Characterization and generation of high-power multi-axis vortex beams by using off-axis pumped degenerate cavities with external astigmatic mode converter

P. H. Tuan; Y. H. Hsieh; Y. H. Lai; K. F. Huang; Y. F. Chen

doi:10.1364/OE.26.020481

1. Introduction

The developments of several frontier technologies, such as precision control of microparticles [1], non-contaminated manipulation in μ-bioengineering [2], quantum information [3], quantum cryptography [4], and fabrication of nanostructures with chirality [5] have made high-power optical vortex beams to be highly in demand. The Laguerre-Gaussian (LG) beams which possess the helical wavefront and central phase singularity can be regarded as the simplest form of optical vortices. Belonging to the eigenstates of optical resonators under paraxial approximation, LG modes can be straightforwardly generated in laser systems with cylindrical symmetry [6] or by external beam transformation of Hermite-Gaussian (HG) modes from systems with rectangular symmetry [7]. LG beams generated by astigmatic transformation of high-order HG modes have been confirmed to show flexibility for precisely controlling the beam transverse order during power scaling [8]. Recently, an analytical model which characterizes the propagation evolution of HG beams via the astigmatic mode converter (AMC) [9] has been built to offer useful insights for creating various generalized Hermite-Laguerre-Gaussian (HLG) beams [10] by beam transformation. Combined with the passively Q-switched (PQS) technique [11], the state of the art of generating vortex beams through beam transformation has been further proved to be a mature and promising way for realizing high-power optical vortices. However, the central order n for the pure HG_n,₀ or HG_0,_n eigenstates is limited to be less than 15 if a clear transverse pattern is required under high power operation [12]. To obtain vortex beams with large and well-defined orbital angular momentum (OAM), it is prerequisite to stabilize the spatial structures of high-order transverse modes during power scaling. Instead of pure HG or LG beams, the geometric modes composed by a group of cavity eigenstates in degenerate resonators have been confirmed to show high resistance against the pump-induced thermal instabilities [12]. It has been shown that the circular geometric modes generated in an off-axis pumped degenerate resonator with external astigmatic transformation can easily achieve high-peak-power PQS operation with clean beam structures [13]. Despite that the circular geometric mode can serve as a promising light source with abundant high-order vortex structures, it may be more desirable to realize vortex beams with multiple vortex centers for practical applications such as multidimensional manipulation of micro-particles [14,15] and quantum information by singular optics [16,17].

In this work, an Nd:YVO₄ degenerate laser resonator under two dimensional off-axis pump scheme is utilized to generate the generalized geometric modes to be further converted into vortex beams. The theoretical analysis of beam transformation shows that the generalized geometric mode with the transverse tomography consisting of several high-order HG modes instead of fundamental Gaussian spots can be one-to-one converted into the multi-axis LG beams via a SU(2) cylindrical mode converter [18]. The multi-axis LG beam with the transverse pattern of several uniformly arranged LG modes on the circumference can serve as a promising source to offer multi-center vortex structures. The mode order of each vortex center of the multi-axis LG beam is demonstrated that can be flexibly controlled by simply adjusting the off-axis displacement of pump beam. This flexibility of easier varying the OAM distribution of vortex beams is fairly beneficial for the applications like holographic optical manipulation and optical information processing [19–21]. More importantly, the experimental results reveal that both the generalized geometric modes and the corresponding multi-axis vortex beams by beam transformation can preserve quite clean beam structures even when the average output power has been scaled up to several watts. The overall average output power for the generated multi-axis vortex beams with non-deformed structures can reach up to beyond 3 W at a pump power about 8 W. Finally, the propagation evolution of the generalized geometric modes is further analyzed to provide useful insights in creating vortex beams with multi-center structures. The good agreement between the experimental and theoretical results once again confirm the feasibility of using the proposed method to realizing various kinds of high-power multi-center vortex beams.

