1. Introduction

The injection-locked pulsed laser [1] is a powerful tool for studying nonlinear optical processes such as four-wave mixing and Raman scattering, particularly in isolated atoms or molecules [2, 3]. Injection-locked laser systems have also been employed in various applications such as laser radar and differential absorption lidar [4].

Recently, an extension of the injection-locked laser system to a dual-wavelength-oscillation was reported [5, 6]. (Closely related works with the dual-wavelength injection-locked laser system: see Ref. 7–9.) The advantages of an injection-locked pulsed laser, specifically, the coexistence of high power and spectral purity, and the precise control of the oscillation wavelength, were extended into the dual-wavelength regime. As a significant additional advantage, the automatic temporal and spatial overlap of the two-wavelength pulses were also successfully demonstrated.

The next essential step would be the establishment of a design method capable of precisely realizing various output-performances, such as a specific combination of two wavelengths and their pulsed power ratio. The purpose of this letter is to demonstrate a procedure for establishing a basic model of a dual-wavelength injection-locked pulsed Ti:sapphire laser, and to demonstrate the ability to generate precise, predictable outputs over a wide spectral region and a full dynamic range through comparisons between calculations and measurements.

2. Dual-wavelength injection-locked pulsed laser system

 

Fig. 1. Conceptual schematic of the dual-wavelength injection-locked pulsed laser. Ti:s: titanium sapphire laser crystal, PZT: piezoelectric transducer, OC: output coupler. The insets show the reflectivity of the output coupler as a function of wavelength and a typical spectrum for dual-wavelength injection-locked pulsed output, respectively.

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A conceptual schematic of the dual-wavelength injection-locked pulsed laser studied here is shown in Fig. 1. (Greater detail is provided in Ref. 5) The laser system consists of a single power oscillator and two independent seed lasers. The power oscillator is a pulsed Ti:sapphire laser with a triangle resonator. The two seed lasers are a continuous-wave Ti:sapphire laser (Seeder1, Λ₁; Coherent 899–21) and a home-made external-cavity-controlled diode laser (Seeder2, Λ₂). The wavelengths of the two seed lasers are stabilized at peaks of longitudinal modes of the power oscillator by way of a dual feedback loop [5].

We used a second harmonic of a Q-swiched Nd:YAG laser for pumping the Ti:sapphire crystal. The typical pump energy was 37 mJ per pulse, resulting in a total pulsed energy of 13 mJ for the two-wavelength outputs. The measurements described below were carried out at this pump energy. As seen in Fig. 1, the pulsed output energy was completely concentrated at the two seed wavelengths. The performance of this laser system, such as the selection of the two wavelengths and their pulsed energy ratio, was essentially controlled by the seed lasers only [5].

3. Model for dual-wavelength injection-locked pulsed laser

3.1 Master equation

The model for this dual-wavelength injection-locked laser is simply based on a set of rate-equations using the population inversion, ΔN(t), and intracavity photon numbers, n_Λ1(t) and n_Λ2(t), for the two seed wavelengths. These formulas are shown above. It is assumed that the generated pulses are longer than the cavity length. As a result, the population inversion and the photons of the two wavelengths are distributed homogeneously inside the cavity, resulting in the above expressions in which ΔN, n_Λ1, and n_Λ2, are functions of a single-parameter, time t.

The laser medium is represented in the above equations by the population-inversion, ΔN(t), only. This is due to approximating the decay of the lower level of the laser transition as immediate. In addition, the initial population inversion is also approximated to be instantaneous at t = 0, since the pumping process (~ 5 ns) is shorter enough than the typical buildup-time (~ 25 ns) of the injection-locked pulses. This initial population-inversion, ΔN(0), decays through stimulated and spontaneous (3.2 μs) emission processes. In this model, the important assumption regarding the homogeneous broadening over the relevant wide spectral region of the laser transition is introduced. This is expressed in the coupling of the two photon wavelengths to the common population inversion by the respective gain-coefficients,$G\left({\Lambda }_i\right)=\frac{\mathrm{c\sigma }\left({\Lambda }_i\right)}{\mathrm{LA}}$, (i = 1 or 2). Here, σ (Λ_i), A, c, and L are a cross-section of the stimulated emission, a cross-sectional-area of the cavity-mode, the velocity of light: 3×10¹⁰ cm/s, and a cavity length: 23.4 cm, respectively.

