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Abstract
A simple method to generate second-harmonic conical waves with tunable cone angles (apex angles) only using a cube uniaxial nonlinear crystal is proposed. The formation mechanism is based on the birefringence phase-matching among a plane fundamental wave, a spherical fundamental wave, and a conical second-harmonic wave. The cone angle in our demonstration of the method could be continuously adjusted from 0 to 30° by just rotating the crystal, which is the largest reported tunable range. Moreover, a tunable range of 0–68° could be achieved by using several crystals with different cut angles. The polarization and the longitudinal distribution of the generated second-harmonic wave are theoretically analyzed. The measured conversion efficiency was around 7%, and it decreased with the increasing of the cone angle.
© 2017 Optical Society of America
1. Introduction
Second-harmonic (SH) generation (SHG) is a basic and important optical parametric process in nonlinear optics and its applications. Recently, some novel non-collinear optical parametric interactions that can be used to generate SH waves with specific spatial shapes, such as conical, Bessel, and ring-shaped SH waves, have been discovered and proposed by using different types of materials and geometries. The first method, which is very popular recently, is to generate a SH conical wave by an incident fundamental harmonic (FH) wave and an additional scattering FH wave in a χ2 one- or two-dimensional periodically poled photonic crystal based on nonlinear Cerenkov radiation and/or the quasi-phase-matching condition [1–6]. For example, Piskarskas et al. [1] excited a ring-shaped SH wave by a Bessel-like FH wave in a periodically poled KTiOPO4 (KTP) crystal, Xu et al. [2] discovered conical SHG in a two-dimensional hexagonally poled LiTaO3 crystal, and Saltiel et al. [3] generated a SH conical wave by nonlinear Bragg diffraction within two-dimensional annular periodically poled stoichiometric lithium tantalate. These types of methods require periodically poled photonic crystals with one- or two-dimensional microstructures, whose application is challenging because of the fabrication difficulty and the economic cost of such photonic crystals. The second popular method is based on non-collinear birefringence phase-matching among an incident FH wave, its own scattering wave, and the generated SH conical wave in a nonlinear crystal, in which the scattering FH wave is produced by structural imperfection, intrinsic composition inhomogeneity, impurity ions, etc. on the surfaces or inside the crystal [7–12]. Generally, the SH conical wave generated by this method is weak and non-uniform owing to the weak and non-uniform scattering FH wave. Moreover, the SH conical wave cannot be conveniently adjusted because of the limited controllability of the scattering FH wave. The third method uses the conical refraction phenomenon in a biaxial crystal to generate the FH or SH conical wave [13–16]. Generally, due to the position-dependent polarization distribution of the FH conical wave, the generated SH conical wave possesses a non-uniform intensity distribution [14]. The fourth method reported by Chen et al. is based on the surface complete phase-matching in a bulk nonlinear medium with anomalous dispersion to generate SH ring-shaped waves [17–20]. However, in practice, the required anomalous-dispersion-like environment limits the wavelength range of the FH wave, and accordingly its applications. Furthermore, for the above four kinds of methods, the cone angles of the generated SH conical waves are not easy to adjust or the adjustment ranges are very limited, which is a disadvantage in applications [13, 21]. To overcome this problem, for these methods, the period of the photonic microstructure, wavelength of the FH wave, the temperature of the crystal, etc. needs to be adjusted. However, such parameters are usually invariable or difficult to change.
In the above four methods, the second method, which is based on the birefringence phase-matching, does not need any photonic microstructures, additional elements, specific properties or environments, etc. Moreover, the arrangement geometry is relatively simpler. In this condition, based on this method we make a simple improvement to enhance the intensity and the controllability of the scattering FH wave, and accordingly improve the characteristics of the generate SH conical wave. Compared with the previous methods, the cone angle of the generated SH conical wave can be conveniently adjusted in a wide range by just slightly rotating the crystal. In this new method, because the birefringence phase-matching rather than the quasi-phase-matching or the nonlinear Cerenkov radiation is used, the optional ranges of the nonlinear crystals and the FH wavelengths are widened a lot. The performance of this method was theoretically analyzed and experimentally demonstrated, and the two results agree well with each other. Besides that, the space-dependent polarization and longitudinal evolution of the generated SH conical wave were also discussed.