2. Generalized geometric modes V.S. multi-axis vortex beams: theoretical consideration

At first the coherent superposition of cavity eigenstates in an off-axis pumped degenerate resonator is analyzed to find the fundamental principle for realizing the generalized geometric modes. The HG eigenstates in a typical laser resonator formed by a spherical mirror at z = -L and a plano mirror at z = 0 can be given by:

Φ_{n, m, s}^{(HG)} (x, y, z) = \sqrt{\frac{2}{L}} ψ_{n} (\tilde{x}) ψ_{m} (\tilde{y}) \cdot \sin [k_{n, m, s} \tilde{z} - (n + m + 1) θ_{G} (z)]

with

ψ_{n} (\tilde{x}) = \frac{1}{\sqrt{2^{n} n! \sqrt{π}}} H_{n} (\tilde{x}) e^{- {\tilde{x}}^{2} / 2}; ψ_{m} (\tilde{y}) = \frac{1}{\sqrt{2^{m} m! \sqrt{π}}} H_{m} (\tilde{y}) e^{- {\tilde{y}}^{2} / 2},

where

θ_{G} (z) = t a n^{- 1} (z / z_{R})

is the Gouy phase, z_R is the Rayleigh range,

H_{n} (\cdot)

is the Hermite polynomials of order n,

k_{n, m, s} = 2 π f_{n, m, s} / c

, f_n,m,s is the eigenmode frequency, c is the light speed,

\tilde{x} = \sqrt{2} x / ω_{x} (z)

,

\tilde{y} = \sqrt{2} y / ω_{y} (z)

,

\tilde{z} = z + [(x^{2} + y^{2}) z] / [2 (z^{2} + z_{R}^{2})]

. Note that for a system without astigmatism ω_x(z) = ω_y(z) =

ω_{o} \sqrt{1 + {(z / z_{R})}^{2}}

where ω_o is the beam waist radius. The eigenmode frequency of the resonator can be expressed by the longitudinal mode spacing Δf_L and transverse mode spacing Δf_T as

f_{n, m, s} = s \cdot Δ f_{L} + (n + m + 1) \cdot Δ f_{T}

, where s is the longitudinal mode index, and n and m are the transverse mode indices. In practice, the high-order HG beam with mode indices n_o and m_o can be experimentally generated by using end pumping with the off-axis displacement (Δx, Δy) along the x- and y-directions to satisfy the relations of

Δ x = \sqrt{n_{o}} ω_{g}

and

Δ y = \sqrt{m_{o}} ω_{g}

, where ω_g is the cavity mode size on the gain medium [22,23]. For a spherical resonator, the ratio of

Δ f_{T} / Δ f_{L}

can be expressed as

Δ f_{T} / Δ f_{L} = (1 / π) \cdot \tan^{- 1} (L / z_{R})

. Once the cavity length of the spherical resonator is adjusted to be

L = R \sin^{2} (P π / Q)

, the frequency ratio between the transverse and longitudinal mode spacing can be found to be simply given by

Δ f_{T} / Δ f_{L} = P / Q

, where R is the radius of curvature of the spherical mirror, P and Q are co-prime integers. This degenerate condition considering the longitudinal-transverse mode coupling will lead a group of HG modes

Φ_{n_{o}, m_{o} + Q \cdot K, s_{o} - P \cdot K}^{(HG)} (x, y, z)

to constitute a family of frequency degenerate states with minimum transverse order n_o and maximum longitudinal order s_o. With a sufficient large off-axis displacement, several degenerate HG modes

Φ_{n_{o}, m_{o} + Q \cdot K, s_{o} - P \cdot K}^{(HG)} (x, y, z)

will be simultaneously excited to contribute to the output emission. This collective contribution by the degenerate HG eigenstates can lead the lasing mode to be described by a SU(2) coherent state as:

Ψ_{n_{o}, m_{o}}^{M} (x, y, z; ϕ_{o}) = \frac{1}{2^{M / 2}} \sum_{K = 0}^{M} \sqrt{\frac{M!}{K! (M - K)!}} e^{i K ϕ_{o}} Φ_{n_{o}, m_{o} + Q \cdot K, s_{o} - P \cdot K}^{(H G)} (x, y, z),

where M + 1 denotes the number of HG modes in the superposition and ϕ_o is the relative phase. If only one-dimensional off-axis pumping is considered to lead the central mode order to be (0, m_o), the wave patterns of

| Ψ_{0, m_{o}}^{M} (x, y, z; ϕ_{o}) |

can be confirmed to nicely manifest the periodic orbits of geometric rays in the degenerate resonator [24]. Because of its feature of ray-wave correspondence,

Ψ_{0, m_{o}}^{M} (x, y, z; ϕ_{o})

is frequently named as the planar geometric modes. Figures 1(a)-1(d) show the intracavity traces and the corresponding far-field transverse patterns for the planar geometric modes with P/Q to be 1/4, 2/7, 1/3, and 2/5, respectively. The number and mutual separation of fundamental Gaussian spots at the far-field tomography of planar geometric modes can be seen to be respectively determined by the propagating directions and divergence angles of the ray periodic orbits. These planar geometric modes has been confirmed to be important in many applications such as optical delay lines, on-chip infrared spectroscopy, and design of ultrafast lasers [25–27]. Next an additional off-axis displacement along the x-direction is considered to examine how the morphologies of these planar geometric modes will be changed with the gradually increasing mode order n_o. Figures 2(a)-2(d) show the far-field patterns of

| Ψ_{n_{o}, m_{o}}^{M} (x, y, z; ϕ_{o}) |

at z = 2L with M = 10, m_o = 36, and increasing n_o from 0 to 35 for degenerate conditions of P/Q = 1/4, 2/7, 1/3, and 2/5, respectively. Note that the relative phase for the case of P/Q = 1/4 is given to be ϕ_o = π, whereas ϕ_o = 0 for other cases. It can be clearly seen that the increasing mode order n_o only leads the localized centers to be changed from fundamental Gaussian spots to high-order

{HG}_{n_{o}, 0}

modes while leaving the distribution along y-direction to be nearly unchanged. Because the mode expansion for each Gaussian spot center is independent of the formation of localized rays, these generalized geometric modes are quite useful to be transformed into vortex beams with easily-controlled OAM distribution as seen in later discussion.

Fig. 1 The intracavity ray traces and the corresponding far-field transverse patterns for the planar geometric modes with P/Q to be (a) 1/4, (b) 2/7, (c) 1/3, and (d) 2/5, respectively.

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Fig. 2 The calculated far-field patterns of $| Ψ_{n_{o}, m_{o}}^{M} (x, y, z; ϕ_{o}) |$ at z = 2L with M = 10, m_o = 36, and increasing n_o from 0 to 35 for degenerate conditions of P/Q = (a)1/4, (b) 2/7, (c) 1/3, and (d) 2/5, respectively. (aʹ)-(dʹ) show the corresponding multi-axis LG beams via astigmatic mode conversion. The final columns of (aʹ)-(dʹ) are the corresponding phase structures for the cases with n_o = 35.

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In order to derive the relation between the generalized geometric modes and their corresponding transformed modes after astigmatic conversion, the theoretical model in Ref. [9] for characterizing the transition from HG-based beams to LG-based beams is briefly reviewed. For a forward propagating HG mode:

Φ_{n, m}^{(HG +)} (x, y, z) = [ψ_{n} (\tilde{x}) e^{- i (n + 1 / 2) θ_{G, x} (z)} e^{i k x^{2} / 2 R_{x} (z)}] \cdot [ψ_{m} (\tilde{y}) e^{- i (m + 1 / 2) θ_{G, y} (z)} e^{i k y^{2} / 2 R_{y} (z)}]

passing through a single-lens AMC, the wave function

ψ_{n} (\tilde{x}) ψ_{m} (\tilde{y})

which mainly determines the spatial pattern should be re-projected onto the new coordinate system