The injected photon numbers of the seeds are noted by n_Λis, producing the initial intracavity photon numbers, n_Λi(0)=n_Λis τ_c(Λ_i) . These seeds are amplified with a gain-coefficient, G (Λ_i), and at the same time decay with the cavity-lifetime τ_c (Λ_i). τ_c (Λ_i) is defined by $-\frac{L}{c\bullet \ln R\left({\Lambda }_i\right)}$, where R (Λⁱ) is the reflectivity of the output coupler (See the inset of Fig. 1, 0.622 and 0.628 at 764.4 and 784.4 nm, respectively). The reduction energies for τ_c (Λ_i) correspond to the injection-locked pulsed outputs, P (Λ_i).

3.2 Procedure for determining the effective parameters of the model

In order to carry out the calculations based on this model, precise values of the initial population inversion, ΔN(0), and the wavelength-dependent gain coefficient, G(Λ_i), are further required in addition to the other parameters above. These parameters, however, strongly depend on the alignment of the pumping YAG laser, the Ti:sapphire crystal quality, and other experimental factors. It is not realistic to determine all of these parameters ab initio including these experimental factors. In the present model, effective values for these parameters were used based on careful comparisons with the actual laser performance.

First, the pulsed waveform was measured with the laser system oscillating at a single-frequency of 784.4062 nm. The initial population inversion, ΔN(0) = 5.096×10¹⁶, corresponding to 49% of the actual pump energy of 37 mJ, was obtained from the total photon number of the pulsed outputs. Using this value for ΔN(0), the effective gain coefficient at this wavelength, G(784.4)=3.257×10^-8 s^-1, was determined so that the calculated pulsed waveform coincides with this measurement. This pumping layout was maintained and these parameters, ΔN(0) and G(Λ_i), were scaled proportionally for other pumping energies.

The injection-locked laser was then switched to the dual-wavelength mode. Dual-wavelength oscillations were observed at various combinations of the two wavelengths over 760–830 nm, while one of the wavelengths, Λ₁, was fixed at 784.4062 nm. The wavelength-dependent effective gain coefficients, G(Λ_i), were determined so that the calculated pulsed energy ratios coincide with the measurements (3.028×10^-8 s^-1 at 764.4 nm or 3.299×10^-8 s^-1 at 807.8 nm).

 

Fig. 2. Frequency dependence of the effective gain coefficient, G(Λ) (normalized at the peak). Circles are the effective gains reduced from the model with one wavelength fixed at 784.4062 nm (solid circle). Lorentz fitting is shown by the solid curve, together with the dotted curves but shifted by ±1% . Blue curve is reported in Ref. 10.

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Figure 2 plots the effective gain coefficient, G(Λ), normalized at the peak, as evaluated using this procedure. The solid curve is a fitting using a Lorentz function, with the dotted curves representing shifts in this function by ±1% . As shown in Fig. 2, the gain coefficients evaluated from the measurements varied within this ±1% region. This is not a fluctuation, but rather is due to the residual wavelength dependence of the present laser system caused by a small mismatch between the c axis of the Ti:sapphire crystal and the polarization axis of the seeds. The blue curve is from a previous report [10]. Taking into account for variations in Ti:sapphire crystal quality, the gain coefficient, G(Λ), obtained using this procedure is reasonably accurate.

4. Demonstration of precisely predictable laser performance

On the basis of the model established here, it is possible to precisely predict many aspects of the laser performance. For example, it is possible to ascertain the seed powers, $n_{{\Lambda }_{\mathrm{is}}}\frac{\mathrm{hc}}{{\Lambda }_i}\equiv S\left({\Lambda }_i\right)$, required for generating an equal pulsed output energy. It is a common requirement to set an equal pulsed energy for arbitrary combinations of two wavelengths.