2. Method and demonstration
The geometry of the proposed method is shown in Fig. 1(a) (lower left). A 5 mm (x) × 5 mm (y) × 15 mm (z) type I beta barium borate (BBO) crystal with a 22.9° cut angle (the angle between the optic axis c and the crystal normal direction) was used. Only the input and output surfaces of the crystal were precisely polished, and the other four were unpolished rough surfaces (similar with surfaces of ground glass). A 1034 nm FH pulse was incident on the crystal. The parameters of the FH pulse were about 2 mm in diameter, about 70 µJ pulse energy, 500 fs duration (full width at half maximum, FWHM), and linear polarization parallel to the y–z plane. The FH beam refracted into the crystal from the input surface, and near the output surface whose beam center was set at about the edge of the cube crystal. Therefore, part of the beam transmitted through the crystal and the rest was reflection scattered by the rough surface inside the crystal. In this condition, a SH conical wave was generated by the transmission FH wave and the scattering FH wave based on type I non-collinear birefringence phase-matching. On a screen positioned about 90 mm away from the BBO crystal, as well as a SH spot generated by the collinear phase-matching SHG of the transmission FH beam, we observed ring-shaped SH light. In the experiment, we slightly rotated the BBO crystal around the y axis in the clockwise direction, as shown in Fig. 1(b), and the diameter of ring-shaped SH light (i.e., the cone angle α of the SH conical wave) significantly increased. Besides that, as shown in Fig. 1(a) (lower right), similarly, a scattering FH wave could also be produced by refraction scattering of the rough surface (i. e., from the air into the crystal), which produces the same result so we did not repeat the experiments under this condition. Thus, we produced a SH conical wave using only a single uniaxial crystal, and the cone angle could be conveniently adjusted.
Fig. 1 (a) Experimental setup for generation of SH conical waves. c is the optic axis, ϕ is the angle from the z axis to the optic axis, α is the cone angle (in air) of the SH conical wave, and β is the propagation (or central axis) direction (in air) of the SH conical wave relative to the z axis. (b) Evolution of the observed SH rings on the screen while the BBO crystal was rotated in the clockwise. The ϕ values are about (i) 21.6°, (ii) 22.5°, (iii) 23.5°, (iv) 24.5°, (v) 25.5°, and (vi) 26.5°.
Combination of the FH waves that contribute to the SH conical wave inside the crystal is schematically illustrated in Fig. 2(a). The transmission FH beam is considered to be a plane wave propagating in the positive direction of the z axis, and the optic axis c of BBO is in the x–z plane. The scattering FH beam is considered to be a spherical wave, and both its source and propagation direction (central axis) are in the x–z plane. In our model, for simplicity, a line-shaped source and a point-shaped source are used for the plane and spherical waves, respectively. The actual source can be considered to be the superimposition of lots of line- and point-shaped sources. Two FH waves intersect in the three-dimensional space with an elliptical intersection region. The FH plane wave is polarized parallel to the y–z plane (ordinary ray), and the polarization of the FH spherical wave is relatively complex. Each optical ray of the FH spherical wave polarizes in the plane formed by its wave vector and the normal vector of the x–z plane, and the polarization is also perpendicular to its wave vector. In the intersection region of the two FH waves, the birefringence phase-matching condition can be satisfied at an elliptical curve, as shown by the black line in Fig. 2(a). If we consider two arbitrary couples of optical rays from the FH plane wave and FH spherical wave, when the intersection points are located on the black elliptical curve, SH waves can be generated as a result of phase-matching, otherwise no SH waves are observed because of phase-mismatching. Therefore, a SH conical wave would be generated, and it can be considered to originate from a point source located at the apex in the x–z plane. Figure 2(b) shows the polarization of the generated SH conical wave, and the polarization of each optical ray is in the plane containing the optic axis and its wave vector (extraordinary ray), which is also perpendicular to its wave vector. Moreover, as shown in the inset of Fig. 2(b), the polarization of the SH conical wave is symmetric with respect to the x–z plane, and the angle between the polarization direction and the x–z plane increases with increasing position separation from the x–z plane.
