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Beam quality factor of aberrated Laguerre-Gaussian optical beams

Chemist M. Mabena, Teboho Bell, Nokwazi Mphuthi, Justin Harrison, and Darryl Naidoo

Open Access Open Access

Abstract

The influence of aberrations on the beam quality factor of Laguerre-Gaussian beams is investigated. We derive analytical expressions for the beam quality factor due to astigmatism and spherical aberration. We show that the width of a Laguerre-Gaussian beam is a significant parameter that determines the aberration effects on the beam quality factor. For each aberration, we derive an expression for the width that separates the region where the beam quality factor changes infinitesimally and where it changes drastically. The validity of the analytical expressions is assessed by performing numerical simulations. There is excellent agreement between the analytical and numerical results.

© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement

1. Introduction

The influence of phase aberrations on light fields is a critical area of interest resulting in the development of a large body of knowledge [1–8]. A phase aberration may be mathematically represented as a complex phase function that will modify the wavefront of a light field. The most fundamental aberrations include tilt, defocus, astigmatism, coma, and spherical aberration. For light fields that are described as laser beams, such aberrations can be present in the atmosphere, optical elements, and optical crystals, and may influence the intensity distribution and propagation characteristics of a laser beam in free-space. While the effect of defocus is present in optical lenses and often employed to focus laser beams, the rest of the aberrations typically represent unwanted perturbation effects and can lead to degradation of the intensity profile and the laser beam quality. Knowledge of phase aberrations in optical systems toward their mitigation has found use in various fields such as imaging [7], ophthalmology [9], atmospheric turbulence [10], and high-power laser systems [11].

The beam quality factor ($M^2$) is a useful parameter that can be used to measure the quality of a laser beam. It gives information on how tightly a laser beam can be focused and information about the propagation dynamics of laser beams [12–14]. The beam quality factor of a laser beam is invariant when passing through an unaberrated optical system and when it passes through a medium with a linear or quadratic index of refraction such as a lens. However, physical optical elements or systems are not completely free from aberrations which can be induced due to thermal blooming, misalignment, refractive index changes, and manufacturing imperfections. For a laser beam propagating through a physical optical system with the presence of aberrations, the beam quality factor may be influenced according to the nature of the aberrations which will result in a modification of the laser beam intensity distribution.

To this extent, much work has been done to characterize and understand the effect of phase aberrations on the quality of a laser beam [13,15–19]. Siegman presented a theoretical analysis of the effect of aberration of the quartic phase caused by spherically aberrated optical components on the quality factor of the laser beam [13]. Subsequently, this analysis was experimentally demonstrated to show that the beam quality of a laser beam should remain unaffected until the incident beam exceeds a certain critical width relative to the physical width of the lens [15]. As an expansion to the work by Siegman, George et al. performed a theoretical analysis of the effect of the quartic phase aberration due to spherical aberration of a lens on the beam quality of a higher-order transverse-mode [18]. Alda et al. presented a formalism for the analysis of phase aberrations on the characteristic parameters of a laser beam [6]. More work has since been done to relate the effect of general phase aberrations on the laser beam quality factor. Jeong et al. described a simpler method to accurately determine the quality factor of the laser beam, which uses a Fourier Transform [16]. Using the method of moments, Mafusire and Forbes proposed a model to quantify the effect of phase aberrations on the quality factor of a Gaussian laser beam [17].

Recent years have seen an outburst in the generation of light tailored in its spatial degrees of freedom, viz. amplitude, phase, and polarisation. This type of light can be generated both intra- and extra-cavity and is commonly referred to as structured light. The extra-cavity generation of structured light has been driven by significant advances in beam shaping tools such as spatial light modulators [20,21], geometric phase liquid crystals [22], metasurface elements [23], and diffractive optical elements [24]. In parallel with the extra-cavity milestones in structured light generation, a variety of techniques have been developed to generate structured light lasers [25–29]. The breakthroughs which have been made in the generation of structured light have made possible a myriad of applications [30–34]. Consequently, with the fast-growing ubiquity of structured light, it has become apparent the need to address the influence of phase aberrations on its propagation characteristics. In particular, the influence of phase aberrations on the beam quality factor.

In this study, we utilized the generating function of Laguerre-Gaussian (LG) optical beams to model the beam quality factor due to different aberrations: $0^{\circ }$ astigmatism, $45^{\circ }$ astigmatism, $x-$triangular astigmatism, $y-$triangular astigmatism, and spherical aberration. The use of the generating function alleviates the calculations and makes it trivial to obtain a general expression for the beam quality factor of the entire space of LG optical beams. While experimental methods for measuring the beam quality factor are well-established, analytical expressions are essential for understanding qualitative and quantitative features, particularly for parametric studies. Therefore, we present analytical expressions for the beam quality factor of LG optical beams under different aberrations. Our calculations are performed individually for each aberration to establish a toolbox for the convenient characterization of optical systems that are dominated by a specific aberration. This work contributes significantly to the understanding and design of high-quality optical systems that use structured light and has practical applications in areas such as laser technology and imaging.

