Second Harmonic Generation of Twisted Vector Vortex Beams Using aβ-BaB₂O₄ Crystal

Abstract

In this study, we demonstrate both theoretically and experimentally second harmonic generation (SHG) of a twisted vector vortex optical field (TVVOF) using a nonlinear type-II phase-matched β-BaB₂O₄ (BBO) crystal. Our study introduces a novel method to manipulate SHG by independently modulating the two orthogonal polarization components of a TVVOF. This flexibility in controlling SHG can be achieved through accurate experimental adjustments of the polarization components. Furthermore, we reveal that the SHG can be dynamically tuned by varying the angle between the polarization direction of the optical field and the principal axis of the BBO crystal via rotation. These findings provide a new approach for the flexible manipulation of SHG in structured vector optical fields, which have potential applications in optical communication, quantum optics, and photonic device engineering.

1. Introduction

Second harmonic generation (SHG) has been extensively studied due to its fundamental physics and potential applications [1,2,3,4,5]. Remarkably, in the nonlinear optics region, a material system’s electric dipole moment (or nonlinear polarization) strongly depends on the spatial distribution profiles of the strength, phase, and states of polarization (SoP) of an applied optical field. Thus, a structured light field involving light–matter interactions provides additional freedom to manipulate nonlinear optical processes. The nonlinear polarization driven by the structured light field with a spatial variant SoP is sensitively dependent on the vectorial structure of an applied field. Therefore, the light–matter interaction with a structured light field extends our understanding and fundamental functionalities in nonlinear optics. On the other hand, the structured light field could be flexibly manipulated in multiple dimensions by the nonlinear optical process in both spatial and frequency domains [2,3]. In particular, the second-order nonlinear optical interaction requires the material to be able to birefringence to meet the phase-matching conditions. Thus, it is appreciated that the structured light fields possessing customized SoP vector structures and special spatial phase distributions provide the desired platform to shape and control the SHG.

The twisting phase, a non-separable quadratic phase within partially coherent light fields, has received widespread attention, due to its innovative attributes and promising practical uses [6,7] since Simon and Mukunda first discovered it in 1993 [8]. A novel form of the twisting phase with the capability of being generated experimentally through the use of a spatial light modulator (SLM) has been suggested [9,10]. At the same time, several prospective uses for the newly introduced twisted beams have been identified, including the manipulation of micro-particles [11] and the measurement of vortex optical fields [12,13]. In the field of structured light, space-time vector light sheets and space-time wave packets represent advanced optical fields that are characterized by non-separable spatio-temporal degrees of freedom. They exhibit unique properties such as near-diffraction-free propagation, self-healing, controllable group velocities, and exotic refractive phenomena, which can be achieved through innovations in spatio-temporal Fourier synthesis and open new avenues in beam optics and ultrafast optics [14,15]. Recently, twisted vector vortex fields have been experimentally generated, and their propagation and tightly focusing properties have been studied [4,16]. In addition, the effects of twisted caustic phases [17] and higher-order twisted phases on the polarization state of the vector light field [18] have also been studied experimentally and theoretically. Nonetheless, the SHG of a twisted vector vortex beam with spatially varying SoP has not been explored. In particular, the effect of twisting phases on the SHG may result in intriguing phenomena.

In this work, the SHG of twisted vector vortex optical fields (TVVOF) using a type-II phase-matched β-BaB₂O₄ (BBO) crystal is demonstrated theoretically and experimentally. The SHG of TVVOF can be flexibly manipulated by the two orthogonal polarization components of a fundamental wave of TVVOF that can be independently modulated experimentally by both the twisting phases and the vortex phases. In particular, the SHG can be manipulated with the various angles between the polarization direction of the optical field and the principal axis of BBO by rotating the BBO. In addition to demonstrating the manipulation of SHG, this study explores its potential applications in the manipulation of higher-order beam shaping and light–matter interactions, optical communication, quantum optics, photonic device engineering, and micro-processing. These applications leverage the precise control of SHG to enhance performance and efficiency in various technological domains.

