Generating high-power asymmetrical Laguerre-Gaussian modes and exploring topological charges distribution

Y. H. Hsieh; Y. H. Lai; M. X. Hsieh; K. F. Huang; Y. F. Chen

doi:10.1364/OE.26.031738

1. Introduction

Generally, the mathematical representation of scalar waves are complex-valued function, which can be in terms of their real and imaginary parts. The intersections of nodal lines of real and imaginary parts are isolated zeros, so-called phase singularities [1]. Singularities in wave field have been attracted wide interests in the fluid dynamics, plasma physics, atmospheric physics, and optics [2]. Singularities can be quantitatively described by the topological charges, namely the net change of phase in a circuit enclosing the zeros. In optics, the momentum flow around singularities can twist to form vortices, which carry the angular momentum [3]. The vortices can be used in the applications such as micro- and nano- particles manipulation [4,5], the fabrication of twisted nanostructure [6–8] as well as the sub-millimeter helical fiber [9], and the quantum communication technologies [10,11]. Moreover, it is intriguing to explore localized and extended singularities in the structured lights [2] since the strong variation of vortex fields can make the lights sensitive to small changes in the medium. This property can be utilized for high-resolution imaging [12], bio-sensing [13], spectroscopy [14], and astronomical observation [15]. One of the most famous optical vortex beams is the Laguerre-Gaussian (LG) beam carrying the orbital angular momentum (OAM). The OAM and topological charge of LG modes are directly determined by its azimuthal quantum number. The LG modes can be generated by using the astigmatic mode converter (AMC) [16–20], the spiral phase plates [21,22], the metasurface with nanoscale inclusions [23], or the spatial light modulator (SLM) [24,25]. In addition to the LG beams with the radial symmetry, the LG modes with asymmetrical intensity distribution are proposed by A. A. Kovalev et al [26]. They generalize a family of asymmetrical LG modes by applying a complex-valued shift to symmetrical LG modes in the Cartesian plane. The asymmetrical LG modes for zero radial index with the crescent-like shape are experimentally generated by using the SLM. The crescent-shaped LG modes can be used in the applications which include the manipulation of biological cells [27] and the generation of the entanglement states for quantum communication [28]. For the practical applications, the scalability of the crescent-shaped LG modes is a crucial issue. However, it is difficult to obtain high-power asymmetrical LG modes by using the SLM due to its low damage threshold for the incident pumping power.

Recently, the high-power LG-based beams [29,30] are realized by exploiting the off-axis pumped solid-state lasers [31] and the AMC. In this work, the off-axis displacement and the output coupler for the Nd:YVO₄ laser are controlled to generate the asymmetrical Hermite-Gaussian (HG) modes with different transverse orders. The asymmetrical HG modes are then transformed to asymmetrical LG modes with the crescent-like shape by using an AMC. As the reflectance of the output coupler decreases, the asymmetry of the LG modes is found to increase. The average output power for all the asymmetrical LG modes can exceed 1W at a pump power of 4W. Furthermore, the theoretical analysis based on the inhomogeneous Helmholtz equation with localized source distribution is exploited to verify the experimental results. The agreement between numerical and experimental wave patterns confirm the importance of the system loss on the formation of asymmetrical LG modes. In addition, the phase structure is explored to reveal that, as the loss increases, the singularities at the origin will be rearranged and additional singularities are formed in the outside region with unchanged average OAM.

2. Experimental results and discussion

The experimental setup for observing the variation of the transverse patterns on the reflectance of output coupler is shown in Fig. 1. The laser was composed of a concave-plano cavity with a 9mm a-cut 0.25-at. % Nd:YVO₄ crystal as gain medium. Both sides of the Nd:YVO₄ crystal were coated for antireflection at 1064 nm (reflection <0.1%). The laser crystal was wrapped with indium foil and mounted in water-cooled copper holders with temperature stabilized at 16 °C. The gain medium was placed near the concave mirror with a separation about 2-3 mm. The input concave mirror had a high-reflectance coating at 1064 nm (reflection >99.8%) with the radius of curvature being R = 30 mm. The output coupler was a flat mirror with a reflectance $R_{o c}$ in the range of 30-90% at 1064 nm. The pump source was an 808-nm fiber-coupled laser diode with a core diameter of 100 μm, a numerical aperture of 0.16, and a maximum output power of 20W. A focus lens with a 38-mm focal length and 94% coupling efficiency was used to reimage the pump beam into the laser crystal. The pump radius was estimated to be 130 μm. The pumping spot was initially aligned at the transverse center of the laser cavity to excite the fundamental Gaussian mode. Subsequently, the pumping beam was displaced in the x- direction by ∆x to generate high-order modes. With an external AMC, HG modes were transformed into LG modes [19,20]. The transverse pattern were observed in the far field region at z = 40 cm by using the CCD camera.

