Second-Harmonic Generation of the Vortex Beams with Integer and Fractional Topological Charges

Abstract

The single-pass second-harmonic generation (SHG) of a vortex beam under low fundamental wave depletion is systematically studied. Vortex modes at 1064 nm with integer topological charges from ±1 to ±9 and fractional ones at ±0.75 are generated by modulating the fundamental Gaussian beam with different spiral phase plates. The frequency doubling of these fundamental vortex modes is realized via single-pass SHG through the KTP. A detailed theoretical model is set up in the single-pass SHG of the vortex beams. Theoretical analysis indicates that the higher the order of the vortex beams, the lower the SHG efficiency, when the beam waists and fundamental power are given. The experimentally measured SHG output characteristics verify those obtained via theoretical analysis. Conservation of the orbital angular momentum during the SHG process is also verified, regardless of the fractional or integer vortex beams. SH LG_0,2l vortex beams with high mode purity are obtained. The beam waists of fundamental/SH in KTP measured using a 4f system demonstrate that the Rayleigh ranges of the fundamental wave and SH wave are the same. The paper comprehensively presents some basic laws in the single-pass SHG of a vortex beam. In addition, it also indicates that SHG is an effective method to improve the mode purity of vortex beam.

1. Introduction

The research on vortex beams has been a hot topic, since Allen discovered the relationship between orbital angular momentum (OAM) and the Laguerre–Gaussian beam (LG_p,l; p is the radial mode index, and l is the azimuthal mode index) in 1992 [1]. Here, l is also known as topological charge (TC), and the OAM of a LG mode is lћ per photon (ћ is the reduced Planck constant). Usually, the vortex beam contains an exp(ilφ) term in its expression. It usually includes a LG vortex beam [2,3], Bessel–Gaussian vortex beam [4,5], ring Airy–Gaussian vortex beam [6], anomalous vortex beam [7], perfect vortex beam [8,9], and so on. Vortex beams display a wavefront helical phase and doughnut structure, and constructs a high-dimensional Hilbert space; therefore, they are widely applied in the manipulation of micro-particles [10,11,12], spatial resolution imaging [13,14], quantum optics [15,16,17], classical and quantum optical communications [18,19], etc. Based on OAM conservation in nonlinear interactions, a vortex beam at any wavelength can be realized [20,21]. As the most common nonlinear frequency conversion technology, second-harmonic generation (SHG) is widely used to obtain vortex beams with large TCs at a short wavelength.

The study of OAM transformation in nonlinear optics began from the SHG of LG vortex beams. As early as 1996, K. Dholakia et al. reported the SHG of LG_0,l vortex beams via a birefringent crystal, potassium titanyl phosphate (KTiOPO₄, KTP), and demonstrated that the OAM of each photon in the SH mode is twice that in the fundamental mode [22]. This is the first time OAM conservation was verified during a SHG process in an experiment. Then, in 1997, they showed that the frequency-doubling of high-order LG_p,l (p > 0) vortex beams in KTP; OAM conservation in SHG was verified again [23]. The walk-off effect of the birefringent crystal limits the use of a longer crystal, leading to the low efficiency of SHG and distorted beam spots [24]. The use of the quasi-phase matching method effectively solved these problems [25]. G. H. Shao et al. employed periodically poled lithium niobite (PPLN) as the frequency-doubling crystal in the frequency conversion of a vortex beam, and avoided the undesired walk-off effect to reserve high-quality LG modes [26]. In order to further increase SHG efficiency, external-cavity-enhanced frequency-doubling technology are used in the SHG of vortex beams [27,28,29]. Z. Y. Zhou et al. used periodically poled KTP as the SHG crystal and set up an external cavity to enhance the frequency conversion efficiency to 10.3% [27]. Recently, in 2020, J. Heinze et al. compared the cavity-enhanced SHG of LG_0,0 and LG_3,3 modes both theoretically and experimentally. Their study demonstrated that nonlinear crystals have higher absorption losses and lower effective nonlinearities for higher-order modes, resulting in lower SHG efficiency for high-order modes [28,29]. The modes (TCs) of two fundamental beams in SHG process are the same in the previous introduction, and this makes the TC value of the SH beam even-order. Some interesting phenomena occur in the SH beam when the TCs of two fundamental beams are different. In 2013, S. M. Li et al. explored the SHG behaviors of a pair of fundamental fields carrying arbitrary OAMs per photon in a noncolinear type-I phase matching configuration, and realized the flexible management of the SH field’s OAM via the arbitrary combination of the two fundamental fields’ OAMs, in particular, fractional and odd-order OAMs [30]. In 2017, J. P. Leonardo et al. investigated the nonlinear mixing of OAM in type-II SHG with arbitrary TCs imprinted on two orthogonally polarized beams [31]. In addition to the investigation of the SHG of LG vortex beams, researchers also focused on other vortex beams, such as BG beams [32,33,34,35,36,37], composite vortex beams [38,39], fractional vortex beams [40,41,42,43], perfect vortex beams [44], etc. As far as we know, few researchers have studied the variation of the waist radius in single-pass SHG and the effect of the vortex order (or the TC) on the SHG efficiency in experiments.

