Investigating the generation and propagation evolution of orange optical vortices using continuous-wave KGW Raman lasers with astigmatic mode transformations

J. C. Tung; P. L. Li; M. H. Hou

doi:10.1364/OE.470070

1. Introduction

High-order vortex beams (HVBs) with multiple singularities or high-order topological charges carrying orbital angular momentum have been widely used in various applications in the past decades, such as manipulations of particles [1–3], optical communication [4,5], quantum entanglement [6], optical display [7] and, in particular, optical microscopy [8]. In addition, the complexity of HVBs has been manifested to be of specific interest for the effective realization of high-resolution imaging [9]. To resolve HVBs with nanoscale resolution, the nano-surfaces of rhodamine and sulforhodamine dye molecules, relying on their sensitivity to the amplitude, phase and polarization of light fields, are widely known to be efficient light-harvesting systems [9]. Specifically, the fluorescent dye of sulforhodamine 101 with its maximum absorption at 586 nm has been applied to improve the signal-to-noise ratio of cancer cell imaging [10]. Therefore, the investigation of HVBs near 586 nm is an important task for determining the structure of biological molecules and the detection of sick organisms in vivo. Thanks to the advent of solid-state stimulated Raman scattering (SRS), diode-pumped neodymium Nd-doped lasers combining SRS with sum frequency generation (SFG) or second-harmonic generation (SHG) have been confirmed as a promising approach for generating orange light near 586 nm [11–14]. Even though the HVBs with high-order topological charges in 586 nm has been obtained [15], the orange HVBs with multiple singularities have not been reported yet.

Thus far HVBs have been demonstrated from several special cases of laser modes with diverse methods, such as Hermite–Laguerre–Gaussian (HLG) modes generated in the astigmatic mode converter (AMC) system [16,17], helical-Ince–Gaussian modes realized in special cavities with a spatial light modulator [18], and geometric modes formed in a degenerate cavity [19–21]. However, the generation of HVBs are often accompanied by the creation and annihilation of phase singularities during propagation. For instance, the HLG mode exists m singularities with topological charge one in the near field, but the singularities evolve into a single vortex with topological charge m in the far field [17]. Interestingly, the singularity preservation in HVBs has been demonstrated by adding trace amounts of other wave components [22]. It is believed that the generation of HVBs that maintain their singularities during propagation is considered interesting and of great value for applications of independent vortices embedded in a beam.

In this work, we employ a selectively off-center pumped Nd:YVO₄/KGW Raman laser with intracavity SHG to produce various high-order orange beams (HOBs), which refer to high-order transverse modes at 588 nm. We experimentally confirm that the HOBs can achieve comparable persistence at the pump power of 2.88 W, where the average output powers are generally from 300 mW to 160 mW with increasing the off-center displacements from 0.14 mm to 0.21 mm. The lasing HOBs are subsequently converted by single cylindrical lens AMC [23,24] to generate a variety of the vortex structured lights. We demonstrate for the first time the effect of the SHG technique on the propagation evolution of converted lasing modes by constructing a theoretical model. Theoretical analyses are performed in detail to compare with experimental results. Based on the excellent agreement between numerical calculations and experimental patterns, the phase structures of the transformed HOBs in the propagation are further calculated to manifest the robustness of singularity evolution.