Fig. 1 Configuration of the concave-plano cavity with employing the off-axis pumped Nd:YVO₄ laser to generate HG modes. The matching lens and the a single AMC are used to convert HG modes to LG modes. The CCD camera is used to record the far-field transverse patterns.

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The variation of wave patterns on output reflectance for ∆x = 0.07 mm (n = 2) are displayed in Fig. 2. Note that the off-axis pumping scheme with the off-axis displacement ∆x is also shown in here. As the output reflectance decreases to 50%, it can be seen that both the HG and LG modes display an asymmetrical intensity distribution. More importantly, the asymmetry can be increased by decreasing the output reflectance. In addition to the wave patterns, we explore the influence of the output coupler on the lasing performance. In Fig. 2, the dependence of the average output power on the output reflectance for several incident pump power $P_{i n}$ is demonstrated. Here we set the incident pump power to be smaller or equal to 4W for avoiding the pattern distortion. The results indicate that average output power of asymmetrical LG modes with $R_{o c} = 50 %$ as well as 30%, albeit being lower than the output power of symmetrical LG modes with $R_{o c} \geq 70 %$ , is able to reach as high as 1.55W and 1.45W at $P_{i n} = 4$ W, respectively.

Fig. 2 The dependence of wave patterns and average output power on output reflectance for ∆x = 0.07 mm (n = 2). The scheme of off-axis pumping along x direction is illustrated at the top-left corner.

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Next, we discuss another case of wave patterns versus output reflectance for ∆x = 0.13 mm (n = 6). Note that the HG modes with higher order can be converted to LG modes with OAM given by $n ℏ$ . Figure 3 shows the experimental wave patterns of generated HG modes as well as LG modes with the variation of output reflectance. Like the case in Fig. 2, the wave patterns have asymmetrical intensity distribution when $R_{o c} \leq 50 %$ . The average output power of asymmetrical LG modes with $R_{o c} = 50 %$ and 30% in Fig. 2 can reach up to 1.43W and 1.29W at $P_{i n} = 4 W$ , respectively. Finally, we consider the generation of HG modes with indistinguishable order by setting large off-axis displacement as 0.3 mm. Figure 4 displays the dependence of experimental transverse patterns and the average output power on output reflectance. Once again, the asymmetrical HG modes can be generated when $R_{o c} \leq 50 %$ and be further converted to asymmetrical LG modes. With low output reflectance as 50% and 30%, the average output power of asymmetrical LG modes can be up to 1.35W and 1.02W at $P_{i n} = 4 W$ , respectively. Consequently, the symmetrical and asymmetrical LG modes can be generated by simply using high and low output reflectance. The achievement of watt-level asymmetrical LG modes is believed to be valuable in practical applications. In the following, the theoretical resonant modes derived from the inhomogeneous Helmholtz equation with the localized source distribution are exploited to verify and analyze the experimental transverse wave patterns.

Fig. 3 The dependence of wave patterns and average output power on output reflectance for ∆x = 0.13 mm (n = 6).

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Fig. 4 The dependence of wave patterns with indistinguishable order and average output power on output reflectance for ∆x = 0.3 mm.

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3. Theoretical analysis

Recently, the theoretical resonant modes based on the inhomogeneous Helmholtz equation with localized source distribution are derived as a superposition of degenerate and nearly degenerate eigenmodes to reconstruct experimental lasing transverse modes [32]. The resonant mode can be further utilized to explore the influence of output reflectance on the morphologies of structured laser beam [29]. Here we brief review the derivation of resonant modes, and use it to verify the experimental transverse patterns. Since the lasing resonant modes are subject to the same boundary condition as the homogeneous Helmholtz equation for the concave (at $z = - L$ )-plano (at $z = 0$ ) cavity, the eigenmodes which involve in the superposition can be written as the HG modes with quantum number n and m representing the transverse orders in x- and y- directions. Considering m = 0 for off-axis pumping along x- direction, the eigenmodes are given by