({\tilde{x}}^{'}, {\tilde{y}}^{'})

given by the active (xʹ) and inactive (yʹ) axes of the cylindrical lens. Here the Gouy phases θ_G,x(z) and θ_G,y(z) as well as the wavefront curvatures R_x(z) and R_y(z) for x- and y-directions are separated for convenience in later derivation. Note that here the wave number k_m,n,s is denoted by k for simplicity and the longitudinal index s is neglected without losing generality. It has been verified that the projection of the original state

ψ_{n} (\tilde{x}) ψ_{m} (\tilde{y})

onto the new basis

ψ_{n^{'}} ({\tilde{x}}^{'}) ψ_{m^{'}} ({\tilde{y}}^{'})

of the new coordinate system can be expressed as a basis expansion followed the SU(2) algebra as [28]:

ψ_{n} (\tilde{x}) ψ_{N - n} (\tilde{y}) = \sum_{n^{'} = 0}^{N} B_{n^{'}, n}^{N} (α) ψ_{n^{'}} ({\tilde{x}}^{'}) ψ_{N - n^{'}} ({\tilde{y}}^{'}),

where

\begin{array}{l} B_{n^{'}, n}^{N} (α) \\ = \sum_{ν = \max (0, n^{'} - n)}^{\min (n^{'}, N - n)} \frac{{(- 1)}^{ν} \sqrt{n^{'}!} \sqrt{(N - n^{'})!} \sqrt{n!} \sqrt{(N - n)!} {(\cos α)}^{N - n + n^{'} - 2 ν} {(\sin α)}^{n - n^{'} + 2 ν}}{(n^{'} - ν)! ν! (n - n^{'} + ν)! (N - n - ν)!} . \end{array}

Here the constraint of

N = n + m = n^{'} + m^{'}

should be satisfied and α is the angle between the new coordinate axis xʹ and the original coordinate axis x. By further considering the astigmatic effect from the cylindrical lens on the beam radii, Gouy phases, and wavefront curvatures for the xʹ and yʹ-directions [9], the final expression for propagation evolution of the pure HG modes transformed by the single lens AMC can be given by:

{\tilde{Φ}}_{n, m}^{(HG+)} (x, y, z; α) = e^{- i [(N + 1 / 2) θ_{G, y^{'}} (z) + θ_{G, x^{'}} (z) / 2]} {\tilde{φ}}_{n, m} (x, y, z; α) e^{i Θ (x, y, z; α)},

where

θ_{G, x^{'}} (z) = \frac{π}{2} + \tan^{- 1} (\frac{z - z_{R}}{z_{R}}), θ_{G, y^{'}} (z) = \tan^{- 1} (\frac{z + z_{R}}{z_{R}});

{\tilde{φ}}_{n, m} (x, y, z; α) = \sum_{n^{'} = 0}^{N} e^{- i n^{'} β (z)} B_{n^{'}, n}^{N} (α) ψ_{n^{'}} ({\tilde{x}}^{'}) ψ_{N - n^{'}} ({\tilde{y}}^{'});

Θ (x, y, z) = \frac{1}{2 z_{R}} [z ({\tilde{x}}^{'}^{2} + {\tilde{y}}^{'}^{2}) - z_{R} ({\tilde{x}}^{'}^{2} - {\tilde{y}}^{'}^{2})];

_{$β (z) = θ_{G, x^{'}} (z) - θ_{G, y^{'}} (z)$} is the Gouy phase difference;

{\tilde{x}}^{'} = \sqrt{2} x^{'} / ω_{x^{'}} (z)

,

{\tilde{y}}^{'} = \sqrt{2} y^{'} / ω_{y^{'}} (z);

ω_{x^{'}} (z) = ω_{o} \sqrt{1 + {[(z - z_{R}) / z_{R}]}^{2}},

and

ω_{y^{'}} (z) = ω_{o} \sqrt{1 + {[(z + z_{R}) / z_{R}]}^{2}} .