In this paper, the SHG of LG_0,l vortex beams with different TCs (TCs from ±1 to ±9 and fractional TCs at ±0.75) generated by modulating the fundamental Gaussian beams with cascaded spiral-phase plates [45] are studied systematically. A theoretical description of the SHG process under low-fundamental-wave depletion is presented in Section 2. The experimental setup and procedures are depicted in Section 3. The experimental results and discussion, including the SH output versus the fundamental vortex order, l_f, beam patterns and TCs of the fundamental (SH) vortex beams, their optical field distributions, and the beam waists of fundamental (SH) vortex beams in KTP are shown in Section 4. The article ends with the conclusion in Section 5. Our results verify that the Rayleigh range of the SH beam is the same as that of the fundamental beam during the SHG process. The lobes of the SH beam obtained via the tilted lens method allow us to determine the TC carried by the fundamental infrared vortex beams [46]. In addition, we demonstrated that SHG is an effective method for obtaining vortex beams with high mode purity.

2. Theoretical Analysis

The nonlinear coupled-wave equations that govern type-II (e + o → e) SHG are given by the following [47,48]:

$$\frac{d{E}_{\mathrm{f}\mathrm{o}}}{dz}=\frac{i4\pi }{n_{\mathrm{f}\mathrm{o}}{\lambda }_{\mathrm{f}}}{d}_{eff}{E}_{\mathrm{S}\mathrm{H}}(z)E_{\mathrm{f}\mathrm{e}}^{*}(z)\exp (i\Delta k_{\mathrm{S}\mathrm{H}\mathrm{G}}z),$$

(1)

$$\frac{d{E}_{\mathrm{fe}}}{dz}=\frac{i4\pi }{n_{\mathrm{f}\mathrm{e}}{\lambda }_{\mathrm{f}}}{d}_{eff}{E}_{\mathrm{S}\mathrm{H}}(z)E_{\mathrm{f}\mathrm{o}}^{*}(z)\exp (i\Delta k_{\mathrm{S}\mathrm{H}\mathrm{G}}z),$$

(2)

$$\frac{d{E}_{\mathrm{S}\mathrm{H}}}{dz}=\frac{i4\pi }{n_{\mathrm{S}\mathrm{H}}{\lambda }_{\mathrm{S}\mathrm{H}}}{d}_{eff}{E}_{\mathrm{f}\mathrm{o}}(z)E_{\mathrm{f}\mathrm{e}}(z)\exp (-i\Delta k_{\mathrm{S}\mathrm{H}\mathrm{G}}z),$$

(3)