2. Experimental observation of HOBs

In the experiment, we used the off-center pumping to excite HOBs in a plano-concave resonator for a diode-pumped Nd:YVO₄/KGW Raman laser with intracavity SHG as shown in Fig. 1. The resonator was formed with a diode-pumped Nd:YVO₄ laser with a KGW and a LBO crystal for intracavity SRS and SHG, respectively. The gain medium was an a-cut 0.25-at.% Nd:YVO₄ crystal with dimensions of 3 × 3 × 10 mm³. The entrance facet of the Nd:YVO₄ crystal was coated to be highly reflective (HR) within 1030-1200 nm (R > 99.9%) and highly transmissive (HT) at 808 nm (T > 95%). The other facet of the Nd:YVO₄ crystal was coated to be HR at 588 nm (R > 98%). The Raman gain medium was a N_p-cut KGW crystal with dimensions of 3 × 3 × 20 mm³. The polarization of the fundamental field at 1064 nm was oriented to be parallel to the N_m axis of the KGW crystal to generate the Stokes wave near 1176 nm, corresponding to the Raman shift of 901 cm⁻¹. The KGW crystal was coated to separate the cavity of the Stokes wave from that of the fundamental field. For the design of a separate cavity, the facet of the KGW crystal toward the Nd:YVO₄ gain medium was coated to be HR at 1176 nm (R > 99.9%) and HT at 1064 nm (T > 99.5%). The other side of the KGW crystal was coated to be HT at 1064 and 1176 nm (T > 99%) and HR at 588 nm (R > 98%) to reflect the orange light in the backward generation. We employed indium foils to wrap the Nd:YVO₄ and KGW crystals and then used copper holders to mount them with water cooling at a temperature of 20 °C. The nonlinear crystal for the SHG of the Stokes wave was a LBO crystal with dimensions of 3 × 3 × 8 mm³ and the cut angle at θ = 90° and φ = 3.9°. The temperature of the LBO crystal was controlled at 24°C by a thermo-electric cooler for the best phase matching. The output coupler was a concave mirror with a radius of curvature of 100 mm. The concave side of the output coupler toward the laser cavity was coated as a dichroic mirror which was HR at fundamental and Stokes wavelengths (R > 99.9%) and HT at the SHG orange wavelength at 588 nm (T > 95%), whereas the other plane side was an antireflection coating for the orange output wavelength (R < 0.2%).

Fig. 1. Experimental setup for generating HOBs with off-center pumping scheme in a plano-concave cavity.

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The pump source was a 3 W fiber-coupled 808 nm laser diode with a core radius of 100 µm and a numerical aperture of 0.22. With a pair of coupling lenses, the pump radius in the gain medium was approximately 130 µm. All crystals and output couplers were arranged compactly with a total cavity length of approximately 50 mm. In the end-pumping scheme, the distance away from the center of the optical axis along the y axis indicated by Δy can be precisely controlled by the manual translation stages. Here we observed the output power of the HOBs versus the input power at different off-center displacements Δy, as shown in Fig. 2(a). The overall output power increased linearly for the input power below 2.9 W. The lasing thresholds of 0.71 W, 0.84 W and 0.96 W were obtained for Δy = 0.14, 0.18 and 0.21 mm, respectively. At a pump power of 2.88 W, the average output power could be up to 300 mW, 240 mW and 160 mW at Δy = 0.14, 0.18 and 0.21 mm, respectively. Moreover, the HOBs can be fairly sustained from lasing threshold up to the pump power of 2.88 W. The experimental far-field transverse patterns corresponding to the three Δy at an incident pump power of 2.88 W were imaged by a CCD camera, as depicted in Figs. 2(b)–2(d).

Fig. 2. (a) Experimental results for average output power of the HOBs versus incident pump power at different off-center displacements Δy. Experimental far-field transverse patterns of the HOBs with (b) Δy = 0.14 mm, (c) Δy = 0.18 mm, and (d) Δy = 0.21 mm at an incident pump power of 2.88 W.

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3. Theoretical analysis of HOBs

Under the paraxial approximation, the eigenmodes for the laser cavity with a plane mirror at z = 0 and a concave mirror at z = L can be expressed as the Hermite-Gaussian (HG) modes:

(1)$$\Phi _{m,n,s}^{}(x,y,z) = \;\frac{{\sqrt {2} }}{{w(z)}}\textrm{X}_m^{}(\xi ,z)\textrm{Y}_n^{}(\varsigma ,z)\,{e^{ - i\,{k_{m,n,s}}\tilde{z}}}{e^{i\,({m + n + \textrm{1}} ){\theta _G}(z)}}, $$

where

(2)$$\textrm{X}_m^{}(\xi ,z) = \frac{\textrm{1}}{{\sqrt {{\textrm{2}^m}\,m!\,\sqrt \pi } }}\;{H_m}(\xi )\;{e^{ - {\xi ^\textrm{2}}/\textrm{2}}}, $$

(3)$$\textrm{Y}_n^{}(\varsigma ,z) = \frac{\textrm{1}}{{\sqrt {{\textrm{2}^n}\,n!\,\sqrt \pi } }}\;{H_n}(\varsigma )\;{e^{ - {\varsigma ^\textrm{2}}/\textrm{2}}}, $$

