However, parametric generation has challenges in the LWIR regime. The poor quantum defect of near-infrared (NIR) pumped LWIR amplification, combined with the generally low quantum efficiency (conversion efficiency of photons) of OPA, results in energy conversion efficiencies at the sub-percent to few-percent level. Broadband phase matching can require short amplifier crystals or large noncollinear incident beam angles that reduce beam overlap, limiting amplifier gain. Limited transparency windows and apertures of bulk crystals are crucial design factors to avoid heating and damage [45–51].
Stimulated Raman scattering (SRS) has been widely used in frequency downconversion, which can be continuous through an intrapulse soliton self-frequency shift [52] or discrete through Stokes-wave generation. Gases are particularly intriguing Raman media because of their high vibrational transition frequencies. However, SRS conversion efficiency can be severely degraded by self-phase modulation (SPM) if SRS takes place in the transient-Raman regime, where the pulse duration is shorter than the vibrational dephasing time (or phonon decay time) [53]. Since gases commonly have ${\gt}\!{100}\;{\rm ps}$ dephasing times, subpicosecond pulses drive transient SRS. The Raman threshold in the transient regime is determined by the pulse energy rather than the peak power as in the steady-state regime [54,55]; therefore, with fixed pulse energy, a shorter pulse experiences more SPM without an increase in the SRS. Pulse stretching by frequency chirping helps reduce the SPM [56]. Still, it suffers from Raman spectral narrowing because only the trailing edge of a pulse (and thus a fraction of the spectrum) undergoes conversion [Fig. 1(a)]. This results in not only lower efficiency but also a longer (dechirped) Stokes pulse. To overcome this problem, a scheme that involves two chirped pulses with orthogonal polarizations was proposed [Fig. 1(b)]. The pulses are temporally separated but by less than the vibrational dephasing time of the molecule. The first pulse should be intense enough to overcome the Raman threshold. It excites phonon waves from which the second pulse scatters; consequently, SRS occurs throughout the entire second pulse (and thus the bandwidth). The second pulse can be pictured as entering a sea of existing phonons, or oscillating molecules, with which it interacts nonlinearly. With this approach, 52-fs Stokes pulses at 1.2 µm have been generated from 48-fs Ti:sapphire pulses in hydrogen [4]. Later, 4.4-mJ and 66-fs Stokes pulses at 1.28 µm were generated [8].
Fig. 1. Techniques for SRS with ultrashort pulses. L: focusing or collimating lens; W: window. A hollow-core fiber, if used, is placed inside a gas chamber; otherwise, the light is focused for efficient interactions with gas molecules. (a) Single-pulse scheme. The pump pulse is transform-limited or chirped. (b) Double-chirped-pulse scheme. The polarizations are represeneted by “s” and “p.” Phonon waves are excited by the first pulse and then seen by the second pulse. The Stokes pulse in the second pulse can be easily extracted with a polarizing beam splitter and a long-pass filter, while the first pulse is dumped.
In this article, we report a theoretical investigation of femtosecond pulse generation into the LWIR regime by SRS. Although the aforementioned single- and double-pulse approaches have been deployed in the NIR regime with great success, extension into the LWIR regime is challenging. Several factors that play out differently in NIR and LWIR Stokes generation will be discussed, such as the competition between vibrational and rotational SRS. In contrast to NIR Raman Stokes generation in gases, which can be efficient in free space [8,57], a waveguide structure will be required to drive Stokes generation to LWIR wavelengths. A waveguide geometry selectively suppresses unwanted Raman transitions through Raman gain suppression, leaving only the one with the largest frequency shift. Therefore, we propose, what we believe, to the best of our knowledge, is a new waveguide-based approach to efficiently produce femtosecond Stokes pulses in the LWIR. It relies on a two-color source, as opposed to conventional single-color approaches. Numerical simulations predict that the proposed approach will be able to generate sub-100-fs pulses at 12 µm with almost 50% quantum efficiency. Advantages and disadvantages of the distinct approaches will also be discussed.
To avoid confusion in the following discussions, quantum efficiency, if not specified, refers to the quantum efficiency with respect to the second pulse. When we mention the property of a LWIR pulse, it also refers to those of the second pulse. For simplicity, peak power refers to that of the dechirped Stokes signal in the second pulse.
2. LWIR GENERATION WITH A SINGLE-WAVELENGTH NIR SOURCE
A. Challenges in the LWIR Regime
To generate a LWIR Stokes pulse, hydrogen is attractive due to its large vibrational frequency shift 124.6 THz ($4155\;{{\rm cm}^{- 1}}$). Loree et al. [58] and Grasyuk et al. [59] obtained 9.2-µm light by exciting hydrogen with strong picosecond pulses from a Nd:YAG laser whose 1.064-µm pulses underwent two vibrational Stokes shifts, but the efficiency of this approach is low. In addition, the temporal phase structure of a pulse can be easily distorted by nonlinear interactions among different colors that temporally overlap, so such a cascaded approach will likely be problematic with the chirped input pulses required to avoid or reduce nonlinear cross-phase modulations (XPM).