Here, $\Delta k_{\mathrm{S}\mathrm{H}\mathrm{G}}=2\pi (n_{\mathrm{S}\mathrm{H}}/{\lambda }_{\mathrm{S}\mathrm{H}}-n_{\mathrm{f}\mathrm{o}}/{\lambda }_{\mathrm{f}}-n_{\mathrm{f}\mathrm{e}}/{\lambda }_{\mathrm{f}})$ is the phase mismatch factor. The subscripts “fo (fe)” and “SH” represent the ordinary (extraordinary) beam of the fundamental wave and the SH wave. E_j is the amplitude of wave j in the type-II SHG crystal. λ_f (λ_SH) is the wavelength of the fundamental (SH) wave in a vacuum. n_j is the refractive index of wave j at wavelength λ_j in the SHG crystal. d_eff is the effective nonlinear coefficient of the SHG crystal.

Assuming a phase-matched interaction ($\Delta k_{\mathrm{S}\mathrm{H}\mathrm{G}}=0$), the fundamental small-signal approximation (the fundamental depletion can be ignored), and collimated Gaussian beam approximation (for short SHG crystal), the amplitude of the SH wave can be expressed as:

$$E_{\mathrm{S}\mathrm{H}}=\frac{i4\pi d_{eff}}{n_{\mathrm{S}\mathrm{H}}{\lambda }_{\mathrm{S}\mathrm{H}}}{E}_{\mathrm{fo}}{E}_{\mathrm{fe}}L,$$

(4)

Here, L is the length of the SHG crystal. The amplitude of LG_0,l vortex beam in a cylindrical coordinate system (r, θ, z) is:

$$E_{\mathrm{j}}=\sqrt{\frac{P_{\mathrm{j}}}{\pi n_{\mathrm{j}}c{\epsilon }_0\left|l_{\mathrm{j}}\right|!}}\frac{1}{w_{\mathrm{j}}}{(\frac{\sqrt{2}r}{w_{\mathrm{j}}})}^{\left|l_{\mathrm{j}}\right|}\exp (-\frac{r^2}{w_{\mathrm{j}}^2})\exp (i{l}_{\mathrm{j}}\theta ),$$

(5)

where P_j is the power of LG_0,lj vortex beam. w_j is the beam spot of the LG_0,0 mode. This indicates the radial position where the amplitude of the Gaussian term falls to 1/e times its on-axis value. ε₀ is the permittivity of the vacuum, and c is the speed of light in the vacuum.

For type-II SHG, the beam parameters for two fundamental vortex beams, E_fo and E_fe, are the same. In other words, $w_{\mathrm{f}\mathrm{o}}=w_{\mathrm{f}\mathrm{e}}=w_{\mathrm{f}}$, $l_{\mathrm{f}\mathrm{o}}=l_{\mathrm{f}\mathrm{e}}=l_{\mathrm{f}}$, and $P_{\mathrm{f}\mathrm{o}}=P_{\mathrm{f}\mathrm{e}}=P_{\mathrm{f}}$. The SH field can be denoted by:

$$E_{\mathrm{S}\mathrm{H}}=\frac{i4d_{eff}L\sqrt{P_{\mathrm{f}o}{P}_{\mathrm{f}e}}}{n_{\mathrm{S}\mathrm{H}}\sqrt{n_{\mathrm{f}o}{n}_{\mathrm{f}e}}c{\epsilon }_0\left|l_{\mathrm{f}}\right|!w_{\mathrm{f}}^2}{(\frac{\sqrt{2}r}{w_{\mathrm{f}}})}^{2\left|l_{\mathrm{f}}\right|}\exp (-\frac{2r^2}{w_{\mathrm{f}}^2})\exp (i2l_{\mathrm{f}}\theta ),$$

(6)

The expression of SH wave shown in Equation (6) has the standard form of the LG vortex beam, and we have l_SH = 2l_f and w_SH = w_f/$\sqrt{2}$. l_SH = 2l_f reflects that the OAM of each SH photon equals to that of two fundamental photons, that is, the OAM is conserved in the SHG process. Considering the Rayleigh range, $Z_{\mathrm{j}}=\pi w_{\mathrm{j}}^2/{\lambda }_{\mathrm{j}}$, we can find that Z_f = Z_SH; that is to say, the Rayleigh range of the SH beam is the same as that of the fundamental beam during the SHG process.