H_m (•) is the Hermite polynomial of order m, $\xi = \sqrt {2} x/w\textrm{(}z\textrm{)}$, $\varsigma = \sqrt {2} y/w(z)$, $w(z) = {w_o}\sqrt {\textrm{1} + {{(z/{z_R})}^\textrm{2}}}$, ${w_o} = \sqrt {\lambda {z_R}/\pi }$ is the beam radius at the waist, λ is the emission wavelength, ${z_R} = \sqrt {L(R - L)}$ is the Rayleigh range, R is the radius of curvature of the concave mirror, θ_G (z) tan^–1(z/z_R) is the Gouy phase, $\tilde{z} = z + [({x^\textrm{2}} + {y^\textrm{2}})z]/[\textrm{2}({z^\textrm{2}} + {z_R}^\textrm{2})]$, k_m,n_,s is the eigenvalue of the wave number, s is the longitudinal index, and m and n are the transverse indices in the x and y directions, respectively.

With the expansion of eigenmodes, the lasing mode can be given by the superposition of HG modes:

(4)$$\Psi (x,y,z) = \sum\limits_{m,n,s}^{} {{c_{m,n,s}}\Phi _{m,n,s}^{}(x,y,z)} , $$

where the coefficient ${c_{m,n,s}}$ solved from the orthonormal property of eigenmodes is expressed as

(5)$${c_{m,n,s}} = \int_{}^{} {\int_{}^{} {\int_{}^{} {\Phi _{m,n,s}^{}(x,y,z)F(x,y,z){\kern 1pt} dx{\kern 1pt} dy{\kern 1pt} dz} } }.$$

The coefficient ${c_{m,n,s}}$ in Eq. (4) indicates that the amplitude of the eigenmode is proportional to the overlap efficiency between the pump region $F(x,y,z\textrm{)}$ and the eigenmode $\Phi _{m,n,s}^{}$. In general, the distribution of the pump source $F(x,y,z\textrm{)}$ in the longitudinal z-direction can be approximated to be uniform. On the other hand, the transverse distribution of the pump source is similar to a Gaussian distribution. Consequently, the pump source $F(x,y,z\textrm{)}$ for the off-center pumping with a transverse displacement Δy in the y-direction can be given by

(6)$$F(x,y,z) = \frac{\textrm{2}}{{\pi {w^\textrm{2}}\textrm{(}z\textrm{)}{L_c}}}\exp \left[ { - \frac{{{x^\textrm{2}} + {{\textrm{(}y - \Delta y\textrm{)}}^\textrm{2}}}}{{{w^\textrm{2}}\textrm{(}z\textrm{)}}}} \right], $$

for | z – z_c | ≤ L_c/2, where z_c and L_c are the location and length of the gain medium, respectively. Since the longitudinal distribution of the pump source is nearly to be uniform, the coefficient ${c_{m,n,s}}$ related to the source term $F(x,y,z\textrm{)}$ can be regarded as independent of the index s in the neighborhood of the central index s_o. Consequently, the coefficient ${c_{m,n,s}}$ determined by the integral in Eq. (5) can be approximately reduced as ${c_{\textrm{0},n,{s_o}}}$ for the off-center pumping in the y-direction. The relationship between the transverse order n of HG modes and the relative lasing probability $\textrm{|}{c_{\textrm{0},n,{s_o}}}{\textrm{|}^\textrm{2}}$ for various off-center pumping has been analyzed in the previous work [25].

With the fundamental lasing modes $\Psi (x,y,z)$ in Eq. (4), the frequency-doubled lasing modes $\tilde{\Psi }(x,y,z)$ can be given by the square of $\Psi (x,y,z)$ and simplified as:

(7)$$\tilde{\Psi }(x,y,z) = \eta {\Psi ^\textrm{2}}(x,y,z) = \eta \sum\limits_{n,n^{\prime}}^{} {{\kappa _{n,n^{\prime}}}\textrm{ }\psi _{n,n^{\prime}}^{}(x,y,z)}, $$

where $\psi _{n,n^{\prime}}^{}(x,y,z) = \Phi _{\textrm{0},n,{s_o}}^{}(x,y,z)\,\Phi _{\textrm{0},n^{\prime},{s_o}}^{}(x,y,z)$, ${\kappa _{n,n^{\prime}}} = {c_{\textrm{0},n,{s_o}}}{c_{\textrm{0},n^{\prime},{s_o}}}$, and η is a constant related to the effective conversion efficiency in the second-harmonic process. At z = 0, the product term of eigenmodes $\psi _{n,n^{\prime}}^{}(x,y,\textrm{0}) = \Phi _{\textrm{0},n,{s_o}}^{}(x,y,\textrm{0})\,\Phi _{\textrm{0},n^{\prime},{s_o}}^{}(x,y,\textrm{0})$ is expressed as:

(8)$$\psi _{n,n^{\prime}}^{}(x,y,\textrm{0}) = \frac{{\textrm{2/(}\pi {w_o}^\textrm{2}\textrm{)}}}{{\sqrt {{\textrm{2}^{n + n^{\prime}}}n\,!\;n^{\prime}\,!} }}{H_n}({\varsigma _o}){H_{n^{\prime}}}({\varsigma _o}){e^{ - \textrm{(}{\xi _o}^\textrm{2} + {\varsigma _o}^\textrm{2})}}, $$

with ${\xi _o} = \sqrt {2} x/{w_o}$ and ${\varsigma _o} = \sqrt {2} y/{w_o}$. In order to generate the vortex structures, $\psi _{n,n^{\prime}}^{}(x,y,\textrm{0})$ in Eq. (8) needs to be represented by HG modes to realize the beam transformation later. Therefore, $\psi _{n,n^{\prime}}^{}(x,y,\textrm{0})$ can be further derived as

(9)$$\psi _{n,n^{\prime}}^{}(x,y,\textrm{0}) = \frac{{\sqrt {n\,\textrm{!}\;n^{\prime}\,\textrm{!}} }}{{{\textrm{2}^{\textrm{(}n + n^{\prime}\textrm{)}/\textrm{2}}}\pi {w_o}^\textrm{2}}}\sum\limits_{u = \textrm{0}}^{\min (n,n^{\prime})} {\frac{{{\textrm{2}^{u + \textrm{1}}}{H_{n + n^{\prime} - \textrm{2}u}}({\varsigma _o}){e^{ - \textrm{(}{\xi _o}^\textrm{2} + {\varsigma _o}^\textrm{2})}}}}{{u\,!(n - u\textrm{)!}(n^{\prime} - u)!}}}. $$

Using HG function to express this result, Eq. (9) can be represented as

(10)$$\psi _{n,n^{\prime}}^{}(x,y,\textrm{0}) = \frac{\textrm{1}}{{{{\tilde{w}}_o}^\textrm{2}}}\;\sqrt {\frac{{n\,\textrm{!}\;n^{\prime}\,\textrm{!}}}{{{\textrm{2}^{n + n^{\prime}}}\pi }}} \sum\limits_{u = \textrm{0}}^{\min (n,n^{\prime})} {{a_u}} \sum\limits_{v = u}^{\bar{n}} {{b_{u,v}}} \;\textrm{ X}_\textrm{0}^{}({\tilde{\xi }_o},\textrm{0})\textrm{Y}_{n + n^{\prime} - \textrm{2}v}^{}({\tilde{\varsigma }_o},\textrm{0})$$

with ${a_u} = \frac{{{{\textrm{(} - \textrm{1)}}^u}{\textrm{2}^{\textrm{2}u}}(n + n^{\prime} - \textrm{2}u)!}}{{u\,!(n - u\textrm{)!}(n^{\prime} - u)!}}$, ${b_{u,v}} = [{\textrm{(} - \textrm{2)}^v}(v - u)!\sqrt {(n + n^{\prime} - \textrm{2}v)!} ]{\textrm{ }^{ - \textrm{1}}}$, ${\tilde{\xi }_o} = \sqrt {2} x/{\tilde{w}_o}$, ${\tilde{\varsigma }_o} = \sqrt {2} y/{\tilde{w}_o}$, and ${\tilde{w}_o} = {w_o}/\sqrt {2}$. The parameter $\bar{n}$ in Eq. (10) is the integer closest and not greater than the value of (n + n′)/2. Consequently, $\psi _{n,n^{\prime}}^{}(x,y,z)$ in Eq. (7) at different z can be given by