ψ_{n, s}^{(H G)} (x, y, z) = \sqrt{\frac{2}{L}} Φ_{n} (x, y, z) \exp [- i k_{n, s} \tilde{z} + i (n + 1) \tan^{- 1} (z / z_{R})],

where

Φ_{n} (x, y, z) = \frac{1}{\sqrt{2^{n + m - 1} π n!}} \frac{1}{w (z)} H_{n} [\frac{\sqrt{2} x}{w (z)}] \exp [- \frac{x^{2} + y^{2}}{w {(z)}^{2}}]

Here

H (\cdot)

is the Hermite polynomials of order n,

k_{n, s} = 2 π f_{n, s} / c

,

f_{n, s}

is the eigenmode frequency,

z_{R} = \sqrt{L (R - L)}

is the Rayleigh range,

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

,

w (z) = w_{o} \sqrt{1 + {(z / z_{R})}^{2}}

, and

w_{o} = \sqrt{λ z_{R} / π}

is the beam radius at the waist. The eigenmode frequency of the spherical cavity is given by

f_{n, s} = [s Δ f_{L} + (n + 1) Δ f_{T}]

f_{n, s} = [s Δ f_{L} + (n + 1) Δ f_{T}]

, where s is the longitudinal mode number,

Δ f_{L}

is the longitudinal mode spacing, and

Δ f_{T}

is transverse mode spacing. For a locally pumped laser system, the resonant modes satisfy the inhomogeneous Helmholtz equation:

(\nabla^{2} + {\tilde{k}}^{2}) Ψ (x, y, z) = η_{c} F (x, y, z),

where

\tilde{k} = k + i α

,

k = 2 π / λ

is the wave number of the emission light, and

η_{c}

represents the conversion efficiency for the pump source. α is a small loss parameter including loss from the scattering, the absorption, and the output coupling. The value of α can be further estimated by the generalized formula:

[λ / w_{o}] [\ln (1 / R_{o c}) + Γ]

, where Г related to nonsaturable intracavity round-trip dissipative optical loss is given by 10⁻². The lasing modes and the source distribution can be written as the expansion of HG modes:

Ψ (x, y, z) = \sum_{n, s} a_{n, s} ψ_{n, s}^{(H G)} (x, y, z),

and

η_{c} F (x, y, z) = \sum_{n, s} b_{n, s} ψ_{n, s}^{(H G)} (x, y, z) .

Substituting Eqs. (4) and (5) into Eq. (3), the coefficient

a_{n, s}

can be deduced to be:

a_{n, s} = b_{n, s} / ({\tilde{k}}^{2} - k_{n, s}^{2})

. With the orthonormal property of eigenmodes, the coefficient

b_{n, s}

is given by

b_{n, s} = η_{c} ∭ ψ_{n, s}^{(H G)} (x, y, z) F (x, y, z) d x d y d z .

The pump source distribution F(x,y,z) is considered to be approximately uniform in the longitudinal z axis. Thus, the coefficient

b_{n, s}

related to the source distribution can be regarded as independent of index s in the neighborhood of the central index s_o and the radius of the pump beam can be viewed as a constant

w (z_{c})

in the range

| z - z_{c} | \leq L_{c} / 2

, where z_c is the location of the gain medium and L_c is the length of the gain medium. On the other hand, the transverse distribution of the source is assumed to be the Gaussian distribution. As a result, the F(x,y,z) for the off-axis pumping of the transverse displacement ∆x can be expressed as

F (x, y, z) = \frac{1}{L_{c}} \frac{2}{π w^{2} (z_{c})} \exp [- \frac{{(x - Δ x)}^{2} + y^{2}}{w^{2} (z_{c})}] .