Equation (7) clearly reveals that the spatial structure of the transformed HG mode

{\tilde{Φ}}_{n, m}^{(HG+)} (x, y, z; α)

is solely dominated by the wavefunction

{\tilde{φ}}_{n, m} (x, y, z; α)

in Eq. (9). It has been known that the transformed modes

{\tilde{Φ}}_{n, m}^{(HG+)} (x, y, z \to \infty; α)

at the far field with β(z)→π/2 correspond to the general HLG modes that contain HG and LG beam families as particular representatives [10]. Furthermore, several theoretical and experimental studies have confirmed that a pure HG mode with mode indices

(n, m)

can be nicely converted into a pure LG mode with mode indices

(\tilde{n}, \tilde{m})

at the far field via the AMC with the operation angle α = π/4 [29]. Using the expression of eigen-energy for a quantum isotropic harmonic oscillator solved under both the rectangular and cylindrical coordinates, the relation between the indices of the HG and LG modes can be determined to be

n = \tilde{n}

and

m = \tilde{n} + | \tilde{m} |

[18]. Following the above discussion and using Eq. (6) and Eq. (9), LG beams by beam transformation can be directly written as the superposition of HG modes as:

\begin{array}{l} {\tilde{Φ}}_{\tilde{n}, \tilde{m}}^{(LG)} (x, y, z) = (\sum_{μ = 0}^{N} e^{- i μ π / 2} B_{μ, \tilde{n}}^{2 \tilde{n} + \tilde{m}} ψ_{2 \tilde{n} + \tilde{m} - μ} ({\tilde{x}}^{'}) ψ_{μ} ({\tilde{y}}^{'})) \\ \times e^{- i [(2 \tilde{n} + \tilde{m} + 1 / 2) θ_{G, y^{'}} (z) + θ_{G, x^{'}} (z) / 2]} e^{i Θ (x, y, z)} \end{array}

with

B_{μ, \tilde{n}}^{2 \tilde{n} + \tilde{m}} = \sum_{ν = \max (0, μ - \tilde{n})}^{\min (μ, \tilde{n} + \tilde{m})} \frac{{(- 1)}^{ν} \sqrt{(\tilde{n} + \tilde{m})! n! (2 \tilde{n} + \tilde{m} - n)! μ!}}{\sqrt{2^{2 \tilde{n} + \tilde{m}}} (μ - ν)! ν! (n - μ + ν)! (\tilde{n} + \tilde{m} - ν)!} .

Since the generalized geometric mode is the coherent superposition of the degenerate HG eigenstates

Φ_{n_{o}, m_{o} + Q \cdot K, s_{o} - P \cdot K}^{(HG)} (x, y, z)

, its transformed wave function via the AMC can be straightforwardly written as the total contribution from the corresponding transformed LG modes

Φ_{{\tilde{n}}_{o}, {\tilde{m}}_{o} + Q \cdot K, s_{o} - P \cdot K}^{(LG)} (x, y, z)

as:

{\tilde{Ψ}}_{{\tilde{n}}_{o}, {\tilde{m}}_{o}}^{M} (x, y, z; ϕ_{o}) = \frac{1}{2^{M / 2}} \sum_{K = 0}^{M} \sqrt{\frac{M!}{K! (M - K)!}} e^{i K ϕ_{o}} Φ_{{\tilde{n}}_{o}, {\tilde{m}}_{o} + Q \cdot K, s_{o} - P \cdot K}^{(L G)} (x, y, z),

