1. Introduction

Luminescent emission, corresponding to the electronic transition from the excited state to the ground state, has been an interesting topic in condensed matter physics over one hundred years since the propose of a discrete Bohr atomic model [1]. As well known, the electronic transition is not isolated in realistic materials and it can be affected by the surrounding lattices with the simultaneous emission and absorption of one or more phonons. This process of energy transfer is governed by the electron-phonon coupling strength and gives rise to a distinct multiphonon fluorescence modulated by increasing phonon numbers step-by-step [2,3]. In 1950, Huang and Rhys built a theoretical model to explain the multiphonon-assisted light emission and absorption on the basis of the Franck-Condon principle. A dimensionless Huang-Rhys factor (S) was proposed to evaluate the electron-phonon coupling strength quantized by the phonon numbers [4]. Till now, electron-phonon coupling effect and corresponding Huang-Rhys factors have been widely studied in the field of photonics, including F-centers, light-emitting diodes, phosphors materials and even semiconductors [5–9].

Laser is created by the light amplification by stimulated emission of radiation, which is inherently linked to spontaneous fluorescence emission. Accordingly, electron-phonon coupling should be also an important factor to modulate lasing emission. In general, transition metal ions with electron configuration of 3d^N, lack shielding of the outer-shell electrons and hold strengthened electron-phonon coupling interaction and fluorescence extension [10]. Since the first laser was realized in ruby (Cr³⁺:Al₂O₃) crystal [11], many broadband vibronic lasers have been developed in Ti:sapphire (Ti³⁺:Al₂O₃), alexandrite (Cr³⁺:BeAl₂O₄), Cr:Colquiriite (Cr³⁺:LiSrAlF₆ and Cr³⁺:LiCaAlF₆) [12–15], etc., which is benefited from great spectral broadening induced by strong electron-phonon coupling effect. Even, our group demonstrated an ultrabroadband multiphonon-assisted laser beyond the fluorescence spectrum over 400 nm [16]. Many intriguing phenomenon, e.g., ultrafast lasers, frequency-comb generation, homoclinic orbits and chaos, self-pulsing, were discovered owing to complex coupling among photons, electrons and phonons [17–21]. Despite of great success, however, their behind physical mechanism for describing the multi-body interaction during multiphonon-assisted lasing process is still unrevealed. At present, almost all vibronic lasers can only be prejudged by the experimental spectroscopy without a reliable guiding theory.

Alexandrite is a broadband tunable laser gain medium with excellent laser performances [22]. Since 2013, the maturity of high-power red laser diode (LD) technology has injected new vitality into the development of highly-efficient and miniaturized alexandrite lasers [23]. A series of achievements have been made in the LD-pumped alexandrite lasers. For example, A. Teppitaksak et al. achieved a continuous laser output of up to 26.2 W in the alexandrite, which is the maximum output power among the LD-pumped alexandrite laser [24]. In 2018, W. R. Kerridge-Johns et al. achieved a tunable alexandrite laser generation from 714 nm to 818 nm, with a tuning range of 104 nm, which is the widest tuning range [25]. In addition, our group developed the mode-locked alexandrite laser with the highest repetition rate of 7.5 GHz and the shortest pulse width of 36 fs, respectively [26,27].

Here, based on the multiphonon-assisted transition theory proposed by Huang [4], we established a direct relationship between the laser performance and dynamic step-by-step multiphonon coupling process. Taking an alexandrite crystal as the example, the tunable laser performances (730-815 nm) for various phonon numbers (p = 2 ∼ 5) were manifested and employed to identify the proposed theoretical relationship. Besides, the temperature dependence was also found and verified. Our work should be helpful for discover the broadband lasing and inspire the study of laser physics in the strong electron-phonon coupled systems.