Substituting the Eqs. (1), (2), and (7) into the Eq. (6) and using the generating function of the Hermite polynomials, the coefficient

b_{n, s}

can be derived to be

b_{n, s} = (η / L_{c})

[\sqrt{2 / π w^{2} (z_{c})}] [{(n_{o})}^{n / 2} e^{- n_{o} / 2} / \sqrt{n!}]

, where

η

is a constant that includes the effective conversion

η_{c}

and the overlap integral in the longitudinal direction, and

n_{o} = {[Δ x / w (z_{c})]}^{2}

. The expression

{(n_{o})}^{n / 2} e^{- n_{o} / 2} / \sqrt{n!}

is the form of the square root of the Poisson distribution which indicates that the maximum contribution in the resonant states is the eigenmode with order n to be the most immediate integer to the value n_o. Note that it also has been found that the high-order HG mode can be generated in the off-axis pumped solid-state lasers with

n_{o} = {[Δ x / w (z_{c})]}^{2}

. The Poisson distribution can be approximately expressed as the Gaussian distribution for large value of n_o from the central limit theorem. Thus, the

b_{n, s}

can be derived as

b_{n, s} = \frac{η}{L_{c}} \sqrt{\frac{2}{π w^{2} (z_{c})}} \frac{1}{\sqrt{\sqrt{2 π n_{o}}}} \exp [- \frac{{(n - n_{o})}^{2}}{4 n_{o}}] .

With the property of Gaussian distribution, the effective range of the mode order n is limited to

| n - n_{o} | \leq N

with

N = 2 \sqrt{n_{o}}

. Substituting the expression of

b_{n, s}

in Eq. (8) into

a_{n, s} = b_{n, s} / ({\tilde{k}}^{2} - k_{n, s}^{2})

with

α < < k

and

k = π [s_{o} + (n_{o} + 1)] / L

, the resonant lasing modes in the Eq. (4) can be further deduced to be

\begin{array}{l} Ψ (x, y, z; γ) = \\ \frac{η λ L}{4 π^{2} L_{c}} \sqrt{\frac{2}{π w^{2} (z_{c})}} \frac{1}{\sqrt{\sqrt{2 π n_{o}}}} {\sum_{s = s_{o} - J}^{s_{o} + J} \sum_{n = n_{o} - N}^{n_{o} + N} \frac{\exp [- {(n - n_{o})}^{2} / 4 n_{o}]}{[(s_{o} - s) + (n_{o} - n) Ω] + i γ} ψ_{n, s}^{(H G)} (x, y, z)}, \end{array}

where J is associated with the effective range of the longitudinal order s and

γ = α L / π

is the inverse quality factor. Ω is ratio between transverse mode spacing to longitudinal mode spacing. For Ω being a rational number, there are many combination of (n,s) can satisfy

(s_{o} - s) + (n_{o} - n) Ω = 0

and contribute to the superposition. In this paper, Ω is considered being an irrational number as

\sqrt{13} / 17

to lead to the result that only one state of

(s, n) = (s_{o}, n_{o})

can satisfy the above condition and dominates the superposition. Consequently, the resonant modes can display the feature of an eigenmode. Moreover, the nearly degenerate eigenstates around central order are also involved in the superposition due to the inverse quality factor

γ

. The larger

γ

is, the more effective modes take part in the superposition. For concave-plano cavity, Ω can be expressed as

Ω = [(1 / π) \sin^{- 1} (\sqrt{L / R})]

[32]. The cavity length L for

Ω = \sqrt{13} / 17

can be estimated to be 11.46 mm accordingly. The first and second rows in Fig. 5 demonstrate the dependence of calculated transverse patterns of

{| Ψ (x, y, z; γ) |}^{2}

on the output reflectance at a constant z = 40 cm for n_o = 2 and 6, respectively. By considering the effective modes involved in the superposition, the index J is determined by

J = ⌊ N Ω ⌋

, where

⌊ n ⌋

denotes an integer smaller or equal to n. Here

γ

is

0.7 \times 10^{- 2}

(R_oc = 90%),

2.1 \times 10^{- 2}

(R_oc = 70%),

3.9 \times 10^{- 2}

(R_oc = 50%), and

6.8 \times 10^{- 2}

(R_oc = 30%). It can be seen that the numerical wave patterns agree very well with the experimental transverse patterns. The third row in Fig. 4 displays the case of the dependence of indistinguishable HG modes on the output reflectance. n_o is estimated to be 16 by using

n_{0} = {[Δ x / w (z_{c})]}^{2}

. Again, the simulation results make a good agreement with the experimental lasing modes. It is worthy to mention that the resonant modes display the feature of asymmetrical HG modes when

γ

is large enough, which can be attributed to the sufficient number of nearly degenerate modes involved in the superposition. Since

γ

is directly determined by R_oc_, the agreement confirms that the cavity loss is the key to cause the asymmetrical distribution for HG modes.