Fig. 2. (a) Steady-state rotational and vibrational Raman gains at 20-bar ${{\rm H}_2}$ pressure for linearly polarized light. (b) Relative Raman gain ($\mathop {\log}\nolimits_{10} ({{G_{{\rm vib}}}/{G_{{\rm rot}}}})$). Since the transient Raman gain is correlated with the steady-state Raman gain ${G_{{\rm steady}}}$ (${| {{A_S}(z)} |^2}/{| {{A_S}(z = 0)} |^2} \sim \def\LDeqbreak{}{{\rm e}^{\sqrt {8{G_{{\rm steady}}}Ez}}}$ for a square pulse in the transient-Raman regime, where $E$ is the pulse energy [62]), we used the steady-state values here not only for illustration but also to ignore the role of pulse shapes. (c) and (d) are counterparts of (a) and (b) in a capillary with a 300-µm core diameter where Raman gain suppression takes place. The black dashed line shows where the two Raman gains are equal. Only the ${\rm Q} (\nu = 0 \to 1)$ vibrational and the S(1) rotational transitions were selected for comparison because they dominate at room temperature. Raman gains with and without suppression were calculated with Eqs. (S46) and (S84) in Supplement 1, respectively.
If a source of femtosecond pulses beyond 1.8 µm is available, one vibrational Stokes shift is enough to generate a LWIR pulse. Even in that case, challenges remain. In addition to the vibrational Raman transition, rotational transitions come into play. They compete with vibrational SRS by consuming available pump energy for vibrational Stokes generation. Since Stokes Raman gain is proportional to Stokes frequency, Stokes-wave generation becomes inefficient in the LWIR [60,61]; furthermore, the decrease in the rotational Raman gain with an increasing wavelength is slower than that for the vibrational transition due to their smaller transition frequencies. As a result, rotational SRS can dominate over vibrational SRS at long wavelengths. Figure 2(a) shows that, at 20-bar pressure, the vibrational Raman gain of Q($\nu = 0 \to 1$) drops faster than the rotational gain of S(1), resulting in rotational SRS dominating beyond 1.83 µm. This feature remains the same for gas pressure above 5 bar [Fig. 2(b)]. The dominant rotational Raman gain becomes the major obstacle to convert NIR pulses to LWIR by SRS.
Despite rotational SRS dominating at long wavelengths, LWIR generation with vibrational SRS is possible through Raman gain suppression [63–66]. In SRS, a Stokes wave is accompanied by phonon creation, whereas an anti-Stokes wave is accompanied by phonon annihilation. Because phonons catalyze Stokes-wave generation, the Raman process is suppressed if no phonon is created, as if the molecules do not interact with the light. This situation occurs when the phonon creation and annihilation rates are precisely balanced. To acquire such a balance, phase matching among anti-Stokes, pump, and Stokes wavelengths must be satisfied to maximize the annihilation rate of phonons during anti-Stokes generation, which becomes possible with the unique S-shaped dispersion curve in a hollow-core fiber [67]. Thus, the phase-matching wavelength can be easily controlled by tuning gas pressure and fiber core size. Because the rotational and vibrational frequencies are not equal, Raman gain suppression can occur at different wavelengths for the two Raman transitions. For example, at a pressure of 20 bar, the phase-matching wavelength in a 300-µm-core-diameter capillary for rotational Raman (2.16 µm) is longer than that for vibrational Raman (1.72 µm), which, therefore, allows vibrational SRS to dominate at longer wavelengths [Fig. 2(c)]. Figure 2(d) shows the dominant Raman process at various gas pressures. Vibrational Raman dominates at 2 µm up to 35 bar. As a result, by operating below 35 bar, LWIR generation through vibrational Raman scattering of a NIR pulse can be achieved. For more information, see the discussion of Raman gain suppression in Supplement 1, which includes treatment of the role of electronic nonlinearity.