2. Beam quality factor

The complex amplitude of an optical beam can be expressed as follows,

$$E(x, y) = u(x, y)e^{{-}i \frac{2\pi}{\lambda} \phi(x, y)},$$
where $u(x, y)$ is the amplitude distribution of the optical beam, $\phi (x, y)$ represents the phase of the optical beam. In this work, we will consider $\phi (x,y)$ solely to represent the phase accrued as a result of aberrations in an optical system. Furthermore, we assume that the propagation of the optical beam along the two principal axes, $x$ and $y$, can be treated as separate problems. With this assumption, we can calculate the beam quality factor [12,17,35] in the $x$ and $y$ directions separately as follows,
$$M^2_{x} = 4 \pi \sqrt{ \langle x^2 \rangle \langle \theta^2_x \rangle - \langle x \theta_x \rangle^2 },$$
$$M^2_{y} = 4 \pi \sqrt{ \langle y^2 \rangle \langle \theta^2_y \rangle - \langle y \theta_y \rangle^2 },$$
where $\langle x^2 \rangle$ and $\langle y^2 \rangle$ are the second-order spatial moments along the $x$ and $y$ directions, respectively; $\langle \theta ^2_x \rangle$ and $\langle \theta ^2_y \rangle$ are the second-order angular moments in each direction; Finally, $\langle x \theta _x \rangle$ and $\langle y \theta _y \rangle$ are the first-order spatial-angular moments in the $x$ and $y$ directions, respectively. The calculation of the terms represented in Eq. (2) and Eq. (3) are given below as follows,
$$\langle x^2 \rangle = \int^{\infty}_{-\infty} \int^{\infty}_{-\infty} x^2 u^2\left(x, y\right)dxdy,$$
$$\langle y^2 \rangle = \int^{\infty}_{-\infty} \int^{\infty}_{-\infty} y^2 u^2\left(x, y\right)dx dy,$$
$$\begin{aligned} \langle \theta^2_x \rangle &= \frac{1}{4 \pi^2} \int^{\infty}_{-\infty} \int^{\infty}_{-\infty}\left[ \left( \frac{\partial u}{\partial x} \right)^2 + \left( u \frac{\partial \phi}{\partial x} \right)^2 \right]dx dy\\ & \quad - \frac{1}{4 \pi^2}\left( \int^{\infty}_{-\infty} \int^{\infty}_{-\infty} u^2 \frac{\partial \phi}{\partial x} dx dy\right)^2, \end{aligned}$$
$$\begin{aligned} \langle \theta^2_y \rangle &= \frac{1}{4 \pi^2} \int^{\infty}_{-\infty} \int^{\infty}_{-\infty}\left[ \left( \frac{\partial u}{\partial y} \right)^2 + \left( u \frac{\partial \phi}{\partial y} \right)^2 \right]dx dy\\ & \quad - \frac{1}{4 \pi^2}\left( \int^{\infty}_{-\infty} \int^{\infty}_{-\infty} u^2 \frac{\partial \phi}{\partial y} dx dy\right)^2, \end{aligned}$$
and finally,
$$\langle x \theta_x \rangle = \frac{1}{2 \pi } \int^{\infty}_{-\infty} \int^{\infty}_{-\infty} u^2 x \frac{\partial \phi}{\partial x} dxdy,$$
$$\langle y \theta_y \rangle = \frac{1}{2 \pi } \int^{\infty}_{-\infty} \int^{\infty}_{-\infty} u^2 y \frac{\partial \phi}{\partial y} dx dy.$$

In the calculations below, we use the square of Eq. (2) and Eq. (3) since it removes the square roots and makes the presentation more convenient.

3. Aberrated Laguerre-Gaussian optical beams

Instead of working directly with the expression of Laguerre-Gaussian (LG) optical beams, we rather use the generating function [36] which is given by

$$\mathcal{G} = \frac{1}{1 - \eta} \exp\left[ \frac{\left(x + i\text{T}y\right)\mu}{\omega_0\left( 1 - \eta \right)} - \frac{\left(x^2 + y^2\right)(1 + \eta)}{\omega^2_0\left( 1 - \eta \right)} \right],$$
where $\mu$ and $\eta$ are the generating parameters for the azimuthal ($\ell$) and radial ($p$) indices, respectively, $\text {T}$ gives the sign of the azimuthal index $\ell$, and $\omega _0$ is the radius of the beam waist. A specific LG optical beam is obtained by the following operation:
$${LG}_{\ell, p} = \frac{\mathcal{N}_\text{LG}}{p!}\left[ \frac{\partial^p}{\partial \eta^p}\frac{\partial^{|\ell|}}{\partial \mu^{|\ell|}} \mathcal{G}\right]_{\eta, \mu = 0},$$
where,
$$\mathcal{N}_\text{LG} = \left[ \frac{2^{|\ell| + 1}p!}{\pi \omega^2_0 (p + |\ell|)!} \right]^{1/2}.$$

Examples of the intensity profiles of selected LG optical beams with their respective phases are shown in Fig. 1. The beam quality factor of aberration-free LG optical beams is given in terms of the azimuthal and radial indices as follows,

$$M^2 = 2p + |\ell| + 1.$$

 figure: Fig. 1.

Fig. 1. Intensity cross section for selected LG modes with indices $p=0,1,2,3$ and $l=0, 1, 2$. The insets illustrate the corresponding phase of the respective mode.

