Parameter Oscillators
4.4 Analytical Model for Simultaneous Emission of Fundamental and Signal Waves
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4.4 Analytical Model for Simultaneous Emission of Fundamental and Signal Waves
The OPO device is simple and provides the wavelength tenability to eye-safe region by phase matching condition [24-26]. The dual-signal OPOs generated in the eye-safe wavelength region, utilizing the quasi-phase-matched periodically poled crystals, have been realized [27,28]. High-pulse-energy solid-state lasers combined with Nd3+-doped lasers operating near 1.0 μm and OPO converted eye-safe lasers are potentially valuable for some applications, especially in target ranging of airborne laser systems. The attainment of high-pulse-energy laser increases the risk of optical damage to cavity components because higher pump level is required to exceed pump threshold. Therefore, the demand for damage threshold of the coatings becomes severer.
Intracavity singly-resonant OPO (SRO) takes the advantages of high photon density of fundamental wave, and the good spatial overlapping of fundamental laser and OPO signal. Moreover, the use of output mirrors with partial reflection at fundamental laser instead of a highly reflective mirror is not only to generate a dual-wavelength laser but also to reduce the risk of optical damage [29].
Rate-equation model for the passively Q-switched IOPO has been used to analyze the temporal behavior of fundamental laser and OPO signal pulses [30,31]. The output energy characteristic of the simultaneous emission including the fundamental laser and the OPO signal, however, has not been analyzed and modeled. Nevertheless, to the best of our knowledge, the systematic investigation of the simultaneous emission based on an intracavity SRO has not also been performed.
In this part, we firstly use the rate-equation model of a passively Q-switched IOPO to calculate the output pulse energies of the fundamental laser and OPO signal as functions of the initial transmission of the saturable absorber and the reflectivity at the fundamental laser wavelength. With a nonlinear regression fit to the numerical calculations, we extend the analytical expression for the pulse energy of passively Q-switched lasers reported by Chen et al. [32] to develop an analytical model for the
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output pulse energies of passively Q-switched IOPOs including both the fundamental laser and the OPO signal. To verify the accuracy of the analytical model, based on a passively Q-switched IOPO using a Cr4+:YAG crystal as the saturable absorber, simultaneous generation of the fundamental laser output at 1.06 µm and the OPO signal at 1.57 µm is performed. Various output couplers with different reflectivity at the fundamental laser wavelength, R, are employed to systematically investigate the variation of the output pulse energies of both fundamental laser and OPO signal wavelengths. Experimental results show that the output energy characteristic of a dual-wavelength (fundamental laser/OPO signal) laser agrees very well with the present model. The present model provides design criteria of the dual-wavelength laser with a passively Q-switched intracavity OPO.
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4.4.1 Theoretical Analysis
In Section 4.3, we have employed the rate equation model developed by Debuisschert et al. [30] to confirm the experimental results of an actively Q-switched intracavity OPO. Here the initial population of the passively Q-switched laser is employed in the same rate equation model to calculate the pulse energies of the fundamental laser and OPO signal outputs for passively Q-switched IOPOs with a shared-resonator configuration. In a passively Q-switched laser, the initial population inversion density in the gain medium, n(0)=ni, can be determined from the condition that the roundtrip gain is exactly equal to the roundtrip losses just before the Q-switch opens, i.e.
where To is the initial transmission of the saturable absorber, L is the round-trip fundamental wave intensity loss in the cavity, and R is the reflectivity of the output mirrors at the fundamental wavelength. Since SRO only resonates fundamental laser and signal fields, the evolution equation of the idler wave is eliminated. The rate equations for the four-level Q switched laser with IOPO are given by:
p
where n is the inversion population density of the gain medium, σ is the stimulated emission cross section of the gain medium, c is the speed of light; ϕp is the fundamental laser photon density, ϕs is the OPO signal photon density, lca is the optical length of the laser cavity, lcr is the length of the gain medium, lnl is the length of the nonlinear crystal, σopo is the effective OPO conversion cross section, tr is the round-trip time in the resonator cavity, Δ ϕp is the spontaneous emission intensity,Δϕs is the noise signal intensity, Ls is the round-trip signal wave intensity loss, and Rs is the output reflectivity at the signal wavelength. The effective OPO cross section, σopo, is used to describe the conversion rate and derived from the parametric gain coefficient for small gains of the single resonator oscillator:
2 2 refractive indices at the idler, signal and fundamental laser wavelengths, respectively;
deff is the effective nonlinear coefficient; ε0 is the vacuum permittivity; Αs and Αp are the mode areas for the OPO signal and fundamental laser, respectively. The output pulse energy can be expressed as [33]
where hv is the photon energy, A is the beam area and ϕ is the above-mentioned photon density. The subscripts j=s, p represents the OPO signal and fundamental laser, respectively.