where the central indices follow the relations of

n_{o} = {\tilde{n}}_{o}

and

m_{o} = {\tilde{n}}_{o} + {\tilde{m}}_{o}

. Using Eq. (13), the morphologies of the transformed wave functions

| {\tilde{Ψ}}_{{\tilde{n}}_{o}, {\tilde{m}}_{o}}^{M} (x, y, z; ϕ_{o}) |

corresponding to the generalized geometric modes in Figs. 2(a)-2(d) are calculated and shown in Figs. 2(aʹ)-2(dʹ). It can be seen that even though the localized centers of the generalized geometric modes may overlap to each other, the transverse patterns of the transformed modes obviously present Q localized centers uniformly distributed on the circumference of a circle under the degenerate condition of P/Q as seen from the cases of n_o = 0. Note that the radius of the circle is determined by the central order for the direction to form the geometric rays, i.e. m_o in this situation. On the other hand, each high-order HG mode extended at the localized center with the increasing n_o is converted into pure LG mode to lead the transformed wave function to have an overall circle-bundle structure. Because the transformed modes possess different central axes for the respective LG vortices, it can be named as the multi-axis LG beams. Moreover, as n_o increases to be sufficiently large, the bundled LG modes on the circle will intersect to each other to form intriguing patterns with high degree-of-freedom phase structures as seen the fourth columns of Figs. 2(aʹ)-2(dʹ) for the cases of n_o = 35. The phase structures can be found to clearly present Q phase singularities at the centers of the respective axes of the LG beams for the cases with degenerate conditions of P/Q. Because the mode orders for these multi-axis LG beams are quite large, the phase distribution at the central regions of the bundled-LG beams exhibits complicated structures with Q-folded symmetry. It is worthy to mention that the wave patterns of multi-axis LG beams correspond to important physics about moving charged particles in crossed electric and magnetic fields that show the dynamics with local and global angular momenta [30]. In next section, the off-axis pumped degenerate laser resonator combined with an external AMC is exploited to realize high-power multi-axis LG beams experimentally.

3. Experimental realization of high-power, multi-axis vortex beams

Figure 3 depicts the experimental setup for generating the multi-axis vortex beams. A spherical cavity formed by a concave front mirror with the radius of curvature R = 30 mm and a plano output coupler was set to satisfy different degenerate conditions of P/Q with the cavity length to be $L = R \sin^{2} (P π / Q)$ . The front mirror was coated antireflection (AR, reflection <0.2%) at 808 nm on the entrance face as well as coated high-reflection (HR, reflection >99.8%) at 1064 nm and high-transmission (HT, transmittance >90%) at 808 nm on the second surface. A 0.2 at. % a-cut Nd:YVO₄ crystal with dimensions of 3 × 3 × 12 mm³ and AR coating at 1064 nm on both end surfaces was used as the gain medium for the laser emission. The Nd:YVO₄ crystal wrapped with indium foil was mounted in a water-cooled copper heat sink at 16°C and was placed near the front mirror with d = 5 mm to have an adequate cavity mode size ω_g for generating high-order HG modes under off-axis pumping. The effective transmittance at 1064 nm for the output coupler is 5% to easier achieve power scaling. The pump source was a 16-Watt 808 nm fiber-coupled laser diode with a core diameter of 100 μm and a numerical aperture of 0.16. A coupling lens set with effective focal lens of 38 mm and a unity magnification was exploited to reimage the pump beam into the gain crystal with an average pump diameter of 130 μm considering the beam divergence. An extra-cavity single-lens AMC composed by a spherical lens with the focal length f_m = 90 mm and a cylindrical lens with effective focal length f_c = 43 mm was used to perform the beam transformation.

Fig. 3 The experimental setup for generating the multi-axis vortex beams.