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Upon using the generating function for the LG optical beams, the second-order spatial moment along the $x$ direction is calculated to give the following,

$$\begin{aligned}\langle x^2 \rangle_\text{LG} &= \frac{\mathcal{N}_{LG}^2}{p!^2}\frac{\pi \omega_0^4 e^{-\frac{\mu_1\mu_2}{2\left( \eta_1\eta_2 - 1 \right)}}}{32 \left( \eta_1\eta_2 - 1 \right)^3} \left[ 2\mu_1\mu_2 + \mu_2^2 \right.\\ & \quad \left. + (2\eta_2\mu_1\mu_2 + 4\eta_2^2 - 2\mu_1\mu_2 - 2\mu_2^2 - 4)\eta_1 \right.\\ & \quad\left. + \eta_2^2\mu_1^2 + ({-}2\mu_1^2 - 2\mu_1\mu_2 - 4)\eta_2 + \mu_1^2 \right.\\ & \quad\left. + ( \mu_2^2 + 4\eta_2 - 4\eta_2^2 )\eta_1^2 + 4 \right] . \end{aligned}$$

For convenience, we break up the expression of the second-order angular moment [Eq. (6)] expression into three parts as follows,

$$\theta_1 = \frac{1}{4 \pi^2} \int^{\infty}_{-\infty} \int^{\infty}_{-\infty} \left( \frac{\partial u}{\partial x} \right)^2 dxdy,$$
$$\theta_2 = \frac{1}{4 \pi^2} \int^{\infty}_{-\infty} \int^{\infty}_{-\infty} \left( u \frac{\partial \phi}{\partial x} \right)^2dx dy,$$
$$\theta_3 = \frac{1}{2 \pi} \int^{\infty}_{-\infty} \int^{\infty}_{-\infty} u^2 \frac{\partial \phi}{\partial x} dx dy ,$$
where the last term is the square root of the third term in Eq. (6). That is, the second-order angular moment expression can be rewritten as follows,
$$\langle \theta^2_x \rangle = \theta_1 + \theta_2 - \theta_3^2.$$

Equation (15) does not depend on the phase aberration, therefore, we can give the general expression, in terms of the generating parameters, as follows,

$$\begin{aligned}\theta_1 &=\frac{ \mathcal{N}_{LG}^2 \exp\left(\frac{\mu_1\mu_2}{2\eta_1\eta_2 - 2}\right) }{32\pi(\eta_1\eta_2 - 1)^3} \times \left[ (4\eta_2^2 + \mu_2^2 + 4\eta_2 )\eta_1^2 \right.\\ & \quad \left. + ({-}2\eta_2\mu_1\mu_2 + 4\eta_2^2 - 2\mu_1\mu_2 + 2\mu_2^2 - 4)\eta_1 \right.\\ & \quad \left. + \eta_2^2\mu_1^2 + (2\mu_1^2 - 2\mu_1\mu_2 - 4)\eta_2 \right.\\ & \quad \left. + (\mu_1 - \mu_2 + 2)(\mu_1 - \mu_2 - 2)\right]. \end{aligned}$$

The other two equations [Eq. (16) and Eq. (17)] depend on the phase aberration. Since we are treating each aberration individually, we do not give the expressions here. Table 1 shows algebraic expressions for spherical aberration and the different types of astigmatism. The phase profiles of the aberrations are shown in Fig. 2. The aberration coefficient can be given as a number with units determined from dimensional analysis [37] or it can be related to a physical system [13]. In this work, we use the former approach. The coefficients shown in Tab 1 have different units according to the dimensionality requirement in Eq. (1).

 figure: Fig. 2.

Fig. 2. Phase profiles of the aberrations in Tab. 1. a) $0^{\circ }$ astigmatism, b) $45^{\circ }$ astigmatism, c) $x-$triangular astigmatism, and d) Spherical aberration.

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Tables Icon

Table 1. Algebraic expressions for some primary aberrations.

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3.1 Beam quality factor of aberrated Laguerre-Gaussian optical beams

3.1.1 $0^{\circ }$ Astigmatism

The beam quality factor of LG optical beams is not affected by the $0^{\circ }$ astigmatism. The $0^{\circ }$ astigmatism has, to some degree, the behavior of the defocus aberration, although without its rotational symmetry. It can be easily shown that upon substitution of the expression for $0^{\circ }$ astigmatism(Tab. 1) into Eq. (16), it becomes the second-order spatial moment, in Eq. (17) the result becomes the first-order spatial moment, which is zero. Substitution of the expression for $0^o$ astigmatism(Tab. 1) into Eq. (8) also results in the second-order spatial moment. Finally, substituting all of the above into Eq. (2) gives,

$$M^2_x = 4 \pi \sqrt{\langle x^2 \rangle \theta_1},$$
which shows that when the beam quality factor is calculated along the principal axes, $0^{\circ }$ astigmatism has no effect on the beam quality factor of arbitrary optical beams. This can be seen from Eq. (20) above since there are no aberration-dependent terms.

3.1.2 $45^{\circ }$ astigmatism

The $45^{\circ }$ astigmatism, on the other hand, has an effect on the beam quality factor of LG optical beams. The aberration dependent terms in Eq. (18) for $45^{\circ }$ astigmatism are given below as,

$$\begin{aligned} \theta^{\text{ast45}}_2 &= \frac{D_{\text{ast45}}^2 \mathcal{N}_{LG}^2 \omega_0^4 \pi \exp\left(-\frac{\mu_1 \mu_2}{2\eta_1 \eta_2 - 2}\right)}{32 \left( \eta_1 \eta_2 - 1 \right)^3 \lambda^2}\\ & \quad \times \left[\left( \eta_2 - 1 \right)^2\mu_1^2 + \left(\eta_1 - 1\right)^2 \mu_2^2 \right.\\ & \quad \left. + 4 \left( \eta_2 - 1 \right) \left(\eta_1 - 1\right) \left( \eta_1\eta_2 - \frac{\mu_1 \mu_2}{4} - 1 \right)\right] , \end{aligned}$$
$$\begin{aligned} \theta^{\text{ast45}}_3 &= \frac{D_{\text{ast45}} \mathcal{N}^2_{LG}\omega_0^3 \pi \exp\left(-\frac{\mu_1 \mu_2}{2 \eta_2 \eta_1 - 2}\right)}{i 8 \left( \eta_1 \eta_2 - 1 \right)^2 \lambda}\\ & \quad \times \left[\left( \eta_1 - 1 \right)\mu_2 + \left( \eta_2 - 1 \right)\mu_1 \right]. \end{aligned}$$