2. Theoretical Model

A twisted vector vortex optical field (TVVOF) can be expressed as follows [4]:

$$E(x,y)=\exp (-r^2/w_0^2)[\exp (i{u}_{1}xy+i{m}_1\theta ){\overrightarrow{e}}_1+\exp (i{u}_{2}xy+i{m}_2\theta ){\overrightarrow{e}}_2],$$

(1)

where r = $\sqrt{x^2+y^2}$ represents the polar radius, w₀ is waist radius, $\overrightarrow{e_1}=\left(\overrightarrow{e_x}+\mathrm{e}\mathrm{x}\mathrm{p}\left(i△\theta \right)\overrightarrow{e_y}\right), \overrightarrow{e_2}=\left(\overrightarrow{e_x}-\mathrm{e}\mathrm{x}\mathrm{p}\left(i△\theta \right)\overrightarrow{e_y}\right)$, $\overrightarrow{e_x}$ and $\overrightarrow{e_y}$ are the unit vectors in x- and vertical y- directions, m₁ and m₂ are the vortex topological charges, and u_1, and u₂ are the twisting strength coefficients. The initial SoP of a TVVOF is closely related to the value of Δθ when Δθ = π/2, and the initial SoP of a TVVOF is locally linear polarization in the cross-section of the field. If Δθ = 0, the SoP is a hybrid SoP distribution composed of the linear, circular, and elliptical polarization components located at different positions in the cross-section. In this work, we mainly focus on a TVVOF with a hybrid SoP distribution. For this case, Equation (1) can be written with Δθ = 0 as follows:

$$\begin{array}{l}E\left(x,y\right)=\exp (-r^2/w_0^2)\exp \left(i\left(\frac{u_1+u_2}{2}xy+\frac{m_1+m_2}{2}\theta \right)\right)\times \\ [\cos \left(\frac{u_1-u_2}{2}xy+\frac{m_1-m_2}{2}\theta \right)\overrightarrow{e_x}+i\sin \left(\frac{u_1-u_2}{2}xy+\frac{m_1-m_2}{2}\theta \right)\overrightarrow{e_y}].\end{array}$$

(2)

Under the condition of phase matching of the x and y polarization components of the structured optical field in the type-II BBO crystal, the SHG excited by the pump field (as shown in Equation (2)) can be derived as follows [19,20]:

$$\begin{array}{l}P(2w)=-i{d}_{22}\exp (-2r^2/w_0^2)\exp (i[(u_1+u_2)xy+(m_1+m_2)\theta )]\\ \sin [(u_1-u_2)xy+(m_1-m_2)\theta ],\end{array}$$

(3)

where d₂₂ is the second-order nonlinear coefficient. In addition, the SHG excited by the two orthogonal polarization components of the pump light field is closely related to the angles between the polarization direction of the optical field and the principal axis of BBO. When the angle between the polarization direction of the optical field and the principal axis of BBO is rotated around the optical axis by an angle ϕ relative to the initial position (ϕ = 0), the two orthogonal polarization components of pump light will change into E′_x = E_x*cos ϕ + E_y sin ϕ and E′_y = −E_x sin ϕ + E_y cos ϕ, respectively. In this case the SHG can be written as follows:

$$p^{\prime }(2w)=d_{22}\exp (-2r^2/w_0^2)\exp [i(m_1+m_2)\theta ][\sin 2\varphi -i\sin (m_1-m_2)\theta \cos 2\varphi ].$$

(4)

For the case of ϕ = 0, Equation (4) degenerates into Equation (3), with u₁ = u₂ = 0. In particular, if m₁ = −m₂, Equation (4) reduces to the simple following form: p′(2w) = d₂₂exp(−2r²/w₀²) [sin(2ϕ) − isin(2mθ)cos(2ϕ)]. When the rotation angle ϕ = (2n − 1) π/4 (n = 1, 2, 3…), the SHG becomes p′(2w) = d₂₂ exp(−2r²/w₀²), and a Gaussian beam appears. The spatially varying polarization of fundamental light and the intensity distributions of orthogonal components of fundamental light and SHG for various angles ϕ are shown in Figure 1.

3. SHG of TVVOF Using a BBO Crystal

The experimental setup of SHG of TVVOF using a type-II phase-matched β-BaB₂O₄ (BBO) crystal is schematically shown in Figure 2. Firstly, a horizontal linear polarized ultrafast pulse laser beam (λ = 800 nm) with controllable power passes through an expanded collimation system, which is composed of lens L1 (focal length 100 nm) and lens L2 (focal length 300 nm). The main function of the expanded collimation system is to uniformly amplify the thin beam emitted by the femtosecond laser (Spectra-Physics Mai Tai SP, 750 mW). Then, it is incident on the spatial light modulator (SLM, Holoeye Photonics LETO 1920 × 1080 pixels, and the unit pixel is 6.4 μm). A computer-generated hologram (CGH) is loaded into the SLM with the following transmittance function: t(x₀,y₀) = 0.5 + γ [cos(2πγx₀ + δ₁) + cos(2πγx₀ + δ₂)]/4, where γ and f₀ are the modulation depth and the calculated hologram spatial frequency, respectively. δ₁ and δ₂ are the additional phases in the horizontal and vertical directions, respectively. In this work, the additional phase distributions in the CHG are set as δ₁ = u₁xy + m₁θ and δ₂ = u₂xy + m₂θ.