2. Theoretical model and fluorescence spectrum

In view of crystal structure, the electron-phonon coupling represents the interaction between the active cations and its surrounding host lattices. The electronic transitions of active cations could be perturbed by its adjacent or sub-adjacent atom displacement in host lattices, thereby introducing some coupled states on intrinsic electronic states and leading to phonon-assisted photon emission. This physical principle is always there and should be evidenced in all laser media. For a laser crystal containing significant electron-phonon coupling effects, the total Hamiltonian H_T of the electron-lattice system can be written as Eq. (1) [28]:

According to Huang’s single-frequency and multi-frequency models [28], the fluorescence spectrum consists of a series of discrete phonon spectral lines under the single-frequency model, while for phonons with different frequencies, its transition will naturally produce a continuous spectrum. Here, we only consider the single-frequency approximation and an average phonon energy is used in the calculation. A classical configuration coordinate model of phonon-assisted electron transitions [29], as shown in Fig. 1(a), was plotted to describe multiphonon transitions. The absorption and emission of photons depend on the vertical transitions between the electron ground state (GS) and the excited state (ES) under Born-Oppenheimer approximation. A series of horizontal lines represent the phonon coupled energy levels under a single-frequency model. The electron-phonon coupling effect is related to lattice relaxation, manifested as a configurational displacement ΔQ between the equilibrium position of GS and ES, as well as some multiphonon-assisted transitions. Therefore, at a limited temperature, the fluorescence emission will consist of the peak of zero phonon (p = 0) and the intrinsic sidebands formed by different phonons participating (p ≥ 1). The phonon-assisted fluorescence lines are determined by the intensity of electron-phonon coupling $S\hbar {\omega _0}$, where S is Huang-Rhys factor and $\hbar {\omega _0}$ is the average phonon energy, which can be calculated by experimental fluorescence spectrum. The intensity of fluorescence reaches the maximum when the number of participating phonons p approaches S, and then decreases when p > S.

 

Fig. 1. Schematic diagram of the electron-phonon coupled laser. (a) Configuration coordinate model of fluorescence. (b) Fluorescence spectrum of alexandrite crystal at 300 K (E//b). (c) Lineshape function of alexandrite crystal at 300 K (average phonon energy $\hbar {\omega _0}$= 524 cm^-1 and S = 2.13).

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For the multiphonon-assisted electronic transitions, we assume that the vibration modulus N interacting with the electron is very large. Under single-frequency approximation, the transition probability from electronic state j to i is determined by the following lineshape function [4]:

So the overlapping integral in Eq. (3) becomes

Expand Eq. (5) with $\frac{{{\Delta _{ji}}}}{{\sqrt N }}$ as a series,

In Eq. (6), ${F_0}\left( {\frac{{{\Delta _{ji}}}}{{\sqrt N }}} \right)$ is the zero phonon line (ZPL) transition process without the participation of phonons, ${F_1}\left( {\frac{{{\Delta _{ji}}}}{{\sqrt N }}} \right)$ is a single-phonon process in which each vibration mode produces or annihilates one phonon, ${F_2}\left( {\frac{{{\Delta _{ji}}}}{{\sqrt N }}} \right)$ is a two-phonon process in which each vibration mode produces or annihilates two phonons, and so on. For the transition of the number of phonons from n to n′, the lowest term in the overlapping integral expansion Eq. (6) is ${\left( {\frac{{{\Delta _{ji}}}}{{\sqrt N }}} \right)^{n^{\prime} - n}}$. Since N→∞ is assumed, so only the transition of the number of phonons of each vibration mode that changes by at most ± 1 can contribute to the transition probability.