The first-order spatial-angular moment for $45^{\circ }$ astigmatism in terms of generating parameters is given as,

$$\begin{aligned} \langle x \theta_x \rangle^{\text{ast45}} &= \frac{ D_{\text{ast45}} \mathcal{N}^2_{LG} \omega^4_0 \pi \exp\left(-\frac{\mu_1 \mu_2}{2 \eta_1 \eta_2 - 2}\right)}{i 32 \left( \eta_1 \eta_2 - 1 \right)^3 \lambda}\\ & \quad \times \left[ \left( \eta_2 - 1 \right)^2\mu_1^2 - \left( \eta_1 - 1 \right)^2\mu_2^2\right]. \end{aligned}$$

Substitution of Eq. (21) and Eq. (22) into Eq. (18), and subsequent substitution into Eq. (2) together with Eq. (23) yields the analytical expression of the beam quality factor due to $45^{\circ }$ astigmatism. The final result is given as follows,

$$M^4_x = \left(2p + \ell + 1 \right)^2\left[\frac{ \pi^2 D^2_{\text{ast45}}\omega^4_0}{\lambda^2 } + 1 \right],$$
where $D_\text {ast45}$ is the coefficient of the $45^o$ astigmatism and is related to the strength of the aberration.

3.1.3 Triangular astigmatism

In addition to $0^{\circ }$ and $45^{\circ }$ astigmatism, optical beams can also be affected by triangular astigmatism. The behavior of $x-$ triangular and $y-$ triangular astigmatism is similar in their effect on the beam quality factor of the LG optical beams. Thus, we only give the general expression for $x-$triangular astigmatism below. The aberration-dependent second-order angular moment terms in Eq. (18) for $x-$triangular astigmatism are given below as

$$\begin{aligned} \theta^{\text{xtri}}_2 &={-}\frac{9 D_{\text{xtri}}^2 \mathcal{N}^2_{LG} \omega_0^6 \pi \exp\left(-\frac{\mu_1\mu_2}{2\eta_1 \eta_2 - 2} \right)}{ 128 \left( \eta_1 \eta_2 - 1 \right)^5 \lambda^2 }\\ & \quad \times \left\{ 16 \left( \eta_2 - 1 \right)^2 \left[ \frac{\mu_1^2\mu_2^2}{8} - \left( \eta_1 \eta_2 - 1 \right)\mu_1 \mu_2 \right. \right.\\ & \quad \left. \left. + \left( \eta_1 \eta_2 - 1 \right)^2\right] \left( \eta_2 - 1 \right)^2 + \left( \eta_2 - 1 \right)^4\mu_1^4 \right.\\ & \quad \left. + \left( \eta_1 - 1 \right)^4 \mu_2^4\right\}, \end{aligned}$$
$$\begin{aligned} \theta^{\text{xtri}}_3 &= \frac{3 D_{\text{xtri}} \mathcal{N}^2_{LG} \omega_0^4 \pi \exp\left(-\frac{\mu_1 \mu_2}{2 \eta_1 \eta_2 - 2} \right) }{16 \left( \eta_1 \eta_2 - 1 \right)^3 \lambda}\\ & \quad \times \left[\left( \eta_2 - 1 \right)^2\mu_1^2 + \left( \eta_1 - 1 \right)^2\mu_2^2\right]. \end{aligned}$$

The first-order spatial-angular moment for $x-$triangular astigmatism in terms of the generating parameters is given as

$$\begin{aligned} \langle x \theta_x \rangle^{\text{trix}} &= \frac{3 D_{\text{xtri}}\mathcal{N}^2_{LG} \omega_0^5 \pi \exp\left(-\frac{\mu_1\mu_2}{2\eta_1 \eta_2 - 2}\right)} {64 \left( \eta_1 \eta_2 - 1 \right)^4 \lambda}\\ & \quad \times \left[\left( \eta_2 - 1 \right) \mu_1 + \left( \eta_1 - 1 \right) \mu_2\right]\\ & \quad \times\left[\left( \eta_2 - 1 \right)^2\mu_1^2 + \left( \eta_1 - 1 \right)^2\mu_2^2 \right.\\ & \quad \left. - 4 \left( \eta_2 - 1 \right) \left( \eta_1 - 1 \right) \left( \eta_1 \eta_2 - 1 \right) \right].\end{aligned}$$

Substitution of Eq. (25) and Eq. (26) into Eq. (18), and subsequent substitution into Eq. (2) together with Eq. (27) yields the analytical expression of the beam quality factor due to $x-$triangular astigmatism. The final result is given as follows,

$$\begin{aligned} M^4_x &= \left(2p + \ell + 1 \right)\left[ \frac{9 \pi^2 D^2_{\text{xtri}}\omega^6_0}{2 \lambda^2 } \left( \ell^2 \right. \right.\\ & \quad \left. \left. + 3\left[2p + 1 \right]\ell + 6p\left[p + 1 \right] + 2 \right) \right.\\ & \quad \left. + \left(2p + \ell + 1\right) \right]. \end{aligned}$$

The coefficient of $x-$ triangular astigmatism is denoted by $D_{\text {xtri}}$.