Fig. 2 (a) Schematic of the beamline inside the crystal for tunable SH conical wave generation. Plane and spherical FH waves are from line- and point-shaped sources, respectively. (b) Position-dependent polarization of the SH conical wave. The inset shows the projection of the polarizations in the normal plane of the SH conical wave. (c) Phase-matching diagram in the x–z plane. (d) Evolution of the phase-matching diagrams while the crystal is rotated in the clockwise around the y axis (c is the optic axis).
To explain the parametric process and the tunable cone angle of the SH conical wave, the phase-matching diagram in the x–z plane is shown in Fig. 2(c). The wave vectors of the FH plane wave, FH spherical wave, and SH conical wave are denoted as k1, k1′, and k2, respectively. According to the refractive-index ellipsoid for the ordinary and extraordinary rays, the tracks of the end points of the wave vectors k1′ and k2 are a circle and an ellipse, respectively. When the circle and the ellipse have two intersection points, in which directions the birefringence phase-matching condition is satisfied, taking the three-dimensional condition into account, a SH conical wave would be generated. As shown in Fig. 2(d), if the BBO crystal is rotated, the ellipse rotates relative to the circle. Under the condition of only one intersection point (tangent), a SH single beam is generated [see Fig. 2(d)(i)]. With rotation of the ellipse, the number of intersection points increases from one to two, and the SH single beam changes to a SH conical wave [see (ii)]. The cone angle of the SH conical wave firstly increases and then decreases with further rotation of the ellipse [see (iii) and (iv)], and it finally becomes a SH single beam again [see (v)] before it disappears [see (vi)]. Situations (i)–(iii) and (iii)–(v) are two mirror-symmetrical processes. The proposed method works in situations from (i) to (iii) owing to the generation method of the scattering FH wave. Most of the previous methods operate near situation (ii) or (iii) or (iv), and the scattering source of the additional FH wave includes structural imperfection, intrinsic composition inhomogeneity, impurity ions, etc. on the surfaces or inside the crystal, which are intrinsic properties and cannot be conveniently changed.
3. Simulation
Previously, analytical equations of the non-collinear SHG involving an unfocused FH wave and its scattered wave have already been proposed by R. Trebino based on the first- and second-order approximation [8]. In this work, the numerical simulation method instead of the analytical method will be used to improve the calculation precision. As shown in Fig. 3, k1 propagates along the z axis, the SH wave k2 and the optic axis c possesses a two-dimensional orientation angle (θx, θy) and (ϕ, 0) respectively, and then the vectors of k2 and c can be written as (k2cosθysinθx, k2sinθy, k2cosθycosθx) and (sinϕ, 0, cosϕ), respectively.
Fig. 3 Phase-matching diagram in the x–y–z space. k1, k1′ and k2 are wave vectors of the FH plane wave, FH spherical wave and SH conical wave. θx and θy are orientation angles of k2.