Fig. 5 The calculated transverse patterns of ${| Ψ (x, y, z = 40 cm; γ) |}^{2}$ and phase structure with small $γ$ for $n_{0} = 2$ , 6, and 16.

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By introducing the astigmatism to the orthogonal axis in Cartesian coordinates (x,y), i.e., using an AMC, the eigenmode of HG beams can be transformed to LG beams with the radial index p and the azimuthal index l. Note that p is given by the smaller one between n and m, and l is given by n-m. Thus, p and l are regarded as 0 and n, respectively, for the LG modes described in this paper. The LG modes are then written as

ψ_{n, s}^{(L G)} (r, ϕ, z) = \sqrt{\frac{2}{L}} {\tilde{Φ}}_{n} (x, y, z) \exp [i k_{n, s} \tilde{z} - i (n + 1) \tan^{- 1} (z / z_{R})]

where

{\tilde{Φ}}_{n} (r, ϕ, z) = \sqrt{\frac{2}{π n!}} \frac{1}{w (z)} {(\frac{\sqrt{2} r}{w (z)})}^{n} \exp [- \frac{r^{2}}{w {(z)}^{2}}] \exp (i n ϕ),

and

(r, ϕ)

are the polar coordinates for the Cartesian coordinates (x,y). Hence, the transformed resonant modes can be obtained by replacing

ψ_{n, s}^{(H G)} (x, y, z)

in Eq. (9) with

ψ_{n, s}^{(L G)} (r, ϕ, z)

:

\begin{array}{l} Ψ (r, ϕ, z; γ) = \\ \frac{η λ L}{4 π^{2} L_{c}} \sqrt{\frac{2}{π w^{2} (z_{c})}} \frac{1}{\sqrt{\sqrt{2 π n_{o}}}} {\sum_{s = s_{o} - J}^{s_{o} + J} \sum_{n = n_{o} - N}^{n_{o} + N} \frac{\exp [- {(n - n_{o})}^{2} / 4 n_{o}]}{[(s_{o} - s) + (n_{o} - n) Ω] + i γ} ψ_{n, s}^{(L G)} (r, ϕ, z)} . \end{array}

The transformed wave patterns of

{| Ψ (r, ϕ, z = 40 c m; γ) |}^{2}

with small

γ

are shown in Fig. 6. The corresponding phase structures are calculated by utilizing

Θ (r, ϕ; γ) =

\tan^{- 1} [Im (Ψ (r, ϕ; γ)) / Re (Ψ (r, ϕ; γ))]

. It is seen that wave patterns display the same behaviors as a pure LG modes. In the phase structures, there is a single n_o-times degenerate on-axis singularity.

Fig. 6 The calculated transformed wave patterns of ${| Ψ (x, y, z = 40 cm; γ) |}^{2}$ and phase structure with small $γ$ for $n_{0} = 2$ , 6, and 16.