3.1.4 Spherical aberration

Spherical aberration is a symmetric aberration that is commonly found in optical elements such as lenses and optical windows. One of the ways in which spherical aberration occurs is through thermal effects. In such instances, it generally depends on the pump profile and also on the temperature dependence of the thermal conductivity of the material. It introduces a quartic radial dependence to the aberrated phase. The aberration-dependent terms for the calculation of the second-order angular moment for spherical aberration are given below as follows,

$$\begin{aligned} \theta^{\text{sp}}_2 &={-} \frac{ D_{\text{sp}}^2 \mathcal{N}^2_{LG} \omega_0^8 \pi \exp\left(-\frac{\mu_1 \mu_2}{2 \eta_1 \eta_2 - 2} \right) \left( \eta_2 - 1 \right)^2 \left( \eta_1 - 1 \right)^2}{32 \lambda^2 \left(\eta_1 \eta_2 - 1 \right)^7}\\ & \quad \times \left[\left( \eta_2 - 1 \right)^2 \mu_1^4 \mu_2^2 + 2 \left( \eta_2 - 1 \right) \left( \eta_1 - 1 \right) \mu_1^3 \mu_2^3 + \left( \eta_1 - 1 \right)^2 \mu_1^2 \mu_2^4 \right.\\ & \quad \left. - 16 \left( \eta_2 - 1 \right)^2 \left(\eta_1 \eta_2 - 1 \right) \mu_1^3 \mu_2 - 36 \left( \eta_2 - 1 \right) \left( \eta_1 - 1 \right) \left(\eta_1 \eta_2 - 1 \right) \mu_1^2 \mu_2^2 \right.\\ & \quad \left. - 16 \left( \eta_1 - 1 \right)^2 \left(\eta_1 \eta_2 - 1 \right) \mu_1 \mu_2^3 + 48 \left( \eta_2 - 1 \right)^2 \left(\eta_1 \eta_2 - 1 \right)^2 \mu_1^2 \right.\\ & \quad \left. + 144 \left( \eta_2 - 1 \right) \left( \eta_1 - 1 \right) \left(\eta_1 \eta_2 - 1 \right)^2 \mu_1 \mu_2 + 48 \left( \eta_1 - 1 \right)^2 \left(\eta_1 \eta_2 - 1 \right)^2 \mu_2^2 \right.\\ & \quad \left. - 96 \left( \eta_2 - 1 \right) \left( \eta_1 - 1 \right) \left(\eta_1 \eta_2 - 1 \right)^3\right] , \end{aligned}$$
and,
$$\begin{aligned} \theta^{\text{sp}}_3 &= \frac{D_{\text{sp}} \mathcal{N}^2_{LG} \omega_0^5 \pi \exp\left(-\frac{\mu_1 \mu_2}{ 2\eta_1 \eta_2 - 2}\right) \left( \eta_2 - 1 \right) \left( \eta_1 - 1 \right) \left( 4\eta_1 \eta_2 - {\mu_1 \mu_2} - 4 \right) }{8 \left(\eta_1 \eta_2 - 1 \right)^4 \lambda}\\ & \quad \times\left[\left( \eta_2 - 1 \right) \mu_1 + \left( \eta_1 - 1 \right) \mu_2\right]. \end{aligned}$$

The first-order spatial angular moment, in terms of generating parameters, is given as,

$$\begin{aligned} \langle x \theta_x \rangle^{\text{sp}} &={-} \frac{D_{\text{sp}} \mathcal{N}^2_{LG} \omega_0^6 \pi \exp\left(-\frac{\mu_1 \mu_2}{2 \eta_1 \eta_2 - 2}\right) \left( \eta_2 - 1 \right) \left( \eta_1 - 1 \right) } {16 \left(\eta_1 \eta_2 - 1 \right)^5 \lambda}\\ & \quad \times \left[ \left( \eta_1 - 1 \right)\left(\mu_1^2 \mu_2^2 - 8 \left( \eta_1 \eta_2 - 1 \right) \mu_1 \mu_2 + 8 \left(\eta_1 \eta_2 - 1 \right)^2\right) \left( \eta_2 - 1 \right) \right.\\ & \quad \left. -3 \mu_1^2 \left( \eta_1 \eta_2 - \frac{\mu_1 \mu_2}{6} - 1 \right) \left( \eta_2 - 1 \right)^2 - 3 \mu_2^2 \left( \eta_1 - 1 \right)^2 \left( \eta_1 \eta_2 - \frac{\mu_1 \mu_2}{6} - 1 \right) \right].\end{aligned}$$

Performing the pertinent substitutions into Eq. (18), and subsequent substitution into Eq. (2), the final result for the beam quality factor due to spherical aberration becomes,

$$\begin{aligned} M^4_{x} &= \frac{4 \pi^2 D^2_{\text{sp}}\omega^8_0}{\lambda^2 }\left[ \left( 2p + 1\right)\ell^3 + \left(3p^2 + 3\left[p + 1\right]^2 + 1\right)\ell^2 \right.\\ & \quad \left. + \left( 2p + 1\right)\left(2\left[p^2 + 1\right] + 2\left[p + 1\right]^2 + 1\right)\ell \right.\\ & \quad \left. + 2\left( p\left[p + 1 \right]\left( \left[p^2 + 4\right] + \left[ p + 1\right]^2 \right) + 1 \right) \right]\\ & \quad + \left( 2p + \ell + 1\right)^2, \end{aligned}$$
where $D_{\text {sp}}$ represents the spherical aberration coefficient.