(11)$$\psi _{n,n^{\prime}}^{}(x,y,z) = \frac{\textrm{1}}{{{{\tilde{w}}^\textrm{2}}(z)}}\;\sqrt {\frac{{n\,\textrm{!}\;n^{\prime}\,\textrm{!}}}{{{\textrm{2}^{n + n^{\prime}}}\pi }}} \sum\limits_{u = \textrm{0}}^{\min (n,n^{\prime})} {{a_u}} \sum\limits_{v = u}^{\bar{n}} {{b_{u,v}}} \;\tilde{\Phi }_{\textrm{0},n + n^{\prime} - \textrm{2}v,{s_o}}^{}(x,y,z)$$

with

(12)$$\tilde{\Phi }_{m,n,s}^{}(x,y,z) = \;\textrm{X}_m^{}(\tilde{\xi },z)\textrm{Y}_n^{}(\tilde{\varsigma },z)\;\;{e^{ - i\,{k_{m,n,s}}\tilde{z}}}{e^{i\,\textrm{(}m + n + \textrm{1)}{\theta _G}(z)}}$$

$\tilde{\xi } = \sqrt {2} x/\tilde{w}\textrm{(}z\textrm{)},\;\;\tilde{\varsigma } = \sqrt {2} y/\tilde{w}(z)\;\;\textrm{and} \;\;\tilde{w}(z) = w(z)\textrm{/}\sqrt {2}.$

4. Investigating propagation evolution of transformed HOBs

The AMC based on a single cylindrical lens [23,24], so called the single cylindrical lens AMC, was discovered to achieve the beam transformation more quickly and effectively than the traditional AMC formed by a matched pair cylindrical lenses [26,27]. Here we employ the single cylindrical lens AMC depicted in Fig. 3(a) to transform the HOBs to generate various vortex light fields in the propagation evolution. As shown in Fig. 3(a), a spherical lens is used to focus the input beam to create a new waist at a distance f_c just ahead of a cylindrical lens with focal length f_c and a new Rayleigh range ${z^{\prime}_R}$ equal to f_c. To derive the HG mode $\tilde{\Phi }_{m,n,s}^{}(x,y,z)$ in Eq. (11) transformed by a single cylindrical lens AMC with arbitrary angle α, the eigenfunction basis needs to change from the xy-Cartesian coordinate system to the x′y′-Cartesian coordinate system in which the x′ and y′ axes are the active and inactive components, respectively, as shown in Fig. 3(b). The original coordinates (x, y) are related to its new coordinates (x′, y′) by

(13)$$\left( {\begin{array}{{c}} {\tilde{\xi }}\\ {\tilde{\varsigma }} \end{array}} \right) = \left( {\begin{array}{{cc}} {\cos \alpha }&{\sin \alpha }\\ { - \sin \alpha }&{\cos \alpha } \end{array}} \right)\;\left( {\begin{array}{{c}} {\tilde{\xi }^{\prime}}\\ {\tilde{\varsigma }^{\prime}} \end{array}} \right). $$

By using the SU(2) algebra [27,28], the state $\textrm{X}_\mu ^{}(\tilde{\xi },z)\textrm{Y}_\upsilon ^{}(\tilde{\varsigma },z)$ of the original coordinate system can be expanded with the basis $\textrm{X}_{\mu ^{\prime}}^{}(\tilde{\xi }^{\prime},z)\textrm{Y}_{\upsilon ^{\prime}}^{}(\tilde{\varsigma }^{\prime},z)$ of the new coordinate system with µ+υ = µ′+υ′ = N, which can be derived as

(14)$$\textrm{X}_\mu ^{}(\tilde{\xi },z)\textrm{Y}_{N - \mu }^{}(\tilde{\varsigma },z) = \sum\limits_{\mu ^{\prime} = \textrm{0}}^N {K_{\mu ,\mu ^{\prime}}^N(\alpha )\textrm{ X}_{\mu ^{\prime}}^{}(\tilde{\xi }^{\prime},z)\textrm{Y}_{N - \mu ^{\prime}}^{}(\tilde{\varsigma }^{\prime},z)} , $$

where

(15)$$K_{\mu ,\mu ^{\prime}}^N(\alpha ) = \sum\limits_{\tau = \max (\textrm{0},\mu ^{\prime} - \mu )}^{\min (\mu ^{\prime},N - \mu )} {\frac{{{{\textrm{(} - \textrm{1)}}^\tau }\sqrt {\mu \,!} \sqrt {(N - \mu )!} \sqrt {\mu ^{\prime}\,!} \sqrt {(N - \mu ^{\prime})!} {{(\cos \alpha )}^{N - \mu + \mu ^{\prime} - \textrm{2}\tau }}{{(\sin \alpha )}^{\mu - \mu ^{\prime} + \textrm{2}\tau }}}}{{(\mu ^{\prime} - \tau )!\tau \,!(\mu - \mu ^{\prime} + \tau )!(N - \mu - \tau )!}}} . $$