Without loss of generality, we calculate the output pulse energies of the
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dual-wavelength laser for a practical example of an intracavity OPO at 1572 nm with a type-II non-critically phase-matched x-cut KTP crystal pumped by a passively Q-switched Nd:YAG laser with a Cr4+:YAG crystal as a saturable absorber. With the properties of the Nd:YAG, KTP crystals and the typical cavity parameters:
hvs=1.26×10-19 J, hvp=1.86×10-19 J, ωi = 5.712×1014 sec-1, ωs = 1.198×1015 sec-1, lcr = 2.0 cm, lnl = 2.0 cm, deff = 3.64 pm/V , ni = 1.771, ns =1.737, np = 1.748, ε0 = 8.854 pF/m, Αs = 0.108 cm2, Αp = 0.16 cm2 , lca = 5.5 cm, Rs =0.25 , L = Ls = 0.01, and c = 3×108 m/s, the calculated results for the output pulse energies with respect to R and To
are depicted by the solid lines in Fig. 4.4.1. It can be seen that for a given To the OPO signal output energy increases with increasing the value of the reflectivity R, whereas the output energy of fundamental laser decreases with increasing the value of the reflectivity R. Physically, both the output energies of the OPO signal and fundamental laser are proportional to the initial population inversion density in the gain medium ni that is an increasing function of the factor ln 1/
(
To2)
+ln 1/(
R)
+ , Las shown in Eq. (1). From the viewpoint of practical engineering applications, it is of great usefulness to express the output energies of the OPO signal and fundamental laser as analytical functions of To and R.
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Fig. 4.4.1. Calculated results for the output pulse energy as a function of the reflectivity R for several values of To; solid lines: theoretical results calculated from Eqs. (1)-(6); dashed lines:
modeling results obtained with the analytical expressions of Eqs. (7)-(15).
0.80 0.82 0.84 0.86 0.88 0.90 0.92 0.94 0.96 0.98 1.00 0
0.80 0.82 0.84 0.86 0.88 0.90 0.92 0.94 0.96 0.98 1.00 0
0.80 0.82 0.84 0.86 0.88 0.90 0.92 0.94 0.96 0.98 1.00 0
0.80 0.82 0.84 0.86 0.88 0.90 0.92 0.94 0.96 0.98 1.00 0
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4.4.2 Analytical Model
For the passively Q-switched laser, the output pulse energy has been numerically calculated as an analytical function of the parameters α, β, To, and ln 1/ R
( )
+ by LChen et al. [32], where α =σgs A γσ Asa, β =σes σgs , A/Asa is the ratio of the effective area in the gain medium and in the saturable absorber, σgs is the ground-state absorption cross-section of the saturable absorber, σ is the stimulated emission cross-section of the gain medium, γ is the inversion reduction factor (γ = 1 and γ = 2 correspond to, respectively, four-level and three-level systems; see Ref. [21]). Based on the model of Chen et al. [32] and using a nonlinear regression to fit the calculated results obtained with Eqs. (1)-(6), the output pulse energy at fundamental laser wavelength in the dual-wavelength operation can be analytically expressed as
(1 )q T( )o
where Ep is the pulse energy in the passive Q-switching operation and its expression in the model of Chen et al. [32] is given by
On the other hand, the output pulse energy at signal wavelength in the dual-wavelength operation can be fitted to be given by
( , ) p T( )o
To reveal the accuracy of the analytical function, we use the analytical model in Eqs.
(7)-(15) to calculate the output pulse energies for the case given in Fig. 4.4.1. The
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calculated results based on Eqs. (7)-(15) are shown by the dashed lines in Fig. 4.4.1.
The results obtained with the analytical model can be found to agree very well with the numerical calculation for all cases. As a consequence, a straightforward model is successfully developed for the design of passively Q-switched dual-wavelength 1.06 µm/1.57 µm lasers.
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4.4.3 Experimental Results and Discussion
To demonstrate the utilization of the analytical model, a dual-wavelength 1064 nm/1572 nm laser based on an IOPO pumped by a passively Q-switched Nd:YAG laser in a shared resonator was performed, as shown in Fig. 4.4.2. The structure of the shared resonator is that the OPO cavity completely overlaps with the fundamental laser cavity. The pump source was a quasi-cw high-power diode stack (Coherent G-stack package, Santa Clara, Calif., USA) which consisted of six 10-mm-long diode bars with a maximum output power of 120 W per bar at the central wavelength of 808 nm. The diode stack was constructed with 400 μm spacing between the diode bars so the whole emission area was approximately 10 mm (slow axis) × 2.4 mm (fast axis).