The SH power can be expressed as follows [20]:

$$P_{\mathrm{S}\mathrm{H}}=2\pi {\displaystyle {\int }_0^{\infty }rdr2n_{\mathrm{S}\mathrm{H}}c{\epsilon }_0}{\left|E_{\mathrm{S}\mathrm{H}}\right|}^2,$$

(7)

Substituting Equation (6) into Equation (7), we can obtain:

$$P_{\mathrm{S}\mathrm{H}}=\frac{2\pi d_{eff}^2}{n_{\mathrm{S}\mathrm{H}}{n}_{\mathrm{f}\mathrm{o}}{n}_{\mathrm{f}e}c{\epsilon }_0}\frac{\left(2\left|l_{\mathrm{f}}\right|\right)!}{2^{2\left|l_{\mathrm{f}}\right|}{\left(\left|l_{\mathrm{f}}\right|!\right)}^2}\frac{L^2}{w_{\mathrm{f}}^2}{P}_{\mathrm{f}}^2={\kappa }_{con}{\kappa }_{l_{\mathrm{f}}}\frac{L^2}{w_{\mathrm{f}}^2}{P}_{\mathrm{f}}^2={\gamma }_{l_{\mathrm{f}}}{P}_{\mathrm{f}}^2,$$

(8)

Here, ${\gamma }_{l_{\mathrm{f}}}$ is defined as the SHG coefficient, and it can be specifically expressed as:

$${\gamma }_{l_{\mathrm{f}}}={\kappa }_{con}{\kappa }_{l_{\mathrm{f}}}{L}^2/w_{\mathrm{f}}^2$$

(9)

where ${\kappa }_{con}=2\pi d_{eff}^2/\left(n_{\mathrm{S}\mathrm{H}}{n}_{\mathrm{f}\mathrm{o}}{n}_{\mathrm{f}\mathrm{e}}c{\epsilon }_0\right)$ is constant, if the SHG crystal and the fundamental wavelength are given. ${\kappa }_{l_{\mathrm{f}}}=\left(2\left|l_{\mathrm{f}}\right|\right)!/\left[2^{2\left|l_{\mathrm{f}}\right|}{\left(\left|l_{\mathrm{f}}\right|!\right)}^2\right]$ is a SHG dimensionless parameter related to the fundamental vortex order, l_f. The higher the vortex order, l_f, the smaller the value of parameter ${\kappa }_{l_{\mathrm{f}}}$. The SHG efficiency is defined as:

$${\eta }_{\mathrm{S}\mathrm{H}}=P_{\mathrm{S}\mathrm{H}}/P_{\mathrm{f}}={\kappa }_{con}{\kappa }_{l_{\mathrm{f}}}{L}^2{P}_{\mathrm{f}}/w_{\mathrm{f}}^2={\gamma }_{l_{\mathrm{f}}}{P}_{\mathrm{f}}$$

(10)

In terms of single-pass SHG, the SHG characteristics of a LG vortex beam is similar to those of a fundamental Gaussian beam, that is, the SHG efficiency is directly proportional to the fundamental power, P_f, and inversely proportional to the square of the fundamental beam waist radius, w_f. The difference is that the SHG efficiency of the vortex beam is lower than that of the fundamental Gaussian beam, and the higher the fundamental vortex order, l_f, the lower the SHG efficiency, η_SH.