Using the law of the angle between two vectors, the angle of k2 with respect to c is given by
In the type I phase-matching, the refractive-indexes of k1 and k1′ (i.e., ordinary rays) can be calculated directly using the Sellmeier Equation, and that of k2 (i.e., extraordinary ray) is given bywhere ne is the principal value of the extraordinary refractive-index. Then, the phase-mismatching can be calculated byAnd then, the amplitude of k2 could be obtained from [22]and where A1, A1′ and A2 are amplitudes of k1, k1′ and k2, deff is the effective nonlinear coefficient, and L is the interaction length.Using Eq. (4), the evolution of the SH conical wave with rotation of the BBO crystal was simulated based on the parameters of the demonstration experiment. Figure 4 shows the two-dimensional spatial intensity distributions of the generated SH waves in the θx–θy space. When the angle ϕ is 21.6° [Fig. 4(a)], a SH single beam without a hollow core is generated, which is very close to situation (i) shown in Fig. 2(d). As shown in Figs. 4(b)–4(f), when ϕ is gradually increased from 21.6° to 22.5°, 23.5°, 24.5°, 25.5°, and 26.5°, a SH conical wave appears, and its cone angle increases with increasing ϕ. This process is phenomenologically illustrated by situation (ii) in Fig. 2(d). The SH spot generated by the collinear phase-matching SHG of the transmission FH plane wave is also marked in Fig. 4. By comparing Figs. 4 with 1(b), the simulation result agrees very well with the experiment. The experiment and simulation results show the cone angle of the SH conical wave is very sensitive to ϕ. As shown in Fig. 4, when ϕ is increased from 21.6° to 26.5° (a 4.9° increase), the cone angle of the SH wave increases from 0° to about 13° (inside the crystal). In addition, when ϕ is increased from 20° to 160°, corresponding to processes (i)–(v) shown in Fig. 2(d), evolution of the two-dimensional intensity distributions of the SH conical waves drastically change, as shown in Figs. 5(a) and 5(b). Evolution of the cone angles (αx in the x–z plane and αy in the y–z plane) and the corresponding propagation directions of the SH conical waves (βx and βy) are given as functions of ϕ in Fig. 5(c). The cone angle of the SH conical wave can be increased from 0° to about 40° and then back to 0°, and it rapidly increases and decreases at the beginning and end of the rotation process, respectively. Moreover, in air, the cone angle can be further increased by about 1.655 times owing to refraction, and thus the tunable range is 0–68° in air. In our demonstration, as shown by points in Fig. 5(c), it could be arbitrarily adjusted from 0° to about 18° in the crystal, and accordingly 0–30° in air. Further adjustment of the cone angle in our demonstration was limited by the BBO cut angle of 22.9°. However, as shown by the different colored regions in Fig. 5(c), this limitation can be conveniently overcome by choosing appropriate crystals with different cut angles, for example around 38°, 52°, 66° and 80°. Compared with previous methods, the proposed method has two advantages: the tunable range of the cone angle is much larger and rotating the crystal is much more convenient than adjusting the wavelength of the FH waves or changing the temperatures of the crystals.
Fig. 4 Intensity distribution of SH conical waves in two-dimensional space defined by orientation angles θx and θy. The ϕ values are (a) 21.6°, (b) 22.5°, (c) 23.5°, (d) 24.5°, (e) 25.5°, and (f) 26.5°. The orientation angles (θx and θy) are those determined inside the crystal. The interaction length of SHG is 0.5 mm. The spots illustrate the position of the collinear phase-matching SHG of the transmission FH plane wave.
Fig. 5 Intensity distributions of SH conical waves in the (A, a) θx and (B, b) θy directions with variation of ϕ. (C, c) Cone angles (αx measured in the x–z plane and αy measured in the y–z plane) and the corresponding propagation directions of the SH conical waves (βx and βy) with variation of ϕ. Points in (c) are experiment results. θx, θy, αx, αy, βx, and βy were determined inside the crystal. The interaction length of SHG is 0.5 mm.
From Figs. 5(a) and 5(b), the peak intensity of the SH conical wave increases firstly and decreases secondly with increasing ϕ. Figures 5(c), 5(a) and 5(b) show the propagation direction βx (central axis) of the SH conical wave changes in the x–z plane during the rotation process of the crystal, similar to R. Trebino’s result [8] and what is illustrated in Fig. 2(d). Figure 5(c) shows the vertical intersection of the SH conical wave is an ellipse rather than a perfect circle, but the ellipticity is very small and it is very difficult to observe in experiments.