The full divergence angles in the fast and slow axes are approximately 35° and 10°, respectively. In the experiment, the diode stack was driven to emit optical pulse durations of 300 μs at a repetition rate less than 30 Hz with a maximum duty cycle of 1%. The pump radiation was delivered into the gain medium with a lens duct that was possessed of the advantages of simple fabrication, high coupling efficiency, and insensitivity to slight misalignment. In our experiment, the lens duct was manufactured with the parameters of r = 10 mm, L = 29 mm, H1 = 12 mm, H2 = 3.5 mm, and H3 = 3.5 mm. The coupling efficiency of the lens duct was experimentally measured to be approximately 85%.
The gain medium was a 1.0 at. % Nd:YAG crystal with a diameter of 6 mm and a length of 20 mm. The entrance surface of the laser crystal was coated with high reflection at 1064 nm and 1573 nm (R>99.8%) and high transmission at 808 nm (T>90%). The other surface of the laser crystal was coated with antireflection at 1064 nm and 1573 nm (R<0.2%). The saturable absorber for the passively Q-switching was a Cr4+:YAG crystal with a thickness of 2 mm and an initial transmission of 50% at 1064 nm. The nonlinear crystal for the OPO was a KTP crystal with a cross section of 4 mm × 4 mm and a length of 20 mm. The KTP crystal was x-cut (θ = 90°, and ϕ = 0°) for type II noncritical phase-matching to eliminate walk-off effect between the fundamental, signal, and idler beams. The polarization of fundamental laser has a preferred linear polarization along the y axis of
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KTP crystal which is the ordinary wave. The OPO signal is also polarized parallel to the y axis, whereas the idler is polarized along the z axis. Intracavity pumped by the 1.064 μm radiation, the OPO produces a signal wavelength at 1.57 μm, and the idler wavelength is 3.3 μm. Because of the strong absorption of the idler wave (3−4 μm) in the KTP crystal and in the BK7 glass mirror, the OPO is resonant on the signal frequency only. In general, the reflectivity of the output coupler made from BK7 glass for idler wave is less than 10%. No idler wave was extracted from the cavity and detected. Both surfaces of the KTP and Cr4+:YAG crystals were coated for anti-reflection at 1064 and 1572 nm. All crystals were wrapped with indium foil and mounted in conductively cooled copper blocks. To investigate the dual-wavelength operation, we used several flat output couplers with the reflectivity of 99.8%, 98%, 94%, and 90% at 1064 nm, and the respective corresponding reflectivity at 1572 nm are 10%, 26%, 10%, and 24%. Note that the optimal output reflectivity at the signal wavelength for the IOPO has been verified to be approximately 10-30% in Section 4.3.
The optical resonator was a plane-parallel cavity. The total cavity length was approximately 55 mm. The dielectric mirror coated with high transmission at 1064 nm (T>95%) and high reflection at 1573 nm (R>99.5%) was used to separate the fundamental laser and OPO signal. Two Filters, F1 and F2, were utilized in the experimental measurement. A LeCroy digital oscilloscope (Wavepro 7100; 10 G samples/sec; 1 GHz bandwidth) with the fast InGaAs photodiodes was used to record the pulse temporal behavior at 1064 nm and 1572 nm. The spectral information was monitored by an optical spectrum analyzer (Advantest Q8381A) that employs a diffraction grating monochromator to for measure high-speed light pulses with the resolution of 0.1 nm.
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Fig. 4.4.2. Experimental setup for an intracavity OPO pumped by a diode-pumped passively Q-switched Nd:YAG / Cr4+:YAG laser in a shared resonator.
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Experimental results revealed that the pump threshold energies were 170, 175, 182, and 188 mJ for the reflectivity of 99.8%, 98%, 94%, and 90% at 1064 nm, respectively. The pump threshold energy can be found to increase linearly with decreasing the output reflectivity. It was also found that the laser threshold for the reflectivity of 80% was greater than the maximum pump energy that was approximately 195 mJ from the diode stack through the lens duct. The dependence of the pump threshold energy on the reflectivity can be calculated with [34]
2
where ηp is the pump efficiency including the overlapping efficiency and the absorption efficiency, and hvp is the pump photon energy. With the properties of the Nd:YAG and Cr4+:YAG crystals and the typical cavity parameters: σ = 2.8×10-19 cm2, hvp = 2.46×10-19 J, ηp =0.54, A = 0.16 cm2, R = 99.8%, To = 0.5, and L = 0.01, the theoretical threshold energies were calculated to compare with the experimental results, as shown in Fig. 4.4.3. It can be seen that the experimental results agree very well with the theoretical values. Note that the pump threshold energy with respect to the reflectivity at the fundamental laser wavelength in different To cases can be theoretically calculated with Eq. (16).