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At first, the pump beam was aligned to the transverse center of the resonator to excite the fundamental Gaussian mode under each degenerate condition. Subsequently, the pump beam was offset from the optical axis along y direction with displacement Δy to result in the central order $m_{o} = Δ y^{2} / ω_{g}^{2} = 36$ to generate the planar geometric mode for each degenerate cavity. Then the displacement of pump beam along x direction Δx was gradually enlarged to increase $n_{o}$ from 0 to 35 under the fixed m_o. After the generalized geometric modes at different degenerate conditions have been successfully generated, their output performance with increasing pump power was further studied. Figures 4(a)-4(d) show the dependence of average output power on input pump power for the generalized geometric modes at different degenerate conditions P/Q of 1/4, 2/7, 1/3, and 2/5 with the central mode orders to be (n_o, m_o) = (0, 36), (15, 36), and (35,36), respectively. As n_o increases, the highest achievable output power for these cases can be seen to drop owing to the decrement of overlapping efficiency between the pump and lasing modes with enlarging off-axis displacement [31]. On the other hand, the output power for the cases of P/Q = 1/3 and 2/5 with (n_o, m_o) = (35, 36) seems to exhibit saturation behavior at higher pump level because of the relatively stronger thermal effects on the cavity stability at comparably longer cavity length. Nevertheless, the overall performance of average output power for these generalized geometric modes can easily reach up to over 3 W at a pump power of 10 W with the optical-to-optical conversion efficiency higher than 30%. Figures 5(a)-5(d) display the far-field patterns of generalized geometric modes corresponding to those in Figs. 4(a)-4(d) at different pump power. It can be found that the beam structures for the cases of P/Q = 1/4 and 1/3 can remain almost unchanged even at the high pump level with P_in = 16 W since the degenerate resonator with smaller Q allows more degenerate eigenstates participating in the superposition to resist the thermal-induced perturbation. On the other hand, the patterns for the case of P/Q = 2/5 can be found to become blurred and impure as the pump power increases because there exist larger diffractive losses to influence the localized geometric rays for the longer cavity. However, all these generalized geometric modes can be seen to preserve quite clean beam structures at a pump power of 8 W.

Fig. 4 Average output power versus input pump power for generalized geometric modes at different degenerate conditions P/Q of (a) 1/4, (b) 2/7, (c) 1/3, and (d) 2/5 with central mode orders (n_o, m_o) = (0, 36), (15, 36), and (35,36), respectively.

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Fig. 5 (a)-(d) Experimental far-field patterns of generalized geometric modes corresponding to the output power performance shown in Figs. 4(a)-4(d).

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Next the single-lens AMC was utilized to transform the generalized geometric modes into the multi-axis vortex beams. The operation principle for performing astigmatic transformation via the single-lens AMC is to make the Rayleigh range of the reimaged beam waist by the spherical lens to be just equal to the focal length of the cylindrical lens f_c as shown by Fig. 3. Following this principle and using ABCD law for Gaussian beam, the guideline for determining the mutual distances L_f and L_c in the setup for a successful astigmatic transformation can be derived in terms of f_m and f_c as [32]:

L_{f} = f_{m} (1 + \sqrt{\frac{z_{R o}}{f_{c}} - \frac{z_{R o}^{2}}{f_{m}^{2}}}) and L_{c} = \frac{z_{R o}^{2} + L_{f} (L_{f} - f_{m})}{z_{R o}^{2} + {(L_{f} - f_{m})}^{2}} f_{m},

where

z_{R o} = π ω_{o}^{2} / λ

is the Rayleigh range given by the beam waist ω_o of the laser resonator and λ is the lasing wavelength. By setting the mutual distances of the single-lens AMC to Eq. (14) and adjusting the operation angle α to be π/4, the beam transformation from generalized geometric modes into multi-axis LG beams was performed. Figures 6(a)-6(d) show the transformed beams via the single-lens AMC corresponding to the generalized geometric modes in Figs. 5(a)-5(d). Even though the aberrations for high-order transverse modes in paraxial optical systems may lead the experimentally transformed modes to present certain asymmetry, the generated multi-axis LG beams show morphologies in good consistency with the numerical calculations. The experimental multi-axis LG beams can be also seen to remain fairly stable structures under high-power operation except for the cases of P/Q = 2/5 which show some distortion as the pump power is over 8 W. The good stability for maintaining clean spatial structures during power scaling confirms the multi-axis LG modes to be a nice candidate for high-power multi-center vortex beams.