3. Experimental Setup

The schematic of the experimental setup is shown in Figure 1. A continuous-wave, high-power (as high as 15 W), single-frequency (linewidth of <100 kHz) Yb fiber laser and amplifier (NKT Photonics, Koheras Y10) at 1064 nm is used as the laser source. An optical isolator is employed to prevent the reflected beam from feeding back to the laser system. A lens with focal length of 500 mm is used to recollimate the laser beam. The power control system, composed of half-wave plates (HWP_2,3) and polarizing beam splitters (PBS_1,2), allows us to adjust the pump power in the subsequent optical paths. The cascaded spiral phase plates (UPO Labs) are used to covert the fundamental Gaussian beam into the vortex beam (l_f = ±1~±9, and l_f = ±0.75) [45]. In the SHG experiment, a 6 mm×6 mm×3 mm KTP is used as the SHG crystal. The KTP is mounted in the copper oven, which is installed on a rotatable bracket. The KTP is cut for the type-II (e + o → e) phase-matching process for SHG at 1064 nm at room temperature. A lens (f_2,4 = 30 mm) is used to focus the fundamental beam onto the center of the KTP crystal. A lens (f_3,5 = 50 mm) is used as the collimating lens. The fundamental beam and SHG beam are then separated by the DM_1,2 (dichroic mirrors; R > 99.5% for 532 nm and T > 99% for 1064 nm). The narrowband filter is placed after the DM_1,2 to further obstruct the fundamental beam. At last, the generated SH vortex beam is projected on a screen and then recorded using a CMOS camera (CinCam, CMOS-1202; pixel pitch: 5.3 μm × 5.3 μm). The power of the generated SH vortex beam and the transmitted fundamental vortex beams is directly measured using two power meters (Thorlabs PM 100D).

Both the off-axis interference method [49] and tilted lens method [46,50] are used to measure the TCs carried by the fundamental/SH vortex beams. For the former method, the TEM₀₀ Gaussian beam is used as the reference to interfere with the generated vortex beam with a small included angle. For the latter method, a tilted lens (f₆ = 100 mm) is used to achieve mode transformation from a LG beam into a Hermite–Gaussian (HG) beam. The fundamental beam waist radius in the KTP crystal is about ten micrometers, which is close to the pixel pitch of the CMOS camera, resulting in severe distortion of the beam spots directly measured. Therefore, an improved 4f system consisting of lenses f₇ (f₇ = 25.4 mm) and f₈ (f₈ = 300 mm) is used to amplify and measure the beam waist of the fundamental/SH wave in the SHG crystal. The magnification ratio of the beam waist is 11.8.

4. Results and Discussion

4.1. The Single-Pass SHG Characteristics of LG_0,l Vortex Beams with Different Vortex Orders

According to the theoretical analysis in Section 2, the difference of the single-pass SHG between the LG_0,l vortex beam and the fundamental Gaussian beam is the SHG coefficient, ${\gamma }_{l_{\mathrm{f}}}$ (SHG dimensionless parameter, ${\kappa }_{l_{\mathrm{f}}}$). The value of ${\kappa }_{l_{\mathbf{f}}}$ is 1 for the fundamental Gaussian beam, and <1 for the LG_0,l vortex beam, as shown in

Table 1. The value of ${\kappa }_{l_{\mathrm{f}}}$ for different fundamental vortex orders, l_f.

lf	0	1	2	3	4
${\mathit{\kappa }}_{{\mathit{l}}_{\mathit{f}}}$	1	$\frac{\mathbf{1}}{\mathbf{2}} \mathbf{=} \mathbf{0.5}$	$\frac{\mathbf{3}}{\mathbf{8}} \mathbf{=} \mathbf{0.375}$	$\frac{\mathbf{5}}{\mathbf{16}} \mathbf{\approx } \mathbf{0.313}$	$\frac{\mathbf{35}}{\mathbf{128}} \mathbf{\approx } \mathbf{0.273}$
lf	5	6	7	8	9
${\mathit{\kappa }}_{{\mathit{l}}_{\mathbf{f}}}$	$\frac{\mathbf{65}}{\mathbf{256}} \mathbf{\approx } \mathbf{0.246}$	$\frac{\mathbf{231}}{\mathbf{1024}} \mathbf{\approx } \mathbf{0.226}$	$\frac{\mathbf{429}}{\mathbf{2048}} \mathbf{\approx } \mathbf{0.209}$	$\frac{\mathbf{6435}}{\mathbf{32}\mathbf{,}\mathbf{768}} \mathbf{\approx } \mathbf{0.196}$	$\frac{\mathbf{12}\mathbf{,}\mathbf{155}}{\mathbf{65}\mathbf{,}\mathbf{536}} \mathbf{\approx } \mathbf{0.185}$

. As the vortex order, l_f, increases from 0, ${\kappa }_{l_f}$ first rapidly decreases and then slowly decreases.