Figure 4.4.4 shows the calculated and experimental results for the output pulse energy with respect to the reflectivity at the fundamental wavelength of 1064 nm.
The calculated results obtained from theoretical Eqs. (1)-(6) and from the analytical expressions of Eqs. (7)-(15), respectively, are also shown in Fig. 4.4.4 for comparison. It can be seen that the analytical model is in good agreement with the experimental data and the theoretical calculation. Lowering the reflectivity at 1064 nm can be found to lead to an increase in the output pulse energy at 1064 nm and a relative decrease in the signal output pulse energy. With the reflectivity of 99.8%, 98%, 94%, and 90%, the output pulse energies at 1064 nm were 0.1, 1.9, 5.3, and 8.9 mJ, respectively and the corresponding pulse energies at 1572 nm were 10.8, 9.1, 7.7, and 6.6 mJ, respectively. As a result, the total pulse energy of fundamental and
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signal outputs increased from 10.9 mJ up to 15.5 mJ for the reflectivity decreasing from 99.8% to 90%. Divided by the pump threshold energy, the overall conversion efficiency was enhanced from 6.4% to 8.2% for the reflectivity decreasing from 99.8% to 90%. In other words, the dual-wavelength output efficiency for the signal and fundamental waves was considerably greater than the output efficiency for only the OPO signal wave.
Typically temporal shapes for the depleted fundamental laser and OPO signal pulses were simultaneously detected for the reflectivity of 99.8%, 98%, 94% at 1064 nm, and 90%, as shown in Figures 4.4.5 (a)-(d). It can be seen that the pulse width of the fundamental laser output conspicuously decreases with decreasing with the reflectivity at the fundamental laser wavelength. On the other hand, the pulse shape of the OPO signal output generally displayed a sharp peak accompanied by a much longer long tail. Experimental results revealed that the long tail mainly determined the magnitude of the OPO signal pulse energy, whereas the sharp peak had a dominant influence on the peak power. Figure 4.4.6 shows the output peak powers for the fundamental laser and OPO signal with respect to the reflectivity at 1064 nm. The peak powers were precisely deduced by the numerical integration for the measured temporal pulse profiles to fit the experimental pulse energies. It can be seen that the peak power at 1064 nm is enhanced up to 0.3 MW by lowering the reflectivity at 1064 nm. In contrast, the peak power at 1572 nm could be maintained in the range of 0.62 to 0.82 MW on account of the sharp peak. In general, the central part of the fundamental laser beam converts to the OPO signal beam only, which acts as a naturally spatial filter. Higher order transverse modes were not observed in this experiment.
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Fig. 4.4.3. The pump threshold energy with respect to the reflectivity at the fundamental laser wavelength of 1064 nm in To= 50% case; solid lines: theoretical results; symbols:
experimental values.
To=50%
0.80 0.82 0.84 0.86 0.88 0.90 0.92 0.94 0.96 0.98 1.00 140
150 160 170 180 190 200 210
Pump threshold energy (mJ)
Reflectivity at 1064 nm
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Fig. 4.4.4. Calculated and experimental results for the output pulse energy with respect to the reflectivity at the fundamental laser wavelength of 1064 nm in To= 50% case; solid lines:
theoretical results calculated from Eqs. (1)-(6); dashed lines: modeling results obtained with the analytical expressions of Eqs. (7)-(15); symbols: experimental results.
To=50%
0.80 0.82 0.84 0.86 0.88 0.90 0.92 0.94 0.96 0.98 1.00 0
5 10 15 20 25 30
1064 nm
1572 nm
Output pulse energy (mJ)
Reflectivity at 1064 nm
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Fig. 4.4.5. Experimentally temporal shapes of the fundamental laser (1064 nm) and OPO signal (1572 nm) pulses generated in To= 50% case for the reflectivity of (a) 0.998, (b) 0.98, (c) 0.94, and (d) 0.9.
(d) R=0.9
10 ns/div (b) R=0.98
1572 nm
(c) R=0.94 (a) R> 0.998
10 ns/div 10 ns/div
10 ns/div
1064 nm
1064 nm
1064 nm 1064 nm
1572 nm
1572 nm 1572 nm
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Fig. 4.4.6. The output peak power as a function of the reflectivity at the fundamental laser wavelength of 1064 nm in To= 50% case.
0.89 0.90 0.91 0.92 0.93 0.94 0.95 0.96 0.97 0.98 0.99 1.00 0.0
0.2 0.4 0.6 0.8 1.0 1.2
1572 nm 1064 nm
Output peak power (MW)
Reflectivity at 1064 nm, R
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