An additional challenge to LWIR generation is that the decrease of Raman gain with wavelength implies the use of a long fiber. However, the high loss at LWIR wavelengths limits the fiber length. Typical Kagomé and anti-resonant hollow-core fibers are made with silica, which is lossy beyond 6 µm. Capillaries suffer loss from the inherently lossy waveguide mode, which scales as $\Re [{{\nu _n}}]{\lambda ^2}/{a^3}$ and becomes significant in LWIR [68]. Note that $\Re [{{\nu _n}}]$ considers only the real part of ${\nu _n}$, ${\nu _n} = 1/\sqrt {{{({{n_{{\rm material}}}/{n_{{{\rm H}_2}}}})}^2} - 1}$, $a$ is the core radius, and $n$ is the complex refractive index. These effects sum up to a ${\sim}50$-dB/m loss at 10 µm. To mitigate waveguide loss, capillaries are coated with AgI/Ag [69–71]; the loss can be as low as 1 dB/m at 10 µm for a capillary with 300-µm core diameter. These AgI/Ag-coated capillaries are also commercially available. For more information, see the discussion of waveguide loss in Supplement 1. In addition to the fiber loss, at high pressures, hydrogen can be lossy around 2–3 µm and 7–50 µm due to its ${\rm Q}(\nu = 0 \to 1)$ vibrational [72,73] and S(1) rotational Raman transitions [74,75]. Since a hydrogen molecule possesses no permanent dipole moment, it does not interact directly with infrared light; however, collisions between hydrogen molecules can distort the electron distribution and induce a dipole, leading to pressure-induced absorption at Raman transition frequencies. This type of loss scales quadratically with pressure; at 10 µm, it can be 0.64 dB/m at 20 bar or as high as 16 dB/m at 100 bar. For more information, see the discussion of pressure-induced absorption in Supplement 1.
B. LWIR Generation in a Waveguide
To verify the feasibility of LWIR generation with a NIR pulse in a waveguide, we simulated the SRS process, at room temperature, with a unidirectional pulse propagation equation [76]. This takes into account the frequency dependence of the physical parameters over the large frequency window required to cover a few orders of Raman Stokes and anti-Stokes frequencies. It includes both vibrational and rotational Raman scattering [77], and we assume a 1-m-long AgI/Ag-coated capillary with 300-µm core diameter. We chose 300 µm for demonstration with reasonable pulse energies, acceptably low loss, and commercial availability. The propagation constants for the capillary modes were obtained as in [68]. To model the effect of the AgI/Ag coating, the capillary waveguide loss was ignored and replaced with an experimentally realizable 1 dB/m over the entire spectral range. In addition, pressure-induced absorption at Raman transition frequencies was considered. Input-pulse shot noise (modeled semi-classically by having one phonon per spectral discretization bin) and spontaneous Raman scattering are also considered [78,79].
Before investigating femtosecond Stokes generation, we determined the Raman thresholds at 0.8 and 2 µm, not only as reference points for subsequent analysis but also to verify that the Raman gain is weaker at longer wavelengths. A single-shift process with an input pulse at 2 µm to generate a LWIR Stokes wave was modeled and compared to the result that would be obtained with a 0.8-µm input pulse. In each case, a 10-ps chirped pulse with an 50-fs transform limit is launched, and a pressure of 20 bar is assumed. The pulse is chirped to avoid SPM in the energy range investigated here. As a result of the suppression of Raman gain and its frequency dependence, the gain at 0.8 µm is 30 times higher than that at 2 µm (Fig. 3). Consequently, the Raman threshold at 0.8 µm is 30 times lower than the threshold at 2 µm.
Fig. 3. Stokes-generation quantum efficiency (QE) versus pulse energy for 0.8-µm and 2-µm pulses. The decrease in efficiency at high energies for 0.8 µm results from its early saturation in pulse propagation and subsequent linear propagation loss. Inset: Zoom-in on a portion of the graph at low pump energy.
To obtain femtosecond Stokes pulses, we studied the two conventional schemes by launching one or two 2-µm chirped pulses into the capillary. Figures 4(a) and 4(b) depict the Stokes pulse generated with a single 10-mJ pulse that is strong enough to drive the Raman process (Fig. 3). It suffers from Raman spectral narrowing, so it can only be dechirped to 155 fs with 450-µJ pulse energy (23% quantum efficiency). To study the double-chirped-pulse scheme [Figs. 4(c) and 4(d)], a 15-mJ pulse is launched prior to the 10-mJ pulse by 40 ps. The second pulse undergoes more complete Raman shifting; it can be dechirped to 86 fs with 830-µJ pulse energy (43% quantum efficiency). The Strehl ratios of dechirped peak power to transform-limited peak power are ${\sim}0.4$ for both cases due to significant energy in the long (${\sim}1 {\text -} {\rm ps}$), low pedestal. For both cases, there are significant fluctuations in their Stokes spectra and peak powers; these result from the anti-Stokes interference and will be discussed later.
Fig. 4. (a)–(b) Stokes generation with the single-chirped-pulse scheme. (a) Stokes spectrum and (b) dechirped temporal profile. (c)–(d) Counterparts for the second pulse of a double-chirped-pulse scheme. Simulations are repeated 20 times to demonstrate the mean values (center lines) and standard deviations (shaded areas). PSD: power spectral density; TL: transform-limited pulse; D: dechirped pulse.