The experimentally measured SH power versus the fundamental power for the vortex order l_f = 0, 1, and 2 is shown in Figure 2. The SHG power shown here is the direct measurement SHG power, which is lower than the actual SHG output, since the transmission loss of the narrowband filter is not considered. However, this factor does not affect our analysis here. The experimentally measured SHG power can be well-fitted using Equation (8). The fitting SHG coefficient ${\gamma }_{l_{\mathrm{f}}}$ satisfies the following relationship: γ_1/γ₀ = 0.503, and γ₂/γ₀ = 0.387. These results are consistent with the value of ${\kappa }_{l_{\mathrm{f}}}$ shown in Table 1.

4.2. Beam Patterns and TCs of the Fundamental/SH Vortex Beams

The combination of three pieces of spiral-phase plates with vortex order 1 and three pieces of spiral-phase plates with vortex order 2 allows us to generate the vortex beams with TCs from ±1∼±9 in the experiment. The beam patterns and the images of the measurements of TCs for the generated fundamental vortex beams and their SH vortex beams are shown in Figure 3 and Figure 4. As shown in row (a) of Figure 3 and Figure 4, the multi-ring fundamental vortex beam is generated. In addition to the brightest annular spot representing the LG_0,l mode, there exist outer rings in its surroundings. These outer rings are high-order LG_p,l (p > 0) modes [45]. With the increasing number of cascaded spiral phase plates, the TC value of vortex beams increase, and the high-order LG_p,l (p > 0) modes gradually become obvious. Row (d) of Figure 3 and Figure 4 is the beam pattern of the generated SH vortex beam. It exhibits a single-ring structure, representing the LG_0,l mode. This indicates that higher-order LG_p,l (p > 0) modes are suppressed during the SHG process. Rows (b) and (c) of Figure 3 and Figure 4 are the intensity patterns transformed with a tilted convex lens. These intensity patterns contain several bright spots and dark fringes, which are HG-like light spots. With the increasing number of TCs, the value of the fringes is also increases. The relationship between the number of bright spots, n, and the TC, l, is n = |l| + 1. Meanwhile, the positive and negative TCs can be distinguished via the intensity distribution patterns with different orientations. e.g., the orientation of the bright spot tilts from the bottom left to top right as shown in Figure 3b,e, corresponding to the positive TC, while the orientation of the bright spot tilts from the top left to bottom right as shown in Figure 3b,e, corresponding to a negative TC. All these intensity patterns in rows (b) and (c) of Figure 3 and Figure 4 demonstrate that the TC of the SH beam is twice that of the fundamental beam. Rows (c) and (f) of Figure 3 and Figure 4 are the off-axis interferograms between the generated vortex beam and the reference TEM₀₀ Gaussian beam. The fork number relates to the TC value of the vortex beam, and the fork direction determines the sign of the TC. The fork number of the SH interferogram is also twice that of the corresponding fundamental interferogram, and their fork directions are the same.

In order to further illustrate the effect of the SHG process on the fundamental modes, the intensity distributions of the beam spots for fundamental/SH vortex beams are shown in Figure 5 and Figure A1. As shown in rows (a) and (b) of Figure 5 and Figure A1, for the fundamental vortex beam, in addition to a pair of axisymmetric peaks in the middle part, there exist some small peaks on the outer side, circled by the blue circles. This indicates that the fundamental vortex beam is composed of multiple LG_p,l modes, and the LG_0,l mode dominates in these modes. For the intensity distributions of the SH vortex beams shown in rows (c) and (d) of Figure 5 and Figure A1, except for a pair of axisymmetric peaks, no other peak can be observed. This shows that there exists only one LG_0,l mode in the SH vortex beam. As a whole, the mode purity of the vortex beam is significantly improved during the SHG process. The lower proportion and larger beam spots of high-order modes in the fundamental wave result in extremely low SHG efficiency for high-order modes, as shown through Equation (10).