Fig. 6 (a)-(d) The transformed beams corresponding to the generalized geometric modes in Figs. 5(a)-5(d) via astigmatic mode conversion.

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For better understanding the realization of multi-center vortex beams by the proposed systems, the propagation evolution of generalized geometric modes via astigmatic transformation is further studied. The first rows of Figs. 7(a)-7(c) show the propagation evolution of the transverse patterns for generalized geometric modes of P/Q = 2/7, 1/3, and 2/5 via the astigmatic transformation with α = π/4, π/4, and π/7, respectively. Note that to demonstrate that the roles of n_o and m_o can be exchanged with each other to form the generalized geometric modes without losing generality, the central orders for these cases were tuned to be (n_o, m_o) = (36, 35), (36, 35), and (36, 15), respectively. For the cases with α = π/4, it can be seen that the transverse morphologies will gradually transform from the aligned 1D high-order HG modes to patterns with Q bundled ellipses, and finally to multi-axis LG beams with circle-bundle structures in the far-field region. On the other hand, for the case with α = π/7, it can be found that the transverse morphology requires longer propagating distance to transform from linearly symmetric patterns to elliptically symmetric patterns. Moreover, the far-field structure of the transformed mode with α≠π/4 will not become circular symmetric but still present the pattern with Q bundled ellipses. This transformed mode with the pattern of bundled oval rings can be also viewed as a kind of multi-center vortex beams. We further apply the theoretical expression for the transformed mode discussed in Sec. II to reconstruct the experimental results. The second rows of Figs. 7(a)-7(c) show the numerical calculations by substituting Eq. (7) for the transformed HG modes into the relation of degenerate superposition given in Eq. (13) with the experimental parameters. The good agreement between the numerical calculations and the experimental observations again validates the feasibility of using the proposed method to generate various multi-axis vortex beams with high reliability and predictability.

Fig. 7 Experimental results (upper rows) and theoretical reconstructions (lower rows) of the propagation evolution of transverse patterns for generalized geometric modes with operation parameters (n_o, m_o, P/Q, α) to be (a) (36, 35, 2/7, π/4), (b) (36, 35, 1/3, π/4), and (c) (36, 15, 2/5, π/7), respectively.

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4. Conclusion

In conclusion, the generation of high-power vortex beams with multiple vortex centers by using the off-axis pumped degenerate laser resonator with external beam transformation has been explored. The generalized geometric modes obtained in degenerate cavities with two-dimensional off-axis pumping have been theoretically confirmed that can be transformed into the multi-axis LG beams via the astigmatic mode conversion. By utilizing an off-axis pumped Nd:YVO₄ laser in different degenerate conditions combined with an external cylindrical mode converter, various high-power generalized geometric modes have been generated and to be further transformed into the corresponding multi-axis LG beams. It has been verified that both the generalized geometric modes and the transformed multi-axis LG beams can preserve quite stable beam structures even when the output power has reached about 3 W. The features of high stability and easily-controlled beam morphology for the generated multi-axis LG beams confirm the feasibility of using the proposed approach to realize high-power vortex beams with multiple vortex centers for potential applications.

Funding

Ministry of Science and Technology of Taiwan (Contract No. MOST-105-2628-M-009-001).

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Characterization and generation of high-power multi-axis vortex beams by using off-axis pumped degenerate cavities with external astigmatic mode converter

Abstract

1. Introduction

2. Generalized geometric modes V.S. multi-axis vortex beams: theoretical consideration

3. Experimental realization of high-power, multi-axis vortex beams

4. Conclusion

Funding

References and links

Cited By

Figures (7)

Equations (14)

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