3. EFFICIENT LWIR GENERATION WITH A TWO-COLOR SOURCE
A. Concept
Although it seems feasible to use 2-µm pulses to generate 12 µm, the need for the 15-mJ first pulse, which is simply dumped and thus wasted, is a major disadvantage. Another potential concern is the thermal load from hydrogen phonons, which may affect the nonlinear pulse propagation [80]. To overcome these issues, it is preferable to use a first pulse with a shorter wavelength. First, this reduces the heat load per photon. In conversion from 2 to 12 µm, 83% of the energy becomes heat, while in conversion from 0.8 to 1.2 µm, only 33% of the energy becomes heat. Second, it has a higher Raman gain. This provides two advantages: The energy of the first pulse can be reduced, and the earlier initiation of the Raman process allows the use of a shorter (hence, lower-loss) fiber.
To generate LWIR pulses more efficiently, we propose what we believe, to the best of our knowledge, is a new scheme based on a two-color source (Fig. 5). The first pulse should have a large Raman gain for efficient phonon excitation, while the second pulse is chosen to reach the desired wavelength with its Stokes wave. For purposes of illustration, we choose 0.8 µm for the first pulse and 2 µm for the second pulse. Each pulse has 50-fs transform-limited pulse duration and is chirped to 10 ps. Such pulses can be obtained from a Ti:sapphire-laser-pumped parametric amplifier [81] or second-harmonic generation from a Tm-doped fiber laser [82]. A 40-ps time delay (larger than the pulse duration and smaller than the vibrational dephasing time) between the pulses is assumed.
Fig. 5. Two-color and two-pulse approach for efficient Stokes generation. The first pulse ($\lambda _{{\rm short}}^P$) has a wavelength shorter than that of the second pulse ($\lambda _{{\rm long}}^P$) to more efficiently excite phonons. These phonons drive the SRS in the second pulse for LWIR generation ($\lambda _{{\rm LWIR}}^S$). Two pulses can be combined with a dichroic mirror, while the LWIR pulse can be extracted with a long-pass filter.
We simulated this approach with a 2-mJ first pulse and a 5-mJ second pulse. The energy of the first pulse is enough to excite phonon waves (Fig. 3). Although 5 mJ is below the Raman threshold for 2 µm, phonon waves help drive the Stokes process [Fig. 6(a)]. The phonon waves are excited from the trailing edge of the first pulse and later scatter the second pulse, starting from its leading edge. This yields a smooth Stokes pulse at 12 µm, which can be dechirped to 88-fs duration with almost no pedestal [Figs. 6(b) and 6(c)]. Such a short pulse comprises less than 3 cycles of the electric field. The quantum efficiency reaches 48% with 420-µJ Stokes pulse energy. Compared to excitation with two 2-µm pulses [Figs. 4(c) and 4(d)], this approach generates a cleaner 12-µm dechirped pulse, with a Strehl ratio equal to 0.8, while dumping less energy. Additionally, we do not observe significant fluctuations in the Stokes spectrum and its peak power. The parameter space to obtain such a good pulse quality is also much larger, as shown in Figs. 7 and 11.
Fig. 6. LWIR generation with a 2-mJ pulse at 0.8 µm and a 5-mJ pulse at 2 µm. (a) Photon-number evolutions of two pulses. T: total photon number; AS, P, and S: photon numbers of anti-Stokes, pump, and Stokes pulses, respectively. (b) Spectrum and (c) dechirped temporal profile of the 12-µm Stokes component in the second pulse. (b)–(c) Simulations are repeated 20 times to demonstrate the mean values (center lines) and standard deviations (shaded areas).
Fig. 7. Optimization of the two-color and two-pulse approach. (a) Quantum efficiency, (b) relative peak power, and (c) Strehl ratio of the 12-µm Stokes pulse for various pulse energies. The black circles indicate where the maximum relative peak power occurs. Each simulation was repeated five times, and the mean values were calculated. (d) Dechirped Stokes pulse with (5 mJ, 0.8 µm) and (10 mJ, 2 µm) input pulses.
B. Role of Phonon Waves
In any two-pulse approach, the first pulse excites phonons and is dumped. It is desirable to optimize the process for the highest total quantum efficiency while maintaining the quality of the generated LWIR pulses. To tackle this problem, we start from two perspectives: the first-pulse energy, which determines the generated phonon field, and its wavelength, which determines the interaction with the second pulse.