Intensity variation in the propagation direction is one of the most important characteristics for the optical parametric process in the three-wave interaction. In our above simulation, we assumed that the interaction length of the FH waves and the SH wave was 0.5 mm owing to the large non-collinear angles and the beam aperture. The intensities of the FH plane wave and FH spherical wave were 1 GW/cm2 and 0.1 GW/cm2, respectively. Based on Eq. (4) and the same parameters used in Figs. 4(a)-4(c), we ignored spatial separation among the three waves and simulated the longitudinal intensity variation of the SH wave in the x–z plane. Figures 6(a)–6(c) show that when the angle ϕ is increasing, the cone angle increases, and the intensity of the SH wave increases along the propagation direction. However, as shown in Figs. 6(d)-6(f), when ϕ is reduced to 21.4°, 21.2° and 21.0°, which is too small to satisfy the birefringence phase-matching condition, a clear periodical oscillation of the intensity along the longitudinal direction appears. In this condition, the spatial intensity distribution of the generated SH wave is directly influenced by the interaction length. Figure 7 shows the vertical intensity distributions (parallel to the x–y plane) of the SH waves for different interaction lengths and ϕ = 21.4°, which corresponds to the situation in Fig. 6(d). As shown in Fig. 7(a)–7(f), when the interaction length increases from 0.2 mm to 0.4 mm, 0.6 mm, 0.8 mm, 1.0 mm, and finally 1.2 mm, the central pattern of the SH wave is reduced in space and becomes a multi-ring-shape. However, this multi-ring-shaped SH wave generally cannot be observed in experiments because of a limited interaction length. For the cases of large ϕ, especially when a SH conical wave appears, as shown in Figs. 6(a)–6(b), this phenomenon and the influence are significantly reduced and cannot appear in experiments.
Fig. 6 Longitudinal intensity variation of SH conical waves in the x–z plane. The ϕ values are (a) 21.6°, (b) 22.5°, (c) 23.5°, (d) 21.4°, (e) 21.2°, and (f) 21.0°. The orientation angle θx is that inside the crystal.
Fig. 7 Vertical intensity distributions of SH waves for interaction lengths of (a) 0.2 mm, (b) 0.4 mm, (c) 0.6 mm, (d) 0.8 mm, (e) 1.0 mm, and (f) 1.2 mm. The ϕ value is 21.4°. The orientation angles θx and θy are those measured inside the crystal.
4. Conversion efficiency and possible applications
The conversion efficiency from FH waves to the SH conical wave is an important parameter for such methods, which directly influences their potential applications. For the four kinds of methods classified in the introduction section, the conversion efficiency depends crucially on generation mechanisms, materials, environments, and so on. In the first method (i. e., based on photonic crystal materials and nonlinear Cerenkov radiation or quasi-phase-matching), the reported efficiency is around 10% [2, 3]. However, for the second method (i. e., based on non-collinear birefringence phase-matching among a FH wave and its scattering wave), due to an extremely weak scattering FH wave, the conversion efficiency generally is very low, and we did not find any experiment results in the previous references [7–12]. For the third method (i. e., based on biaxial crystals and conical refraction), the reported efficiency is the highest, and which could reach to 60% [13]. And for the fourth method (i. e., based on the surface complete phase-matching), the demonstrated efficiency is around 15% [18, 19]. In this paper, although the mechanism belongs to the second method, the unpolished rough surface is utilized to generated an additional scattering FH wave. In this condition, compared with structural imperfection, intrinsic composition inhomogeneity, impurity ions, etc., the intensity of the FH scattering wave, accordingly that of the generated SH conical wave, would be significantly increased. The experimental measurement was based on the laser condition explained in the section 2, and the evolution of the conversion efficiency for various ϕ is shown in Fig. 8. In the experiment, after each rotation, the crystal position was slightly changed, which would influence the interaction length and the energy of the generated FH scattering wave (accordingly that of the SH conical wave). In this