1. Introduction

Second-harmonic generation (SHG) is an effective way to extend the laser wavelengths from infrared to visible or violet. A phase-matching condition for the interacting fundamental and second-harmonic lasers is required to achieve efficient conversion. In addition, the conversion efficiency is also proportional to the thickness of the nonlinear optical crystal and to the input fundamental intensity [1]. However, the usable crystal thickness is also limited by the walk-off effect when the critical phase-matching style is used [2,3]. The effective SHG can be achieved with an external-cavity scheme [4], an intra-cavity scheme [5], and a single or multipass scheme [6–13]. For the fundamental laser produced from a Q-switched pulse Nd-doped laser, in which the pulse duration is of the order of nanoseconds corresponding to the repetition rate of the order of kilohertz, the single-pass or multipass SHG is also useful because the fundamental intensity is enough to obtain high conversion efficiency.

In the conventional two-pass SHG scheme, the collinear phase-matching configuration is generally used, in which the directions of the fundamental and the second-harmonic lasers in the second pass through the nonlinear optical crystal are opposite to those in the first pass. Such collinear phase-matching configuration has advantages such as a longer thickness of the nonlinear optical crystal and effective compensation for the walk-off effect. For the Q-switched nanosecond-kilohertz Nd-doped laser, the collinear phase-matching configuration will enlarge the thermal effect in the overlapping area of the fundamental and second-harmonic beams therefore a stable second-harmonic laser is difficult to be obtained.

In this paper, we extend the multipass SHG from the collinear phase matching to non-collinear phase matching and describe the experimental results for the power enhancement in the second-harmonic laser. Because of the use of the non-collinear phase-matching configuration, the fundamental and second-harmonic laser beams pass through different parts of the nonlinear optical crystal, so that the thermal effect due to the beam overlap can be reduced and damages on the crystal surfaces can be effectively avoided. Acting in the same way as the collinear phase-matching configuration, the two-pass SHG with the non-collinear configuration has the advantages such as increasing the crystal thickness and compensating for the walk-off effect, so that the enhancement of the second-harmonic laser can be expected.

2. Principle of multipass non-collinear phase-matching SHG

The principle of the multipass non-collinear SHG in a critical phase-matching nonlinear optical crystal is shown in Fig. 1(a) , where k is the propagation axis for the first pass through the crystal, c is the optical axis of the crystal, o and e denote the ordinary and extraordinary polarization directions respectively. In the first pass through the crystal (Fig. 1(b)), the input fundamental laser propagates along the direction of vector k_1ω and the unconverted fundamental and second-harmonic lasers are along k_1ω-1st and k_2ω-1st respectively. The phase-matching condition is achievable by properly electing the phase-matching angle θ_m between the optical axis c and vectors k_1ω, k_1ω-1st, k_2ω-1st to obtain high conversion efficiency. After the first pass, the unconverted fundamental and second-harmonic lasers are then reflected by a mirror with high reflectivity at the fundamental and second-harmonic wavelengths. The second pass through the crystal is carried out by tilting the propagation directions k_1ω-2nd and k_2ω-2nd of the reflected laser beams from the k axis by an angle α in the e-k plane, and then rotating the propagation directions along the o-axis by an azimuth angle β. The phase-matching condition in the second pass can be satisfied if the angle between the propagation directions k_1ω-2nd and k_2ω-2nd and the backward direction c’ of the optical axis c is equal to the phase-matching angle θ_m (Fig. 1(c)). If the transverse size of the crystal is large enough, the third pass through the crystal is achievable by using another mirror to further reflect the unconverted fundamental and second-harmonic lasers so that these beams enter the crystal again. The propagation directions of the unconverted fundamental and second-harmonic lasers in the third pass are the same as the propagation directions k_1ω-1st and k_2ω-1st in the first pass. As the propagation distance through the crystal becomes longer, high conversion efficiency is expected in comparison with that of the single-pass scheme.