1. Strength of Phonons
We first studied how the strength of the phonon wave affects the Stokes-wave generation in the second pulse by varying the energies of the two pulses (Fig. 7). When the energy of the 0.8-µm pulse is above 1 mJ, its generated phonon waves can efficiently drive Stokes generation for the 2-µm pulse with energies above 2 mJ [Fig. 7(a)]. For fixed 0.8-µm energy, the quantum efficiency increases monotonically with increasing 2-µm pulse energy [Fig. 7(a)]; however, this is not true for the peak power. Figure 7(b) shows the relative peak power of the Stokes signal in the second pulse, which is the ratio of the Stokes peak power to the theoretical maximum peak power if the pump is perfectly converted to the Stokes wave. The maximum peak power lies in the center of Fig. 7(b). Increasing the 2-µm pulse energy too much can induce SPM in the pump, which distorts the temporal phase. This distorted phase will be transferred to the Stokes wave and reduce the pulse quality [Fig. 7(c)]. Increasing the 0.8-µm pulse energy, on the other hand, should facilitate the generation of high-quality Stokes pulses because more phonons are generated to drive the second-pulse SRS. However, it will also induce cascaded Raman processes in the first pulse that excite phonons with multiple wave vectors. As will be discussed in the next section, mismatched wave vectors lead to a strong anti-Stokes signal that can interact nonlinearly with the pump pulse. This distorts the temporal phase and again reduces the Stokes pulse quality [Fig. 7(c)]. Figure 7(d) shows the dechirped Stokes pulse with the highest input energies investigated. Compared to Fig. 6(c), where the Strehl ratio is 0.8, it decreases to 0.5. Although the pulse quality drops at energy extremes, the regime with both good pulse quality (Strehl ratio ${\gt} {0.7}$) and efficient Stokes generation spans a wide range of energy values and can thus be easily reached.
2. Phase-Matching Relation
Prior work has shown that phonon waves in hydrogen play a key role in frequency downconversion (Stokes generation) and upconversion (anti-Stokes generation) [67]. Figure 8 illustrates the core ideas of two Raman processes. The wave vector of the phonons determines the process that will occur. To study this effect, different wavelengths of the first pulse and different gas pressures were considered because they are the primary determinants of the phonon wave vector. To simplify the notation in the following discussions, the propagation constants of pump, Stokes, and anti-Stokes pulses include nonlinear contributions, where, without pump depletion, ${\beta _P} \leftarrow {\beta _P} + \gamma P$, ${\beta _S} \leftarrow {\beta _S} + 2\gamma P$ and ${\beta _{\textit{AS}}} \leftarrow {\beta _{\textit{AS}}} + 2\gamma P$, respectively, due to SPM and XPM. $\gamma$ is the nonlinear coefficient of ${{\rm H}_2}$, and $P$ is the pump peak power.
Fig. 8. Illustration of Raman processes. When phonons are absorbed efficiently, anti-Stokes waves can be produced. When the phonons produced from the Raman-Stokes generation of the pump have the same wave vector as the interacting phonons, the pump essentially emits the same phonons. The pump thus acts as a phonon amplifier for the seed phonon, as well as producing Stokes waves.
We first discuss the effect of phonons on frequency upconversion. This might come into play in downconversion and interfere with the Stokes generation. When the phonon wave vector is the same as the difference of the propagation constants of the pump and anti-Stokes waves (${\beta _{{\rm phonon}}} + {\beta _P} = {\beta _{\textit{AS}}}$), an efficient path for energy transfer to the anti-Stokes wave opens. The pump can efficiently absorb phonons and transform into its anti-Stokes pulse [67]. This process performs as a “delayed” four-wave mixing among first color (P1), first color’s Stokes (S1), second color (P2), and second color’s anti-Stokes (AS2), mediated with phonon waves whose wave vector is determined by the propagation constants of two colors in the first pulse. In the analogy, “phase matching” is used to represent the wave-vector requirement for maximum Raman generation efficiency, which is $({\beta _{{\rm phonon1}}} \equiv {\beta _{P1}} - {\beta _{S1}}) = {\beta _{AS2}} - {\beta _{P2}}$ for the frequency upconversion here. Such phase matching will induce Raman gain suppression if both phonon creation and annihilation occur simultaneously in the same pulse; the separation of two processes, however, allows for Raman generation. To study this effect, we varied the wavelength of the first pulse and monitored the anti-Stokes signal of the second (2-µm) pulse. In particular, when the first pulse is at 1.09 µm (the anti-Stokes wavelength of a 2-µm pump pulse), its generated phonon waves should be inherently phase matched to the frequency upconversion process of the second pulse [Fig. 9(a)]. However, we do not observe an increase in the anti-Stokes photons with different first-pulse wavelengths [bottom blue line in Fig. 9(c)]. Since phonons can initiate Raman scattering for both Stokes and anti-Stokes signals, Stokes growth dominates if SRS occurs. To observe frequency upconversion through phonon interactions, SRS should be avoided by using a weak second pulse. Phonon interactions are analytically derived in Supplement 1, which also includes a discussion of the case where the second pulse is weak enough to avoid SRS. More details about the nonlinear interactions when the second pulse is strong are also included in Supplement 1.