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Next, we demonstrate the wave patterns and the phase structure with large $γ$ . In Fig. 7, for $n_{0} = 2$ , it is seen that not only the patterns display an asymmetric distribution but newly formed unit charges circled by white dashed lines can be found in the outside region. It is also worthy to note that the vortex around the additional singularity in the dark side of crescent-shaped LG modes can be used to trap the micro-particles into the optical field. The phase structures can be further analyzed by calculating the topological charges from the net change of phase in a closed nonselfintersecting loop C enclosing zero point: $T_{c} = {(2 π)}^{- 1} \oint_{C} \nabla Θ \cdot d \overset{⇀}{l}$ . For the loop chosen to be a circle centered at the origin with radius r, the charges can be as a function of r and $T_{c} (r, γ) = {(2 π)}^{- 1} \int_{0}^{2 π} \nabla Θ (x, y; γ) |_{_{x = r \cos θ, y = r \sin θ}} \cdot (- \sin θ {\hat{a}}_{x} + \cos θ {\hat{a}}_{y}) d θ$ . The calculated results of the topological charges versus $r / r_{0}$ are shown in the right hand side of Fig. 7, where $r_{0}$ is the radius of symmetrical LG modes with $γ \to 0$ . The results indicate that the charges of central singularity is split into an n’-times degenerate intensity null on-axis and isolated nulls with unit topological charge nearby. For $r / r_{0} > 1$ , newly isolated singularities can be found. Larger the $γ$ is, more additional charges can be generated. With a large $γ$ , it is inferred that nearly degenerate states which contribute to the resonant modes can cause not only the asymmetrical intensity distribution of LG modes but the change of intensity nulls in phase structures. Figures 8 and 9 depict other cases of wave patterns, phase structures, and calculated topological charges for $n_{0} = 6$ and 16. The charges distribution displays similar behavior as the case of $n_{0} = 2$ . To confirm the vortex structures of the asymmetrical LG modes, we conduct the interference experiment described in the ref [35]. Figure 10 shows the interference patterns for the asymmetrical LG modes ( $γ = 6.8 \times 10^{- 2}$ ) with $n_{0} = 2$ and 6. The results manifest the numerical vortex structures where the topological charges near the origin are split and rearranged. Although displaying the newly isolated charges outside the LG modes by using the interference experiment is still a challenge work, the present results are believed to confirm the simulated phase structures. Typically, the OAM of LG modes are equal to its topological charges. In the present work, since asymmetrical LG modes can be regarded as a superposition of nearly degenerate LG modes, the OAM of asymmetrical LG modes is not directly determined by its charges but obtained by integrating the poynting vector of the electric fields over all space. We calculate the average OAM for optical vortex beam by using $ℏ \int_{V} \overset{⇀}{r} \times Im (Ψ^{*} (r, ϕ) \nabla Ψ (r, ϕ)) d V$ . It is found that, despite being different in shape, since the distribution of the coefficient $\exp [- {(n - n_{o})}^{2} / 4 n_{o}] / {[(s_{o} - s) + (n_{o} - n) Ω] + i γ}$ in Eq. (12) is symmetric with respect to $(s_{o}, n_{o})$ , all the LG modes have the same OAM given by n_{o $ℏ$}. Consequently, the generation of asymmetrical LG modes will result in the rearrangement and newly formation of phase singularities with unchanged average OAM. The results are useful for the modulation of optical vortices [33] and the formation of multidimensional entangled states [34].

Fig. 7 The calculated transformed wave patterns of ${| Ψ (x, y, z = 40 cm; γ) |}^{2}$ , phase structure, and topological charges $T_{c} (r, γ)$ versus r with large $γ$ for $n_{0} = 2$ . Here $r_{0}$ is the radius of symmetrical LG modes with $γ \to 0$ .

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Fig. 8 The calculated transformed wave patterns of ${| Ψ (x, y, z = 40 cm; γ) |}^{2}$ , phase structure, and topological charges $T_{c} (r, γ)$ versus r with large $γ$ for $n_{0} = 6$ .

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Fig. 9 The calculated transformed wave patterns of ${| Ψ (x, y, z = 40 cm; γ) |}^{2}$ , phase structure, and topological charges $T_{c} (r, γ)$ versus r with large $γ$ for $n_{0} = 16$ .

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Fig. 10 Experimental interference patterns for the asymmetrical LG modes $γ = 6.8 \times 10^{- 2}$ for $n_{0} = 2$ and 6.

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4. Conclusions

In conclusion, we have controlled the off-axis displacement and the output coupler of the Nd:YVO₄ laser to generate the asymmetrical HG modes with various transverse order. Then, we have converted the asymmetrical HG modes to asymmetrical LG modes with the crescent-like shape by using an AMC. The average output power for all the asymmetrical LG modes can exceed 1W at the pump power of 4W. Moreover, the theoretical resonant modes based on the inhomogeneous Helmholtz equation with localized source distribution are exploited to reconstruct the experimental transverse patterns, which confirms the importance of the cavity loss on the emergence of asymmetrical LG modes. The resonant modes also have been used to explore the dependence of phase structures on the system loss. As the loss increases, the singularities at the origin are found to be rearranged, and newly formed singularities can be observed in the outside region with unchanged average OAM. The present results of the watt-level average output power for crescent-shaped LG modes are believed to be useful in further applications.

Funding

Ministry of Science and Technology of Taiwan (Contract No. MOST 107-2119-M-009 −015).

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Generating high-power asymmetrical Laguerre-Gaussian modes and exploring topological charges distribution

Abstract

1. Introduction

2. Experimental results and discussion

3. Theoretical analysis

4. Conclusions

Funding

References

Cited By

Figures (10)

Equations (12)

Optics Express