The main results of this work are given by Eq. (24), Eq. (28), and Eq. (32).

4. Numerical simulation

The beam propagation simulation is performed using the angular spectrum method. First, the Fourier transform of the input function is computed as follows,

$$F(a,b) = \int \int u(x,y)\exp\left({-}2 \pi i \left[a x + b y \right] \right)dx dy.$$

This is subsequently followed by multiplication of the angular spectrum with the propagation phase factor which is given by,

$$\Phi(a, b) = \exp\left({-}2\pi i z \sqrt{a^2 - b^2 - \frac{1}{\lambda^2}} \right).$$

Finally, to obtain the optical field at the specified $z$ position, an inverse Fourier transform is performed to give the propagated optical field,

$$u(x,y,z) = \int \int F(a, b) \Phi(a, b)\exp\left( 2 \pi i\left[a x + b y \right] \right)dadb.$$

To obtain the beam quality factor, we use the method proposed by Siegman [12]. This method is based on the measurement of the beam width along the propagation axis, and the final result is plotted and compared to the equation,

$$\omega^2(z) = \left(\frac{M^2 \lambda}{\pi \omega_0}\right)^2 z^2 - 2z_0\left(\frac{M^2 \lambda}{\pi \omega_0}\right)^2z + \left(\frac{M^2 \lambda}{\pi \omega_0}\right)^2z^2_0 + \omega_0^2.$$

The data are fitted with a quadratic polynomial of the form

$$Y = Az^2 + Bz + C.$$

Upon fitting the polynomial and extracting the pertinent coefficients, the beam quality factor is calculated as follows,

$$M^2 = \frac{\pi}{\lambda}\sqrt{AC - \frac{B^2}{4}},$$
where,
$$A = \left(\frac{M^2 \lambda}{\pi \omega_0}\right)^2,$$
$$B ={-}2 z_0 \left(\frac{M^2 \lambda}{\pi \omega_0}\right)^2,$$
$$C = \left(\frac{M^2 \lambda}{\pi \omega_0}\right)^2z^2_0 + \omega_0^2.$$

5. Results and discussion

The numerical simulation method described in the previous section, Sec. 4, is used here to validate the analytical results obtained in Sec. 3.1. The results for each aberration are discussed separately. All the results are generated with $\lambda = 632 \text {nm}$.

5.1 $45^{\circ }$ astigmatism

Figure 3 shows the beam quality factor as a function of beam width for various LG optical beams in the presence of $45^{\circ }$ astigmatism.

 figure: Fig. 3.

Fig. 3. Beam quality factor due to $45^{\circ }$ astigmatism as a function of beam width. The solid lines represent the beam quality factor due to an aberration coefficient of $0.0005~\text {cm}^{-1}$ and the dashed lines represent an aberration coefficient of $0.05~\text {cm}^{-1}$. The same color is used for both lines to represent the same optical beam.

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The solid lines represent the quality factor of the beam due to an aberration coefficient of $0.0005~\text {cm}^{-1}$ and the dashed lines represent an aberration coefficient of $0.05~\text {cm}^{-1}$. All the plots follow a similar trend: the beam quality factor remains almost unaffected until a certain beam width, which we will call the critical beam width. Once the critical beam width is reached, the beam quality factor starts to change drastically. The critical beam width is given by

$$\omega^\text{ast45}_\text{c} = \sqrt{\frac{\lambda}{\pi D_\text{ast}}}.$$

This is the width that will change the quality factor of the beam by a factor of $\sqrt {2}$. The beam quality factor in Fig. 3 where it changes infinitesimally, can be approximated with the binomial approximation under the condition that $\omega _0 \ll \omega _\text {c}$. Within this approximation, the expression of the beam quality factor for $45^{\circ }$ astigmatism becomes

$$M^2_x \simeq (2p + |\ell| + 1)\left[ 1 + \frac{\pi^2 D^2_\text{ast45} \omega_0^4}{2\lambda^2} \right].$$

For the width where the beam quality factor increases drastically, $\omega _\text {c} \ll \omega _0$, it can be shown that the beam quality factor increases quadratically as a function of $\omega _0$,

$$M^2_x \simeq (2p + |\ell| + 1)\frac{\pi D_\text{ast45} \omega^2_0}{\lambda}.$$

Figure 3 is a log-log plot of the beam quality factor versus beam width, therefore Eq. (44) appears as a straight line with a slope determined by the exponent of $\omega _0$ and the $y-$intercept determined by both the aberration-free beam quality factor and the strength of the $45^{\circ }$ astigmatism. All the plots have the same slope and are only shifted from each other due to different $y-$intercepts: at the same aberration strength, the plots become shifted from each other due to the different aberration-free beam quality factors. Furthermore, due to the different aberration strengths, and consequently different critical beam widths, the plots of the same LG optical beam appear to shift from each other.

The analytical results of the beam quality factor were compared with numerical simulations. Figure 4 shows the beam quality factor of different LG optical beams subjected to $45^{\circ }$ astigmatism. Astigmatism coefficients in the range $[-0.005~\text {cm}^{-1}, 0.005~\text {cm}^{-1}]$ were considered. Figure 4(a) shows the beam quality factor plots of LG optical beams with beam width $\omega _0 = 0.05~\text {cm}$. It can be seen that the effect of $45^{\circ }$ astigmatism is infinitesimal at this beam width. Figure 4(b) shows LG optical beams with beam width $\omega _0 = {2}~\text {cm}$. Here, the effect of $45^{\circ }$ astigmatism on the beam quality factor is clearly discernible. It can further be seen that this effect increases as a function of LG mode order $2p + \ell$.

 figure: Fig. 4.