Fig. 9. (a) Dispersion curve of a capillary with 300-µm core diameter filled with hydrogen at 20-bar pressure. The line of each color (top) represents the wave vector of the phonon waves from the first pulse of various wavelengths. They were then added to the value at 2 µm (bottom) to visualize the phase matching of the phonon waves to the second (2-µm) pulse. The black arrow on the bottom right represents the phonon wave vector generated by the 2-µm pulse. Stokes generation is best phase-matched if the phonon wave vector from the first pulse coincides with this vector (${\beta _{{\rm phonon1}}} = {\beta _{{\rm phonon2}}}$). (b) Phase mismatch of phonons from two pulses ($\Delta {\beta _{{\rm phonon}}} = {\beta _{{\rm phonon1}}} - {\beta _{{\rm phonon2}}}$) [the purple region in (a)]. The phase-matched wavelength is around 750 nm when $\Delta {\beta _{{\rm phonon}}} = 0$. (c) Quantum efficiency of Stokes generation in the second pulse and Stokes peak power with various wavelengths of the first pulse. Other simulation parameters were the same as those in Fig. 6. Simulations were repeated 20 times to demonstrate the mean values (center lines) and standard deviations (shaded areas).
Frequency downconversion triggered by phonons is the foundation of many prior works on gas-based Stokes-wave generation. Bauerschmidt et al. experimentally studied its effect in obtaining a Stokes pulse with a careful choice of the phonon wave vector [67]. The second pulse acts as a phonon amplifier for the incoming seed phonon, which, during amplification, produces more Stokes signals. Its efficiency is maximized when the seed phonon has the same wave vector as the generated phonons; phase matching thus becomes $({\beta _{{\rm phonon1}}} \,\equiv\def\LDeqbreak{} {\beta _{P1}} - {\beta _{S1}}) = ({\beta _{{\rm phonon2}}} \equiv {\beta _{P2}} - {\beta _{S2}})$ for a phonon amplifier. For the conventional single-color and double-pulse approach, the condition of a good phonon amplifier is automatically met; for a two-color approach, however, it requires careful design regarding phonon wave vectors. At 20-bar pressure, a 0.7- to 0.8-µm pump can generate phonons almost phase-matched to those from the 2-µm phonon amplifier [Figs. 9(a) and 9(b)]. However, we observed a monotonic increase in efficiency with a decrease in the wavelength [red line in Fig. 9(c)]. Because a shorter wavelength has a higher Raman gain, it is able to generate a stronger burst of phonons in a given propagation distance; this helps more efficiently generate the Stokes wave of the second pulse. However, such an efficiency increase is not significant because of saturation of the Stokes conversion. Peak power, on the other hand, exhibits rapid variation, and only a first-pulse wavelength around 0.8 µm can deliver a 12-µm pulse with the highest peak power [black dashed line in Fig. 9(c)]. To explain this phenomenon, we derived analytical equations for the Stokes/anti-Stokes generation due to phonon modulations. For more information, see the discussion of phonon waves in Supplement 1. The growth rates of their photon numbers are proportional to their frequencies, so the anti-Stokes wave almost always grows faster than the Stokes wave unless the condition of a perfect phonon amplifier is reached. A strong anti-Stokes wave may distort the temporal phase of the pump pulse through XPM and SRS-induced backconversion (also driven by phonons) to the pump wavelength, along with a pulse temporal walk-off during these processes (which leads to 1.5-ps delay after 1-m propagation at 20-bar ${{\rm H}_2}$). As a result, the Stokes pulse generated with an imperfect phonon amplifier will have a temporal phase that cannot be compensated for by a grating pair. The SRS-induced backconversion of anti-Stokes waves to the pump is affected by the beating pattern of phonon waves resulting from multiple Raman transitions in the vibrational Raman “band.” This beating pattern is different from pulse to pulse due to the noise-seeded phonon generation in the first pulse, which eventually leads to a stronger fluctuation in the peak power of the dechirped Stokes pulse.
We also investigated the effect of gas pressure on Stokes-wave generation. Since phonons generated by the 0.8-µm pulse have the same wave vectors as those from the 2-µm pulse at 20 bar, it gives the highest efficiency and peak power [Fig. 10(a)]. Increasing the gas pressure beyond 20 bar reduces the performance due to the phase mismatch of the phonons along with some pressure-induced absorption. Despite propagation far enough for the Stokes wave to saturate, there is a significant drop in efficiency with reduced pressure. This is due to the more severe Raman gain suppression at a smaller gas pressure for the first pulse [Fig. 10(b)]. With fewer phonons excited by the first pulse, the 5-mJ energy of the 2-µm pulse is below the SRS threshold.