At a finite temperature (T ≠ 0 K), the number of initial phonons will not be zero, so various transitions with both phonon absorption and emission must be considered. Thus, the results will include the phonon numbers of each mode of initial state, which need to be averaged according to statistical probability. Therefore, the lineshape function F(λ, T) of multiphonon-assisted electron transitions can be obtained as Eq. (7) [4,28]:

According to the Huang-Rhys theory [28], S and $\hbar {\omega _0}$ can be calculated by Eq. (10):

Moreover, crystal temperature is a crucial parameter to determine the strength of electron-phonon coupling. The electron-phonon coupling effect will produce a lattice relaxation shift Δ_ji [33]. Under the harmonic approximation, the lattice relaxation shift Δ_ji² ∝$\hbar {\omega _0}$ ∝k_BT [34]. Therefore, it can be obtained that ω₀ ∝T and S ∝ω₀Δ_ji² ∝T². As shown in Fig. 2(a), we measured alexandrite fluorescence at different temperatures from 150 K to 348 K using an Edinburgh fluorescence spectrometer (FLS920). Meanwhile, $S(2\bar{n} + 1)$ at each temperature can be calculated by Eq. (10), as shown in the scatter in Fig. 2(b). Then, the values of ln(I/I_ZPL) can be fitted, as shown the red dash line in Fig. 2(b). It also observed that both S and $\hbar {\omega _0}$ increase with high temperature, thus leading to the enhancement of the spectral intensity (F values at each p) and the broadening of the spectral width of the multiphonon-assisted transitions, respectively.

 

Fig. 2. (a) Fluorescence spectrum of alexandrite at 150-348 K. (b) Experimental and fitting results of ln(I/I_ZPL).

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3. Laser experiments and theoretical calculation

To verify the above theoretical predictions, wavelength tuning experiments were subsequently performed on alexandrite crystal. The device of experiment is shown in Fig. 3(a). The commercialized fiber-coupled 638 nm LD is the pump source, in which the maximum output power was 20 W. The core diameter is 200 µm and the numerical aperture is 0.22. The pump light is focused by a beam coupler with a focusing ratio of 1:1 and a focal length of 65 mm. Due to the polarized absorption of alexandrite, only about two-thirds of the pump light is effectively absorbed by the alexandrite polarized along its b-axis. The gain medium is a c-cut alexandrite crystal with Cr³⁺ doping concentration of 0.2 at.% and dimensions of 4 × 4 × 10 mm³ (a × b × c), and coated antireflection (AR) coating at 638 nm in the both ends. The alexandrite is wrapped by indium foil and installed on the copper holder with a water cycle temperature regulated in the range of 5-50°C. The resonator adopts a three-mirror V-type cavity. The plane input mirror M1 is AR in the wavelength of 550 to 665 nm and highly reflection (HR) in the wavelength of 700 to 900 nm. A plane-concave mirror with a radius of curvature of 200 mm is used as folding mirror, which is HR in the wavelength of 700 to 900 nm. Its folding angle is set to be about 12°. The output coupling mirror M3 is also a plane mirror and HR in the wavelength of 700 to 900 nm (R > 99.9%). In addition, a quartz birefringent filter (BiFi) with a thickness of 0.5 mm is used to wavelength tuning and is placed in the cavity at the Brewster’s angle.

 

Fig. 3. (a) Experimental setup for alexandrite laser. (b) Variation of laser output power with wavelength. (c) Tunable laser spectrum at 323 K.

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Using this experimental device, the wavelength tuning is realized, and the corresponding tunable laser output power and laser spectrum are measured by power meter (Newport, Model 1919-R) and spectrometer (Ocean Optics, HR4000) respectively. The results are shown in Figs. 3(b) and (c), respectively. It can be seen from Fig. 3(b) that the wavelength tuning in the range of 728-789 nm is realized at 283 K, and 730-815 nm at 323 K. In addition, the tunable laser spectrum at 323 K with a tuning bandwidth of 85 nm is shown in Fig. 3(c). It can be seen that the output power and the tuning bandwidth increased with the increase of temperature from 283 to 323 K. Compared to lineshape function in Fig. 1(c), the laser wavelength can be basically attributed to multiphonon transitions with varying phonon numbers (p = 2∼5). According to the previous report on the excited state absorption (ESA) cross section of alexandrite crystal at 25°C [35], we know that the alexandrite crystal is seriously affected by the ESA effect when λ < 750 nm and λ > 810 nm. Therefore, we only focus on laser emission in a wavelength tuning range of 760-810 nm. Notably, the laser performances for p = 2 (λ < 740 nm) is relatively deteriorated due to strong ESA effect at this wavelength.