 

Fig. 1 (a) A critical multipass non-collinear phase-matching configuration of the second-harmonic generation; (b) The first pass phase-matching style; (c) The second-pass phase-matching style by using a mirror denoted as M to reflect the unconverted fundamental and second-harmonic lasers.

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The relation of the tilting angle α, azimuth angle β and phase-matching angle θ_m, satisfying the non-collinear phase-matching condition, is calculated by means of the transformation matrix [14]. The azimuth angle β is expressed by

The phase-matching condition of the second-pass SHG is approximately a conical curve and the optical axis c is its centrosymmetry axis with a conical angle of 2θ_m. The tilting angle α can be therefore changed from 0 to 2θ_m. The maximum angle of β is equal to θ_m with respect to the tilting angle α = θ_m. For conventional critical phase-matching configurations with lithium triborate (LBO), potassium titanyl phosphate (KTP) and beta-barium borate (BBO) crystals, Eq. (1) can be approximately expressed by

Therefore, the tilting angle α, azimuth angle β and phase-matching angle θ_m corresponding to the phase-matching condition can be imaged by a circle. Figure 2(a) shows the calculated results of the azimuth angle β for the crystals of type I BBO, type I LBO and type II KTP at the fundamental wavelength of 1064 nm. The phase-matching angles for the BBO, LBO and KTP crystals are 22.9, 11.7 and 23.5 degree respectively. The data of the refraction indexes of the crystals are taken from Ref. 15.

 

Fig. 2 (a) The azimuth angle β as the function of the tilting angle α for type I BBO, type I LBO and type II KTP crystals to satisfy the non-collinear phase-matching condition in the second pass; (b) The ratio of the displacement Δ_o along the o-axis direction to the distance L between the crystal and mirror M for type I BBO, type I LBO and type II KTP crystals.

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The unconverted fundamental and second-harmonic lasers between the crystal and the mirror denoted as M in Fig. 1(c) pass through the air at different phase velocities, resulting in an optical-path difference between the fundamental and second-harmonic lasers [16]. The optical-path difference is given by$2\pi L(2+\tan \alpha \tan {\theta }_m)[2n_0(\lambda /2)-n_0(\lambda )]/\lambda$, where L is the normal distance between the crystal and mirror M (Fig. 2(c)), n₀ is the refraction index of the air, and λ is the fundamental wavelength. The mirror will also induce a relative phase accumulation [17]. In order to obtain high conversion efficiency, the phase shift between the fundamental and second-harmonic lasers before entering the crystal should be the integral multiple of 2π. This needs to optimize both the tilting-azimuth angles and the propagating distance between the first pass and second pass.

In comparison with the firstly-passed beam position on the crystal, the spatial displacement Δ_o of the beams along the o-axis direction after secondly-passed through the crystal is expressed by

3. Experimental results

As shown in Fig. 1(a), a type I critical phase-matching BBO crystal is used as the nonlinear optical crystal for generating the second-harmonic laser at the wavelength of 532 nm. The fundamental laser at 1064 nm is generated from a commercial Q-switched Nd:YVO₄ laser with repetition rates varying from 20 kHz to 70 kHz. The corresponding pulse duration varies from 13.6 ns to 28.6 ns. The beam divergence of the fundamental laser is 3.0 mrad and is directly applied to generate the second-harmonic laser. The thickness of the BBO crystal is 6 mm with a transverse size of 8 mm$\times$8 mm. The phase-matching angle of the BBO crystal is 22.8 degree. The input and output surfaces of the crystal are coated with high transmitivity at both the wavelengths of 532 nm and 1064 nm. A mirror with a high reflectivity at 532 nm and 1064 nm is placed at the distance of 15 mm away from the crystal to reflect the unconverted fundamental and second-harmonic lasers, so that the second pass through the BBO crystal can be achieved. The distance between the mirror and BBO crystal is accurately adjusted in the experiment to ensure that the second-harmonic beam generated in the second pass has the same phase as that generated in the first pass. In addition, the propagation direction of the second-harmonic beam generated in the second pass is slightly different from that generated in the first pass due to the dispersion of the BBO crystal. Such dispersion effect can be ignored for the crystal thickness of 6 mm used in the experiment. After the second pass, the second-harmonic laser is reflected by another mirror which is rotatable with respect to the second- and third-pass through the BBO crystal if necessary. The fundamental and second-harmonic laser beams are finally separated by a dichromatic mirror.