Fig. 10. (a) Quantum efficiency of the Stokes generation from the second pulse and the Stokes peak power at various gas pressures. Other simulation parameters were the same as those in Fig. 6. Simulations were repeated 20 times to determine the mean values (center lines) and standard deviations (shaded areas). (b) Wavelength at which vibrational Raman gain is suppressed versus gas pressure.
Fig. 11. Optimization of the single-color and double-pulse approach. (a) Quantum efficiency and (b) Strehl ratio of the 12-µm Stokes pulse in the second pulse for various pulse energies. Here, two 2-µm pulses were sent into the capillary with simulation conditions the same as in Fig. 4 except for the pulse energies. Each simulation was repeated five times, and the mean values were calculated. The colorbar data ranges were set the same as Figs. 7(a) and 7(c) for comparison. The bottom left of the Strehl ratio can be divided into three regimes depending on how Stokes pulse is generated: (1) Weak Stokes pulse, with a large Strehl ratio, self-generated by the second pulse because phonons of the first pulse are too weak to drive its SRS. It is at the trailing edge of the second pulse. (2) Weak Stokes pulse, with a large Strehl ratio, generated through phonon-driven SRS. It is at the leading edge of the second pulse. (3) Concurrent Stokes pulses from the self-induced and the phonon-driven SRS, where two independent pulses, one at the leading edge and the other at the trailing edge, result in a ring of low Strehl ratios amid the region of large ones on the bottom left.
C. Comparison of Two-Color and Single-Color Schemes
For two-pulse approaches, the two-color scheme outperforms the single-color scheme in total quantum efficiency while making it easier to achieve good pulse quality. For the single-color scheme, the total quantum efficiency is merely 18% [Figs. 4(c) and 4(d)]; however, it is 41% for the two-color scheme (Fig. 6). This can be advantageous from the source point of view, where almost half of the source photons are converted to the LWIR wavelength. Although we can obtain a short 86-fs Stokes pulse with a 15-mJ first pulse in the single-color scheme [Figs. 4(c) and 4(d)], it exhibits significant fluctuations and its Strehl ratio is only half that achieved by the two-color scheme (Fig. 6). Figure 11 shows the quantum efficiency and the Strehl ratio of the Stokes pulse in the single-color scheme. In the regime where the quantum efficiency is high due to the increased first-pulse energy, the Stokes pulse, however, exhibits poor pulse quality, with a maximum Strehl ratio of ${\sim}0.4$. Increasing the first-pulse energy beyond 20 mJ without increasing the stretching will induce more SPM, which will hinder the generation of the first-pulse Stokes wave and its phonons. On the other hand, reducing the first-pulse energy leads to its slow Stokes-wave generation and thus weak phonons, which allows the accompanying anti-Stokes wave in the second pulse to distort the temporal phase of the Stokes pulse, similar to the imperfect phonon amplifier discussed earlier. For more details on the detrimental effect of anti-Stokes waves on the Stokes pulse, refer to Supplement 1. Although the phase-matching condition of a phonon amplifier is automatically satisfied in the single-color scheme for LWIR generation, the 2-µm first pulse is unable to generate a strong burst of phonons for fast phonon amplification to avoid interference from other processes, such as the SPM of the pump, as well as XPM and backconversion of the anti-Stokes pulse. The interference not only deteriorates the pulse quality but also increases the peak-power fluctuation due to the varying phonon beating pattern from pulse to pulse. The mutually exclusive regimes of high efficiency and good pulse quality illustrate that the single-color and double-pulse approach is best-suited only to generate NIR Stokes pulses.
Of course, a practical disadvantage of the two-color scheme is the requirement of a source of two synchronized high-energy pulses at different wavelengths. This introduces complexity and cost to the overall source of LWIR pulses. Note that the quantum efficiency in this article considers only the schemes per se. The overall quantum efficiency with respect to an initial pulse will depend on the efficiency of generating the second color as well as the efficiency of the Raman process, and could be lower than that of the single-color scheme. For some applications, the single-color scheme may offer adequate performance, with potential advantages of simplicity, stability or reliability, and reduced cost; however, it suffers from higher peak-power fluctuation and poorer pulse quality, as discussed previously.