Figure 3 shows the seed power ratios, $\frac{S\left({\Lambda }_2\right)}{S\left({\Lambda }_1\right)}$, required for obtaining equal pulsed outputs for the two wavelengths. Here, one of the seed wavelengths, Λ₁, was fixed at 784.4062 nm, while the other seed wavelength, Λ₂, was varied over the wide range of 760 ~ 830 nm. The prediction of the model is shown by the solid curve with the blue dotted curves representing shifts of ±1% . The circles are the measured results. As seen clearly in Fig. 3, the prediction accurately reproduces the measured data over a wide spectral-region, even in cases that require the seed-power ratio of more than twenty times, while allowing for ±1% variations in G(Λ).

 

Fig. 3. Seed power ratios giving equal pulsed outputs for various combinations of the two wavelengths with one wavelength fixed at 784.4062 nm (solid circle). The behavior predicted from the model is shown by the solid curve, together with the blue dotted curves representing shifts by ±1% . The circles are measured values.

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In contrast to having the equal pulsed output for various two-wavelength combinations, it would also be significant in terms of practical use to be able to manipulate the pulsed output ratio for the two wavelengths over a wide dynamic range.

Figure 4 shows pulsed energy ratios, defined by $\frac{P\left({\Lambda }_2\right)}{P\left({\Lambda }_1\right)+P\left({\Lambda }_2\right)}$, as a function of the seed power ratio, $\frac{S\left({\Lambda }_2\right)}{S\left({\Lambda }_1\right)+S\left({\Lambda }_2\right)}$, while the two seed wavelengths, Λ_1,2, are kept constant. The black curve shows the predicted values from the dual-wavelength laser model when the two wavelengths were set at 784.4043 and 807.6311 nm for Λ₁ and Λ₂, respectively. These wavelengths have nearly the same gain-loss-product, $\frac{G\left({\Lambda }_2\right)R\left({\Lambda }_2\right)}{G\left({\Lambda }_1\right)R\left({\Lambda }_1\right)}=0.9862$. In such a case, the pulsed energy ratio varies almost linearly with the seed power ratio, as seen in Fig. 4. The black circles indicate the measured values. It is clear from these results that the predicted curve accurately reproduces the actual performance over a full dynamic range.

As the gain ratios deviate far from unity, the predicted curve gradually bends tightly. The green and blue curves are predictions for the gain-loss-product of 0.9556 (Λ₁: 784.4053 nm, Λ₂: 815.9666 nm) and 0.9199 (Λ₁: 784.4063 nm, Λ₂: 764.2493 nm), respectively. The green and blue circles indicate the corresponding measured values. These results demonstrate that even in cases in which the calculated curves deviate far from a linear dependence, the model still accurately reproduces the measurements, and that the pulsed energy ratios can be precisely controlled over a full dynamic range. This laser model also simultaneously provides temporal waveforms. It was also shown that the model precisely reproduced not only the pulsed energy ratio but also the temporal waveform entirely, even when the gain coefficient and the pulsed energy ratio deviated far from unity.

 

Fig. 4. Control of the pulsed output ratios of the two wavelengths by adjusting the seed power ratios. Three cases are shown by the curves (from the model) and circles (measured). Black curve and circles; Λ₁: 784.4043 nm, Λ₂: 807.6311 nm. Green curve and circles: Λ₁: 784.4053 nm, Λ₂: 815.9666 nm. Blue curve and circles; Λ₁: 784.4063 nm, Λ₂: 764.2493 nm..

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The accuracy of the model described in this study also clarifies some important points regarding the performance of the dual-wavelength laser. First, the assumption regarding homogeneous broadening is accurate for Ti:sapphire lasers. Second, the pulsed output ratios are essentially dominated by the stability of the two seed wavelengths as shown in Fig. 4. More precisely, the pulsed outputs from the two wavelengths are stable, and the competition between the two wavelengths does not induce instability [11]. Furthermore, the relative pulsed waveforms are identical and have no relative jitter, even though the combinations of the two wavelengths and/or their power ratios vary widely. These were confirmed consistently by both the actual laser performance and the proposed model.

5. Conclusion

We have studied the characteristics of a dual-wavelength injection-locked pulsed laser. The procedure to establish a corresponding laser model has been described, and the accuracy of this model has been clarified over a wide spectral region and a full dynamic range through comparisons between predictions and measurements. The model established here provides a foundation for extensions of the present laser system to more complex laser systems such as an optical-comb injection-locked laser (to be reported elsewhere), and also to various applications [12] of the present laser system.