The coefficients $K_{\mu ,\mu ^{\prime}}^N(\alpha )$ are exactly the Wigner little-d functions $d_{\mu - N/\textrm{2},\mu ^{\prime} - N/\textrm{2}}^{N/\textrm{2}}(\textrm{2}\alpha )$ [29].

Fig. 3. (a) Configuration of the single cylindrical lens AMC. Two vertical dash lines show the positions of the beam waists produced by the spherical lens and by the active axis of the cylindrical lens with focal length f_c. (b) Relationship between the xy-Cartesian and the x′y′-Cartesian coordinate systems of the cylindrical lens.

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Considering the cylindrical lens at z′ = 0 and the effects of the cylindrical lens in the region of z′ > 0 as shown in Fig. 3(a), the beam waists in the x′ (active) and y′ (inactive) axes are separable and given by

(16)$${\tilde{w}_{x^{\prime}}}(z^{\prime}) = {\tilde{w^{\prime}}_o}{\left[ {\textrm{1} + {{\left( {\frac{{z^{\prime} - {{z^{\prime}}_R}}}{{{{z^{\prime}}_R}}}} \right)}^\textrm{2}}} \right]^{\textrm{1/2}}},\;\textrm{and}\;\;{\tilde{w}_{y^{\prime}}}(z^{\prime}) = {\tilde{w^{\prime}}_o}{\left[ {\textrm{1} + {{\left( {\frac{{z^{\prime} + {{z^{\prime}}_R}}}{{{{z^{\prime}}_R}}}} \right)}^\textrm{2}}} \right]^{\textrm{1/2}}}$$

where ${\tilde{w^{\prime}}_o}$ is the beam radius at the new waist for the beam focused by a spherical lens, and the beam refocused by the cylindrical lens in the x′ (active) axis, as shown in Fig. 3(a). Note that ${z^{\prime}_R}$ is equal to the focal length f_c of the cylindrical lens in the AMC configuration. In addition, the wavefront curvatures in the x′ (active) and y′ (inactive) axes are different and given by

(17)$$\frac{\textrm{1}}{{{R_{x^{\prime}}}\textrm{(}z^{\prime}\textrm{)}}} = \frac{{z^{\prime} - {{z^{\prime}}_R}}}{{{{\textrm{(}z^{\prime} - {{z^{\prime}}_R}\textrm{)}}^\textrm{2}} + {{z^{\prime}}_R}^{\;\textrm{2}}}}, \textrm{and}\;\;\frac{\textrm{1}}{{{R_{y^{\prime}}}\textrm{(}z^{\prime}\textrm{)}}} = \frac{{z^{\prime} + {{z^{\prime}}_R}}}{{{{\textrm{(}z^{\prime} + {{z^{\prime}}_R}\textrm{)}}^\textrm{2}} + {{z^{\prime}}_R}^\textrm{2}}}.$$

The Gouy phases in the x′ (active) and y′ (inactive) axes are also different and given by

(18)$${\theta _{Gx^{\prime}}}(z^{\prime}) = \frac{\pi }{\textrm{2}} + {\tan ^{ - \textrm{1}}}\left( {\frac{{z^{\prime} - {{z^{\prime}}_R}}}{{{{z^{\prime}}_R}}}} \right), \textrm{and}\;\;{\theta _{Gy^{\prime}}}(z^{\prime}) = {\tan ^{ - \textrm{1}}}\left( {\frac{{z^{\prime} + {{z^{\prime}}_R}}}{{{{z^{\prime}}_R}}}} \right).$$

Using Eqs. (16–18) and the theoretical model similar to the frequency-doubled lasing modes $\tilde{\Psi }(x,y,z)$ in Eq. (7), the wave function for the converted frequency-doubled lasing modes can be derived as:

(19)$$\tilde{\Psi }^{\prime}(x,y,z{^{\prime}};\alpha ) = \eta \sum\limits_{n,n^{\prime}}{^{}} {{\kappa _{n,n^{\prime}}}\;\psi ^{\prime}_{n,n^{\prime}}{^{}}(x,y,z^{\prime};\alpha )} , $$

where

(20)$${\psi ^{\prime}_{n,n{^{\prime}}}}(x,y,z^{\prime};\alpha ) = \frac{\textrm{1}}{{{{\tilde{w}}_{x^{\prime}}}(z{^{\prime}}){{\tilde{w}}_{y^{\prime}}}(z{^{\prime}})}}\;\sqrt {\frac{{n\,\textrm{!}\;n^{\prime}\,\textrm{!}}}{{{\textrm{2}^{n + n{^{\prime}}}}\pi }}} \sum\limits_{u = \textrm{0}}^{\min (n,n{^{\prime}})} {{a_u}} \sum\limits_{v = u}^{\bar{n}} {{b_{u,v}}} \;\tilde{\Phi }^{\prime}_{n + n{^{\prime}} - \textrm{2}v,\,\textrm{0},\,{s_o}}{^{}}(x,y,z{^{\prime}};\alpha ), $$

(21)$$\begin{array}{l} \tilde{\Phi }^{\prime}_{\mu ,N - \mu ,s}{^{}}(x,y,z^{\prime};\alpha ) = {e^{i\,[\textrm{(1/2)}{\theta _{Gx^{\prime}}}(z^{\prime}) + (N + \textrm{1/2}){\theta _{Gy^{\prime}}}(z^{\prime})]}}{e^{ - i\,\varphi (x,y,z^{\prime})}}\\ \quad \quad \quad \quad \quad \quad \quad \quad \times \sum\limits_{\mu ^{\prime} = \textrm{0}}^N {{e^{i\,\mu ^{\prime}\beta (z^{\prime})}}K_{\mu ,\mu ^{\prime}}^N(\alpha )\textrm{ X}_{\mu ^{\prime}}^{}(\tilde{\xi }^{\prime},z^{\prime})\textrm{Y}_{N - \mu ^{\prime}}^{}(\tilde{\varsigma }^{\prime},z^{\prime})} \end{array}, $$

(22)$$\varphi \textrm{ }(x,y,z^{\prime};\alpha ) = \textrm{2}{z^{\prime}_R}z{^{\prime}}/{\tilde{w^{\prime}}_o}{^\textrm{2}} + \frac{\textrm{1}}{{\textrm{2}{{z^{\prime}}_R}}}[z^{\prime}({\tilde{\xi }{^{\prime}{^\textrm{2}}}} + {\tilde{\varsigma }^{\prime}{^\textrm{2}}}) - {z^{\prime}_R}({\tilde{\xi }^{\prime}{^\textrm{2}}} - {\tilde{\varsigma }^{\prime}{^\textrm{2}}})], $$