4. PHOTOIONIZATION
We considered femtosecond pulses with millijoule energies in our study, so the peak power might be strong enough to ionize hydrogen. To analyze the effects of photoionization on the SRS, we included a photoionization term in the pulse propagation equation, as discussed in the photoionization section in Supplement 1. In the ionization theory, the Keldysh parameter $\gamma$ is a dimensionless metric to determine whether the interaction of an atom with an electric field is in the tunneling ionization regime ($\gamma \lt 1$) or in the multiphoton ionization regime ($\gamma \gt 1$) [83]. We calculated the Keldysh parameters of two pulses with the highest pulse energy considered here; it is 9.6 for the 5-mJ pulse at 0.8 µm, and is 2.7 for the 10-mJ pulse at 2 µm [Fig. 12(a)]. This shows that they are both in the regime of multiphoton ionization. Despite the simplicity of the Ammosov–Delone–Krainov model [84], it is restricted to around $\gamma \lt 1/2$, which is inappropriate for the conditions considered here. As a result, we chose to implement the Perelomov–Popov–Terent’ev model [85], which works for a wider range of $\gamma$ due to its inclusion of the multiphoton-ionization effect.
Fig. 12. (a) Keldysh parameter $\gamma$, excited free-electron density ${n_e}$, and (b) exponential factor $g/| A |$ in the ionization rate equation. ${n_e}$ is normalized to the hydrogen gas number density ($4.9 \times {10^{26}}\;{{\rm m}^{- 3}}$), while $g/| A |$ is normalized to facilitate comparison of the two pulses. The energy of the first 0.8-µm pulse is 5 mJ, and the energy of the second 2-µm pulse is 10 mJ.
To analyze photoionization, we performed simulations with a 5-mJ first pulse and a 10-mJ second pulse, which were assumed in the calculation of the Keldysh parameters. Although the 2-µm pulse has a higher peak power, it does not excite free electrons as efficiently as the 0.8-µm pulse [${n_e}$ in Fig. 12(a)]. In the regime of multiphoton ionization, a field with a higher frequency requires fewer photons to ionize an atom or molecule, and thus can induce ionization more easily. This can be illustrated with the ionization rate equation $W(t) \propto {{\rm e}^{- Cg(\gamma)/| {A(t)} |}}$, where $C$ is a factor independent of $\gamma$, $| {A(t)} |$ is the field strength, and $g(\gamma) = 3\ln (2\gamma)/2\gamma$ is a correction factor that accounts for the actual effect according to the ionization regime. $g(\gamma)/| {A(t)} |$ determines the strength of ionization due to the exponential dependence of $W(t)$. $g(\gamma)$ decreases monotonically with increasing $\gamma$ ($\gamma \propto \omega$, with $\omega$ the angular frequency of the field), leading to increased ionization at higher frequencies. The strong 2-µm pulse has $1.4$ times larger $g/| A |$ than the 0.8-µm pulse [Fig. 12(b)], resulting in a much weaker ionization rate. Despite the increasing density of free electrons from the first pulse, these results confirm that the density is too low to appreciably influence the nonlinear pulse propagation.
5. CONCLUSIONS
In conclusion, we have studied the generation of femtosecond pulses in the LWIR range by SRS in a hydrogen-filled waveguide. Several challenges must be overcome to efficiently generate high-quality pulses of ${\sim}100 {\text -} {\rm fs}$ duration in this region of the spectrum. Although existing approaches with either single or double chirped pulses can deliver femtosecond LWIR pulses, they suffer from either Raman spectral narrowing or low efficiency; it is also challenging to obtain high-quality Stokes pulses with them. A two-color and two-pulse approach is proposed; a shorter-wavelength first pulse efficiently generates phonons, which help seed the vibrational SRS of a longer-wavelength second pulse. Numerical simulations show that it is possible to generate clean sub-100-fs pulses at 12 µm, with 48% quantum efficiency for the second pulse or 41% quantum efficiency considering both pulses.
Numerical simulations revealed the factors that underlie the excellent simulated performance. Raman gain suppression is required to suppress rotational SRS, and this is possible in a waveguide. The role of phase matching in the generation of phonons was discussed, and optimal results are found to occur if the system is operated as an efficient phonon amplifier.
Although pulse energies in the range of 1 mJ to 10 mJ are the focus of this work, SRS may be effective at lower pulse energies in capillaries with smaller core diameters. In addition to silica-based capillaries, chalcogenide-based hollow-core fibers provide promising options [86–88], even though the loss of a chalcogenide fiber [89] would require all wavelengths to be longer than 1 µm. Pulse energies above 10 mJ, on the other hand, can be easily accommodated by the 300-µm-core-diameter capillary considered here with more temporal stretching. It can also be done with a capillary with a larger core size, but Raman gains of both rotational and vibrational transitions will require careful adjustments through Raman gain suppression to obtain dominant vibrational SRS.
We believe that the results presented here offer promising routes to practical waveguide-based sources of energetic femtosecond pulses at LWIR wavelengths.
Funding
Office of Naval Research (N00014-19-1-2592).
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.
Supplemental document
See Supplement 1 for supporting content.
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