Fig. 4. Beam quality factor as a function of the $45^o$ astigmatism for various LG optical beams. The plots show results for the LG modes with (a) $\omega _0 = 0.05~\text {cm}$; (b) $\omega _0 = 2~\text {cm}$. The solid lines represent the analytical prediction of the beam quality factor. The discrete markers represent the results of the numerical simulation

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5.2 x-triangular astigmatism

Figure 5 shows the beam quality factor as a function of the beam width for various LG optical beams in the presence of $x-$triangular astigmatism. The behavior of the beam quality factor is comparable to that of $45^{\circ }$ astigmatism. The critical width of $x-$triangular astigmatism is given by,

$$\omega^\text{xtri}_\text{c} = \left[\frac{2\lambda^2 \left( 2p + |\ell| + 1 \right)}{9 \pi^2 D^2_\text{xtri} \mathcal{M}^\text{xtri}_{p \ell} } \right]^{1/6},$$
where,
$$\mathcal{M}^\text{xtri}_{p \ell} = |\ell|^2 + 3\left(2p + 1 \right)|\ell| + 6p\left( p + 1 \right) + 2.$$

As in the case of $45^{\circ }$ astigmatism, this is the beam width that leads to a change in the beam quality factor by a multiplicative factor of $\sqrt {2}$.

 figure: Fig. 5.

Fig. 5. Beam quality factor due to $x-$triangular astigmatism as a function of beam width. The solid lines represent the beam quality factor due to an aberration coefficient of $0.0005~\text {cm}^{-2}$ and the dashed lines represent an aberration coefficient of $0.05~\text {cm}^{-2}$. The same color is used for both lines to represent the same optical beam.

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In the case where $\omega ^\text {xtri}_\text {c} \ll \omega _0$, the beam quality factor increases drastically. In addition, it can be shown that the beam quality, in this case, grows cubically as a function of $\omega _0$,

$$M^2_x \simeq \frac{3\pi D_\text{xtri} \left[2p + |\ell| + 1\right] \omega^3_0}{4\lambda} \sqrt{ \mathcal{M}^\text{xtri}_{p \ell} } .$$

The log-log plot of the beam quality factor due to $x-$ triangular astigmatism versus the beam width is shown in Fig. 5. The slope of the linear relationship is given by the exponent of $\omega _0$ and the $y-$ intercept is determined by both the aberration-free beam quality factor and the strength of the $x-$ triangular astigmatism. Furthermore, it can be seen that the slope of the lines in Fig. 5 is steeper than the slope of the lines in Fig. 3. This is due to the different exponents of the beam width.

Figure 6 shows the comparison of the analytical beam quality factor due to $x-$ triangular astigmatism and the numerical simulations. Aberration strengths in the range $[-0.005~\text {cm}^{-2}, 0.005~\text {cm}^{-2}]$ were considered. Figure 6(a) is a graph of the beam quality factor of LG optical beams with beam width $\omega _0 = 0.05~\text {cm}$. It can be seen that the beam quality factor does not show any observable changes at this width. This behavior is consistent across all the displayed LG optical beams. At the beam width $\omega _0 = 2~\text {cm}$, Fig. 6, the effect of $x-$ triangular astigmatism on the beam quality factor becomes noticeable, more so in the case of the LG optical beam $\ell = 3, p = 2$.

 figure: Fig. 6.

Fig. 6. Beam quality factor as a function of the $x-$ triangular astigmatism for various LG modes. The plots show results for the LG modes with (a) $\omega _0 = 0.05~\text {cm}$; (b) $\omega _0 = 2~\text {cm}$. The solid lines represent the analytical prediction of the beam quality factor. The discrete markers represent the results of the numerical simulation.

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5.3 Spherical aberration

Figure 7 shows the beam quality factor as a function of the beam width for various LG optical beams in the presence of spherical aberration. For spherical aberration, the critical width, defined in the same way as for the other aberrations above, is given as follows,

$$\omega^\text{sp}_\text{c} = \left[\frac{\lambda \left( 2p + |\ell| + 1 \right)}{2 \pi D_\text{sp} \sqrt{\mathcal{M}^\text{sp}_{p \ell}} } \right]^{1/4},$$
where,
$$\begin{aligned}\mathcal{M}^\text{sp}_{p \ell} &= \left[ \left( 2p + 1\right)|\ell|^3 + \left(3p^2 + 3\left[p + 1\right]^2 + 1\right)|\ell|^2 \right.\\ & \quad \left. + \left( 2p + 1\right)\left(2\left[p^2 + 1\right] + 2\left[p + 1\right]^2 + 1\right)|\ell| \right.\\ & \quad \left. + 2\left( p\left[p + 1 \right]\left( \left[p^2 + 4\right] + \left[ p + 1\right]^2 \right) + 1 \right) \right]. \end{aligned}$$

 figure: Fig. 7.

Fig. 7. Beam quality factor due to spherical aberration as a function of beam width. The solid lines represent the beam quality factor due to an aberration coefficient of $0.0005~\text {cm}^{-3}$ and the dashed lines represent an aberration coefficient of $0.05~\text {cm}^{-3}$. The same color is used for both lines to represent the same optical beam.