$\tilde{\xi }^{\prime} = \sqrt {2} x/{\tilde{w}_{x^{\prime}}}(z^{\prime})$, $\tilde{\varsigma }^{\prime} = \sqrt {2} y/{\tilde{w}_{y^{\prime}}}(z^{\prime})$, and $\beta (z^{\prime}) = {\theta _{Gx^{\prime}}}(z^{\prime}) - {\theta _{Gy^{\prime}}}(z^{\prime})$ represents the Gouy phase difference. Using Eq. (18), the Gouy phase difference $\beta (z^{\prime})$ can be found to increase from 0 to π/2 for z′ from 0 to ∞. It is worthwhile to mention that the converted basis $\tilde{\Phi }^{\prime}_{\mu ,N - \mu ,s}{^{}}(x,y,z^{\prime};\alpha )$ in the far field z′ → ∞ with $\beta (z^{\prime})$→ π/2 are named as the HLG beams [30]. Specifically, the beam $\tilde{\Phi }^{\prime}_{\mu ,N - \mu ,s}{^{}}(x,y,z^{\prime} \to \infty ;\alpha )$ correspond to a HG and a LG modes for α = 0 and α = ± π/4, respectively. Therefore, the mode components of the converted frequency-doubled lasing modes $\tilde{\Psi }^{\prime}(x,y,z^{\prime};\alpha )$ can be transformed into the corresponding superposition of LG modes with α = ± π/4 in the far field z′ → ∞ [25]. In our experiment, a spherical lens with focal length 75 mm and a cylindrical lens with focal length 50 mm were used to focus and refocused the generated HOBs to achieve the criteria for the AMC shown in Fig. 3(a), respectively. Experimental patterns of the converted HOBs at the off-center displacements Δy = 0.14, 0.18 and 0.21 mm were recorded by using a CCD camera at different z′ in the first row of Figs. 4–6, respectively. The corresponding numerical calculations for the propagation evolution of the converted beam $\tilde{\Psi }^{\prime}(x,y,z^{\prime};\alpha )$ with α = –π/4 are shown in the second row of Figs. 4–6. Experimental observations for all propagating positions are excellently reconstructed by theoretical calculations. To further observe the vortex structures of the transformed beams during propagation, phase structures are conventionally described in terms of the phase angle field $\Theta (x,y,z^{\prime};\alpha ) = {\tan ^{ - \textrm{1}}}\textrm{[Im(}\tilde{\Psi }^{\prime}\textrm{)}\textrm{Re} \textrm{(}\tilde{\Psi }^{\prime})\textrm{]}$, where $\textrm{Re} \textrm{(}\tilde{\Psi }^{\prime})$ and $\textrm{Im(}\tilde{\Psi }^{\prime}\textrm{)}$ are the real and imaginary parts of the field $\tilde{\Psi }^{\prime}(x,y,z^{\prime};\alpha )$. The phase angle field $\Theta (x,y,z^{\prime};\alpha )$ related to transformed beams $\tilde{\Psi }^{\prime}(x,y,z^{\prime};\alpha )$ are numerically calculated in the third row of Figs. 4–6. The phase structure near the central part at z′ ≈ ${z^{\prime}_R}$ can be seen that several isolated singularities come out. The individual singularities embedded in the beams are represented by white circles and correspond to the dark region of the transformed beams in the second row. There are 3, 5 and 7 singularities within the beams with topological charge one preserved during propagation for z′ > ${z^{\prime}_R}$, as seen in the third row of Figs. 4–6, respectively. On the other hand, additional singularities appear in the phase structure outside the central part represented by white triangles depicted in the third row of Figs. 5 and 6, because the relative lasing probability of higher-order eigenmodes increases with larger off-center displacements [25]. Notice that the additional singularities are still remained in propagation, but the excited higher-order eigenmodes lead to the non-circular symmetry of the structured beams in the far field, as shown in the last pattern of the second row of Figs. 5 and 6 for z′ = 20.0 ${z^{\prime}_R}$.

Fig. 4. Experimental results (first row), numerical wave patterns (second row) and phase structures (third row) for the propagation evolution of the converted HOB with Δy = 0.14 mm and α = –π/4.

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Fig. 5. Experimental results (first row), numerical wave patterns (second row) and phase structures (third row) for the propagation evolution of the converted HOB with Δy = 0.18 mm and α = –π/4.

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Fig. 6. Experimental results (first row), numerical wave patterns (second row) and phase structures (third row) for the propagation evolution of the converted HOB with Δy = 0.21 mm and α = –π/4.

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5. Conclusions

In summary, we have generated diverse HOBs at 588 nm by means of exploiting the diode-pumped solid-state Raman laser with intracavity SHG in the off-center pumping scheme. At a pump power of 2.88 W, the average output power of the HOBs can reach from 160 mW to 300 mW with off-center displacements in the range of 0.14–0.21 mm. Furthermore, we have employed a single cylindrical lens AMC to transform the HOBs for producing vortex structures. Theoretical analyses have also been performed to confirm the experimental results and to demonstrate the phase structures that the evolution of phase singularities is maintained in the transformed HOBs. It is believed that the present research not only systematically creates fairly persistent high-order orange vortex beams but also provides an important innovation for the generation of robust optical vortices during beam propagation.

Funding

Ministry of Science and Technology, Taiwan (MOST 110-2112-M-027-003-MY3).

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

References

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Investigating the generation and propagation evolution of orange optical vortices using continuous-wave KGW Raman lasers with astigmatic mode transformations

Abstract

1. Introduction

2. Experimental observation of HOBs

3. Theoretical analysis of HOBs

4. Investigating propagation evolution of transformed HOBs

5. Conclusions

Funding

Disclosures

Data availability

References

Data availability

Cited By

Figures (6)

Equations (22)

Optics Express