The SHG characteristics of the vortex beam with integer TCs have been introduced previously. What will happen for the SHG process when the fundamental vortex beam carries a fractional TC? To answer this question, the SHG characteristics of the vortex beam with a fractional TC are investigated. To generate the fractional fundamental vortex beam, the 1064 nm TEM₀₀ laser illuminates the spiral-phase plate designed for 808 nm with vortex order 1, and the vortex beam with a TC of l_f = ±808/1064 × (n₁₀₆₄ − 1)/(n₈₀₈ − 1) is generated [51]. Considering that the spiral-phase plate used in our experiment is made of quartz, and the refractive indices n₈₀₈ and n₁₀₆₄ of quartz at 808 nm and 1064 nm are 1.45318 and 1.44963, respectively, the fractional vortex beams with a TC of ±0.75 can be generated. The beam spot characteristics and TCs for the generated fractional vortex beams and their SH at the near field are shown in Figure 6. As shown in column 1 for the beam pattern, both the fundamental and SH fractional vortex beams appear as an annular spot with an opening, shaped similarly to the letter C. This is consistent with the images of fractional vortex beams reported in references [40,52]. As shown in column 2 of Figure 6, the value of the bifurcation in the interferogram for the TC value 0.75 is 1, and that for the TC value 1.5 is 2. This confirms the conclusion about measuring the TCs of fractional vortex beams in previous literature [53]. When the fractional vortex beam is mode-transformed with a tilted lens, the HG-like beam spot is generated, similar to that of integer TCs, as shown in column 3 of Figure 6. Briefly, 2(3) bright spots are observed when the TC of fractional vortex beam is 0.75 (1.5). These results indicate that the fractional TCs double during the SHG process. As shown in columns 1, 4, and 5, the high-order modes at the outer part of the fundamental vortex beams are suppressed during the SHG process.

4.3. Beam Waists of Fundamental/SH Vortex Beams in KTP Measured Using 4f System

The 4f system is constructed to amplify and measure the beam waists of the fundamental/SH vortex beams in KTP. Column 1 of Figure 7 and Figure A2a depict the amplified beam patterns of the fundamental/SH vortex beams in the middle of the KTP. The beam pattern of the fundamental vortex beam presents a multi-ring structure that is bright in the middle and dark at the edge, while that of the SH vortex beam exhibits the standard doughnut structure. Columns 2 and 3 of Figure 7 and Figure A2b,c present the intensity distributions of the beam patterns along the x- and y- axes, respectively. By fitting these experimental distribution results using the expression of the LG_0,l beam, the beam waist radii of the fundamental/SH vortex beams in KTP can be obtained. The obtained beam waist radii (w_SH and w_f) and their ratio (w_SH/w_f) versus the TCs, l_f, of the fundamental vortex beams are plotted in Figure 8. The ratio of beam waist radii (w_SH/w_f) is between 0.6 and 0.74 in our study. This is consistent with the theoretical prediction, that the waist radius of the SH vortex beam becomes 1⁄$\sqrt{2}$ times that of the fundamental vortex beam. That is to say, the Rayleigh range of the SH beam is the same as that of the fundamental beam.

5. Conclusions

We have conducted a comprehensive study on the single-pass SHG of a vortex beam under low-fundamental-wave depletion. Our results confirm the conservation of OAM and the unchanged Rayleigh range during the SHG process. In addition, SHG efficiency decreases with the increase in the fundamental vortex order, and mode purity is improved in the SHG process. Our results not only verify some basic laws in the SHG of vortex beams, but also provides a new approach for generating vortex beams of high quality.