Generally, the laser performances should gradually deteriorate with the increase of the involved phonon number p. It is expected that the lasing threshold P_th increases, because the F values decreases with the increase of participating phonons. Here, based on the formula of P_th of quasi-three levels [36] and the relationship between emission and absorption cross sections and lineshape function [35,37], we give a general relationship between the lasing threshold P_th and the spectral lineshape function, such as Eq. (11):

The variations of P_th versus λ at crystal temperatures of 287 K, 307 K and 332 K were shown in Fig. 4. The solid lines represented the experimental results, which were the statistical average of the ten groups of experiments. The dashed lines represented the theoretical calculation with β = 0.15. Due to the influence of ESA effect at p = 2, we only consider the case of λ ≥ 760 nm (p ≥ 3). Figure 4 exhibits P_th increases with the red-shift of wavelength, because the wavelength red-shift means that the increasing phonon numbers participated in multiphonon transitions, thus reducing the transition probability and increasing P_th. This overall threshold trend is consistent with the theoretical prediction, except that the predicted change of the tail wing of P_th with λ is relatively gentle compared with the experimental results. According to the ESA cross section of alexandrite crystal at room temperature [35], we know that the alexandrite crystal is seriously affected by the ESA effect at long wavelength. Therefore, this divergence can be attributed to the ESA of the alexandrite crystal, which leads to a decreased effective gain cross-section and a slight increase on P_th.

 

Fig. 4. Variations of P_th with λ at the crystal temperatures of 287 K, 307 K and 332 K, respectively. The experimental threshold data presented as mean ± s.d. (error bars, for ten crystals). The theoretical values were calculated with β = 0.15.

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Moreover, the temperature dependent thresholds P_th at laser wavelengths of 760 and 789 nm were shown in Fig. 5, corresponding to the cases of phonon number p = 3 and 4, respectively. The experimental data was the statistical average of the five groups of crystals. According to the ground state absorption (GSA) cross section of alexandrite crystal at room temperature [35], the GSA can be ignored in theoretical calculations at room temperature and λ ≥ 760 nm, so we obtained ${P_{th}} \propto {1 / {[{{\lambda^2}{F_{em}}(\lambda ,T)} ]}}$. It can be seen that the experimental results were in good agreement with the theoretical prediction. The P_th decreases with increasing temperature at wavelength of 760 nm and 789 nm. This temperature behavior for each wavelength can be explained by the changes of lineshape function versus T under the participation of different phonon numbers, that is, with the increase of temperature, the phonon-assisted transition probability increases, leading to the decrease of P_th. For alexandrite crystal, the increased number of phonons participated into lasing would bring long laser wavelength, which is very favorable for laser wavelength extension under high temperature because Huang-Rhys S factor is proportional to temperature.

 

Fig. 5. Variations of P_th with T at wavelengths of 760 and 789 nm. The experimental threshold data presented as mean ± s.d. (error bars, for five crystals). The theoretical values were calculated with β = 0.15.

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4. Conclusion

In conclusion, we established a relationship between the spectral lineshape function of the multiphonon-assisted transitions and broadband vibronic laser behavior. As expected, laser output power and P_th exhibit significant wavelength and temperature dependence on the number of participating phonons, as demonstrated in alexandrite crystal. The experimental results were well consistent with theoretical prediction, thus proving the reliability of our theory. In addition, this physical mechanism is also broadly applicable to other transition metal ion doped systems with wide emission spectrum, such as Ti:sapphire, and Fe²⁺:ZnSe, etc. Deciphering the complicated relations between electron, phonon, and photon in a strongly coupled system, would shed light on other physical phenomenon, including superconductor, photo-induced phase transition, and laser cooling materials.