The laser beam is diffracted by the holographic grating of the SLM into multiple diffraction orders. A spatial filter, with double holes being placed between two identical Fourier lenses (L3 and L4 with a focal length of 200 mm), is used to remove all higher diffraction orders, except the +1 orders in the x and the y directions. Two λ/2 wave plates are glued with angles of 22.5° and −22.5° to the horizontal direction, respectively. They are tightly attached to the small holes of the spatial filter to generate two orthogonal horizontally and vertically polarized light beams carrying different phase information, δ₁(x, y) and δ₂(x, y). Finally, the two beams carrying different phase information are combined by customized Ronchi grating (RG) with a grating period of 12 lp/mm, which is placed on the rear focal plane of the fourth Fourier lens. By adjusting the period of CGH to match with that of RG, the two beams allow propagating along the same path, where our vector beam generated. It is noteworthy that these two orthogonal polarization components carry different phase information. In particular, δ₁(x, y) and δ₂(x, y) can be individually controlled flexibly by simply manipulating the holograms in SLM.

The generated TVVOF is sent as a fundamental frequency light field to a type II (e + o →e) phase-matched β-BaB₂O₄ (BBO, 10 mm × 10 mm × 0.5 mm) placed between L5 and L6 (f = 200 mm) in the Fourier plane of the second 4f system to investigate the SHG effect. The function of lens L5 is to focus the intensity of the pump light source incident on the type II phase-matched BBO crystal, thereby increasing the intensity of the generated double-frequency light field. L5 also allows us to focus the generated vector beam with a beam size of w₀ = 2 × 10⁻³ m on the BBO. The function of lens L6 is to image the generated SHG light field onto the charge-coupled device (CCD, IMPERX B1020) after passing through a dichroic mirror (DM).

Compared with one-dimensional holographic gratings that can only control the phase, amplitude, and polarization state of an optical field in a single direction, the grating that we employed in the present study adds a degree of freedom and can control the light field in the x and y directions simultaneously. The phase, amplitude, and polarization states can be manipulated so that more complex vector light fields can be flexibly generated.

4. Vector Manipulation of SHG with TVVOF and Rotating BBO

SHG and the corresponding fundamental fields of TVVOF when the twist coefficients are the same (u₁ = u₂) are shown in Figure 3. For comparison, the experimental and theoretical results of the TVVOF without the twisting phase (i.e., twist coefficient u₁ = u₂ = 0) are shown in Figure 3a. Obviously, the light intensity distributions of the TVVOF without the twisting phase modulation are represented by the reshaped lobes. The number of intensity lobes is defined as the difference between the two vortex topological charges, |m₁ − m₂| [21]. On the other hand, the number of lobes of the SHG is twice that of the corresponding fundamental fields (i.e., 2 × |m₁ − m₂|) which is consistent with the topological transformation rule.

For the case of the TVVOF with the same twist coefficient modulations (u₁ = u₂), the number of intensity lobes on the cross-section of the field is similar to that without the twisting phase modulation, as shown in Figure 3a,b. However, the intensity distribution at the cross-section of the TVVOF rotates relative to the untwisted vector vortex field. Since the two orthogonal polarization components are modulated by the same twisting phase, u × y, the reshaped intensities are compressed along a 45° direction of both the fundamental field and SHG, as shown in Figure 3b. The reason for the compression along 45° is attributed to the result of the twisting phase u × y, which can be regarded as a cylindrical lens placed at the angle of 45° [22].

When the two orthogonal polarization components are modulated by the different twisting phases u₁xy and u₂xy (u₁ ≠ u₂), the SHG of TVVOF with a type-II BBO is dependent on the two orthogonal polarization components. In other words, the SHG is closely related to the twisting coefficients (u₁, u₂) and topological charges (m₁, m₂). To more clearly demonstrate the effect of the twisting coefficients and topological charges of orthogonal polarization components on the SHG, we set u₁ = 0 and m₂ = 0 (see Equation (2)) for simplicity of analysis. The SHG can be expressed as follows: p(2w) = −id₂₂exp(−2r²/w₀²)exp [i(u₂xy + m₁θ)] sin(−u₂xy + m₁θ).