condition, a linear stage was used to precisely shift the crystal without changing its angular orientation to maximize the energy of the SH conical wave. Moreover, the evolution of the cone angle is also given in Fig. 8 for comparison. The result shows that when ϕ equalled to about 21.5°, the conversion efficiency was around 7%. While ϕ was increasing, the cone angle α and the conversion efficiency increased and decreased, respectively. In our measurement, when the cone angle increased to about 12°, the conversion efficiency decreased to around 3.8%. It is clear that the conversion efficiency decreased with the increasing of the cone angle. We believe it was mainly due to the increasing of the non-collinear angle, which reduced the interaction length. Besides that, the detail value of the conversion efficiency also strongly depends on the intensity of the scattering FH wave and the phase-matching condition. For a higher efficiency, the scattering FH wave should be enhanced. And, more importantly, most of the scattering FH wave should satisfy the phase-matching and contribute to the conical SHG.
Fig. 8 Measured evolution of conversion efficiency and cone angle (inside the crystal) with variation of ϕ.
The proposed method, as well as the generated tunable SH conical wave, actually could be utilized for some interesting applications. Because of the conveniently tunable capability and a very large tunable range, this method might be useful for microfabrication, for example it could be used to fabricate a concentric-rings-structure (e. g. the annular periodically poled photonic crystal used in refs [3, 4].). This phenomenon could also guide SHG to achieve the best phase-matching condition. As shown in Fig. 1(b), when the SHG spot located at the SH ring, which is the perfect phase-matching direction, the brightest green spot appeared [see Fig. 1(b)(iv)]. Similarly, due to the high sensitivity of the cone angle α to the angle ϕ, as shown in Fig. 5(c), it could be introduced to precisely detect the optic axis direction of a nonlinear crystal. Besides that, just as introduced by R. Trebino [8], this phenomenon could also be applied for the refractive-index measurement. Meanwhile, in some applications, for example microfabrication, a high beam quality of the SH conical wave is required. In this condition, the intensity uniformity of the scattering FH wave should be improved. In fact, two possible methods could be introduced, one is to improve the scattering quality of the rough surface (e. g. engineered diffuser [23]), and the other one is to generate both FH plane and spherical waves via a special element (e. g. a focal lens with a hole in the center). This part of work will be continued in our next step experiments.
5. Conclusions
We have proposed a simple and efficient method to generate SH conical waves. In this method, birefringence phase-matching is used rather than quasi-phase-matching, nonlinear Cerenkov radiation, or other complicated mechanisms. The only material used is a cube uniaxial nonlinear crystal (cube BBO in this demonstration of the method) without any one- or two-dimensional photonic microstructures, specific properties, specific environments, etc. Importantly, in this demonstration, the cone angle of the generated SH conical wave could be adjusted from 0 to 30° in air, and it could be continually adjusted in the range 0–68° by selecting several crystals with different cut angles. The SH conical wave is little depolarized, and the polarization direction of each ray lies on the plane that contains the optic axis and its wave vector. The longitudinal intensity of the SH conical wave periodically changes during propagation. However, under most conditions, it cannot appear owing to the large non-collinear angle, and accordingly the very short interaction length. The measured conversion efficiency was around 7%, which could decrease while the cone angle was increasing.
Funding
Japan Society for the Promotion of Science Grants-in-Aid for Scientific Research (KAKENHI) (JP25247096).
Acknowledgments
The authors gratefully acknowledge experimental condition support from Dr. Koji Tsubakimoto and Mr. Koichi Tsuji. Z. L. acknowledges helpful discussions with Dr. Xiaoyang Guo and Dr. Zhan Jin.
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