The output average power of the Q-switched nanosecond Nd:YVO₄ laser is controlled by the injection current to the laser system. Figure 3 shows the second-harmonic average power under different fundamental average powers at the repetition rate of 30 kHz. For only using the single pass through the BBO crystal, the second-harmonic average power is less than 2.8 W at the repetition rate of 30 kHz and the single pulse duration of 17.3 ns. By introducing the two-pass configuration, the average power of the second-harmonic laser is approximately enhanced by a factor of 2.7 around the fundamental average power of 12 W and the maximum second-harmonic average power is up to 5 W.

 

Fig. 3 The average power of the second-harmonic laser generated in the BBO crystal as the function of the fundamental average laser at the repetition rate of 30 kHz and the single pulse duration of 17.3 ns.

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Figure 4 presents the average and peak powers of the input fundamental and the second-harmonic lasers under different repetition rates of the input fundamental laser at the injection current of 40 A. The enhancement factor that is calculated by the second-harmonic power ratio of the two-pass configuration to the single-pass configuration is larger than 2 for repetition rates varying from 20 kHz to 70 kHz. The maximum enhancement ratio is 2.5 at the repetition rate of 60 kHz and the fundamental average power of 21.9 W. Further enhancement in the average power of the second-harmonic laser is also observed by adding the third pass through the BBO crystal.

 

Fig. 4 Repetition-rate independence of the second-harmonic laser generated in the BBO crystal at the injection current of 40 A to the laser system. (a) For the average power, peak power, and pulse duration of the fundamental laser. (b) For the corresponding average power of the second-harmonic laser. (c) For the conversion efficiency of the second-harmonic laser.

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The maximum conversion efficiencies for the single-, two- and three-pass configurations are 14.1, 29.7, and 33.6% respectively at the repetition rate of 20 kHz and the fundamental average power of 14.5 W. Although the maximum average power of the second-harmonic laser is obtained at the repetition rate of 30 kHz, the maximum conversion efficiency is achieved at the repetition rate of 20 kHz as the fundamental peak power decreases with the increase in the repetition rate. The conversion efficiency for the second pass can be further improved if the divergence of the input fundamental beam (3 mrad) is reduced and the distance between the mirror and the crystal is further optimized. The beam distribution of the second-harmonic laser after the second pass is almost the same as that of the input fundamental laser. However, the beam quality for the third pass is decreased due to the walk-off effect and the complex optical layout. Although the power enhancement of the second-harmonic laser is achieved by using the BBO crystal, the conversion efficiency is relatively low due to the large absorption of the BBO crystal that we used in the experiment. LBO and KTP crystals are practical candidate nonlinear optical crystals if Nd-doped nanosecond-kilohertz Q-switched lasers are used to efficiently convert the wavelength from infrared to visible.

4. Conclusions

We demonstrated that the enhancement of the second-harmonic generation is possible by using a multipass non-collinear phase-matching configuration for a Q-switched nanosecond-kilohertz Nd:YVO₄ laser. We derived a general relationship for the phase-matching angle as the function of the tilting and azimuth angles of the unconverted fundamental and second-harmonic lasers in the non-collinear phase-matching state. The enhancement factor of the second-harmonic laser in a beta-barium borate crystal with the two-pass configuration is up to 2.5 in comparison with the single-pass configuration. The multipass non-collinear phase-matching configuration has advantages of reducing the thermal effect due to the absorption in the crystal and effectively avoiding the damage on the crystal surfaces, and is suitable for Q-switched nanosecond-kilohertz Nd-doped lasers to effectively convert the wavelength from infrared to visible.