The light intensity distributions of the fundamental field and SHG are the reshaped lobes surrounded by the fringes, as shown in Figure 4. Comparing the SHG and the corresponding fundamental fields, one could find that the number of lobes of the fundamental fields is equal to the angular topological charge m₁. The number of lobes of the second harmonic fields is twice that of the angular topological charge of the fundamental fields. For example, when m₁ = 2 and u₂ = 3 × 10⁶ mm⁻², the second harmonic field has four lobes, whereas the fundamental field has two lobes. It can be seen that there are more fringes in SHG than that found in the fundamental field, due to the interference of SHG with the twice twisting coefficients and topological charges of the fundamental field. In particular, the twisting coefficients and topological charges of the fundamental field can be independently modulated during the experimental generation of TVVOF, providing a flexible approach to manipulate the SHG of TVVOF.

Comparing the figures in the first row with those in the second row and the figures in the third row with those in the fourth row, the case with a larger twist coefficient and the same topological charge has compressed the intensity lobes at the center. In addition, more fringes appear at the edge of the beam cross-section. After having their frequency doubled, the twisting phases and the vortex phases are tangled together, as shown in Equation (3). When the vortex phase m₁ is constant, the number of intensity fringes increases with the increase in the twist coefficient, then the characteristic lobe will be compressed and become finer and thinner.

In addition, when the nonlinear crystal BBO is rotated around the optical axis by a certain angle ϕ relative to the initial position (ϕ = 0), the SHG of a vector vortex beam will sensitively depend on the rotating angle ϕ (see Equation (4)). Therefore, the SHG of a vector beam, vortex, or Gaussian beam may appear at certain rotation angles ϕ. When the BBO crystal is rotated to an angle of ϕ = kπ/2, where k is an integer, the SHG field becomes p′(2w) = −id₂₂ exp(−2r²/w₀²)exp [i(m₁θ + m₂θ)] sin(m₁ − m₁) θ. If the rotating angle is ϕ = (2$k+1)\pi /4$, the SHG field is p′(2w) = ±d₂₂ exp(−2r²/w₀²)exp [i(m₁θ + m₂θ)], a vortex appears, as shown in Figure 5. The intensity distributions of SHG of the vector vortex beam with u₁ = u₂ = 0, m₁ = 3, m₂ = 1, and u₁ = u₂ = 0, m₁ = 1, m₂ = −1 as a function of the rotation angles of BBO are shown in Figure 5a. When m₁ = 1, m₂ = −1, the generated SHG field has four lobes (i.e., 2 × |m₁ − m₂|), except at ϕ = (2k + 1)π/4, where the Gaussian beam appears, as shown in Figure 5a. When m₁ = 3, m₂ = 1, the generated SHG field also has a four lobes (i.e., 2 × |m₁−m₂|), but the vortex is generated at ϕ = (2k + 1)π/4, as shown in Figure 5b.

5. Discussion

The light–matter nonlinear interactions of a structured light beam with spatially varying polarization extend the fundamental nonlinear optics and have opened up new applications in a variety of scenarios [2,3,23,24]. In this work, the SHG of TVVOF with the nonlinear type-II phase-matched β-BaB₂O₄ (BBO) crystal is demonstrated theoretically and experimentally. The SHG can be manipulated based on two orthogonal polarization components of a TVVOF. The two orthogonal polarization components can be independently modulated experimentally to flexibly manipulate the SHG of a TVVOF. The SHG is closely related to the angle between the polarization direction of the optical field and the principal axis of BBO, which can be controlled by rotating the BBO crystal. For the vector vortex beam, when the topological charges satisfy the condition m₁ = −m₂, the second harmonic field will degenerate into a Gaussian beam when the BBO is rotated by ϕ = (2k − 1) π/4.

The ability to dynamically control SHG by manipulating the polarization components offers significant potential for practical applications. In nonlinear optics, the SHG characteristics from the twisted vector optical field and engineered SHG spatial/topological feature obtained by controlling the interacting polarization states can further extend to the manipulation of higher-order beam shaping and light–matter interactions. In optical communication, this method can improve signal processing and noise reduction in wavelength division multiplexing systems. In quantum optics, it can enhance the generation and control of entangled photon pairs for quantum key distribution systems. These findings also have implications for photonic device engineering, where tunable SHG can lead to the development of advanced devices like tunable lasers and optical switches. Additionally, in micro-processing, an accurate control of SHG enables high-precision material modification, with potential applications in micro- and nano-scale manufacturing.

6. Conclusions

The SHG of TVVOF using a type-II (e + o → e) BBO crystal has been investigated. The two orthogonal polarization components can be independently modulated experimentally to flexibly manipulate the SHG of TVVOF. In particular, the SHG can be manipulated with the different angles between the polarization direction of the optical field and the principal axis of the BBO. The SHG of the vector beam, vortex, or Gaussian beam appears as a result of the rotation angle of the BBO. These results extend the vector manipulation of the SHG of a structured vector optical field and may find potential applications in ultraviolet micro-processing, optical field manipulation, and quantum optics.