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For the case where $\omega _\text {c} \ll \omega _0$, it can be shown that the beam quality factor grows quartically as a function of $\omega _0$,

$$M^2_x \simeq \frac{2\pi D_\text{xtri} \omega^4_0 \sqrt{\mathcal{M}^\text{sp}_{p \ell}}}{\lambda} .$$

As with the other aberrations, the logarithm of Eq. (48) is a straight line whose slope is determined by the exponent of $\omega _0$ and the intercept $y-$ is determined by the coefficient of $\omega _0^4$. Inspecting Fig. 3, Fig. 5, and Fig. 7 reveals that for spherical aberration, the beam quality factor deteriorates faster with beam width. This can also be easily seen by comparing the exponents of $\omega _0$ in the corresponding equations [Eq. (44), Eq. (46), Eq. (48)].

Figure 8 shows the beam quality factor of different LG optical beams in the presence of spherical aberration. Aberration coefficients in the range $[-0.005~\text {cm}^{-3}, 0.005~\text {cm}^{-3}]$ were considered. Figure 8(a) is a plot of the beam quality factor of LG optical beams with beam width $\omega _0 = 0.05~\text {cm}$. It can be seen that the beam quality factor does not change at all at this beam width. Figure 8(b) shows the beam quality factor of LG optical beams with beam width $\omega _0 = 2~\text {cm}$. In this case, the effect of spherical aberration on the beam quality factor is clearly visible. The effect is more pronounced for the LG optical beam $\ell = 3, p = 2$.

 figure: Fig. 8.

Fig. 8. Beam quality factor as a function of spherical aberration for various LG modes. The plots show results for the LG modes with (a) $\omega _0 = 0.05~\text {cm}$; (b) $\omega _0 = 2~\text {cm}$. The solid lines represent the analytical prediction of the beam quality factor. The discrete markers represent the results of the numerical simulation.

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6. Conclusions

Using the method of moments and the generating function of LG optical beams, we derived analytical expressions for the beam quality factor of LG beams due to spherical aberration and different types of astigmatism: $0^{\circ }$ astigmatism, $45^{\circ }$ astigmatism, $x$ and $y$ triangular astigmatism. Through an analytical analysis, we have shown that the effect of aberrations on the beam quality factor of LG optical beams is not only a function of the aberration strength but also determined by the width. For each aberration, we derived an expression for the width that separates the region where the beam quality factor is negligibly affected by the presence of the aberration and the region where it deviates drastically from the aberration-free one. We call this the critical width, and it is inversely proportional to the aberration strength. For beam widths that are significantly below the critical width, the effect of each aberration is negligible and thus the optical beam is expected to propagate as though the optical element or system is aberration-free. Conversely, when the beam width is significantly larger than the critical width, we found that the beam quality factor changes drastically. The relationship between the beam quality factor of aberrated LG optical beams and the beam width is different for each aberration type. The beam quality factor has a quadratic, cubical, and quartic relationship with the beam width for $45^{\circ }$ astigmatism, triangular astigmatism, and spherical aberration, respectively. Finally, we performed numerical simulations, based on the angular spectrum method, to validate the derived analytical expressions for the beam quality factor of LG optical beams. The numerical results are in agreement with the results of the analytical expressions. The results of this work will be useful in the design of optical systems that are susceptible to aberrations, such as those operating in the high-power regime.

Funding

National Research Foundation.

Acknowledgements

The research was carried out with the partial support of a grant from the National Research Foundation (NRF). CMM would like to thank Shaun Mabena for his assistance with some of the calculations.

Disclosures

The authors declare that there are no conflicts of interest related to this article.

Data availability

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

References

1. E. Wolf, “The diffraction theory of aberrations,” Rep. Prog. Phys. 14(1), 95–120 (1951). [CrossRef]  

2. A. Bhatia and E. Wolf, “On the circle polynomials of zernike and related orthogonal sets,” in Mathematical Proceedings of the Cambridge Philosophical Society, vol. 50 (Cambridge University, 1954), pp. 40–48.

3. C. Hogge, R. Butts, and M. Burlakoff, “Characteristics of phase-aberrated nondiffraction-limited laser beams,” Appl. Opt. 13(5), 1065–1070 (1974). [CrossRef]  

4. J. Hunt, P. Renard, and R. Nelson, “Focusing properties of an aberrated laser beam,” Appl. Opt. 15(6), 1458–1464 (1976). [CrossRef]  

5. V. N. Mahajan, “Uniform versus gaussian beams: a comparison of the effects of diffraction, obscuration, and aberrations,” J. Opt. Soc. Am. A 3(4), 470–485 (1986). [CrossRef]  

6. J. Alda, J. Alonso, and E. Bernabeu, “Characterization of aberrated laser beams,” J. Opt. Soc. Am. A 14(10), 2737–2747 (1997). [CrossRef]  

7. V. N. Mahajan, Optical imaging and aberrations: Ray geometrical optics, vol. 45 (SPIE, 1998).

8. F. Zernike, “Diffraction theory of the knife-edge test and its improved form: the phase-contrast method,” J. Micro/Nanolithogr., MEMS, MOEMS 1(2), 87–94 (2002). [CrossRef]  

9. G.-m. Dai, Wavefront optics for vision correction, vol. 179 (SPIE, 2008).

10. R. J. Noll, “Zernike polynomials and atmospheric turbulence,” J. Opt. Soc. Am. A 66(3), 207–211 (1976). [CrossRef]  

11. P. Herwig, U. Klotzbach, M. Walther, J. Hauptmann, A. Wetzig, and E. Beyer, “Aberrations induced by high brightness lasers,” Phys. Procedia 12, 779–786 (2011). [CrossRef]  

12. A. E. Siegman, “New developments in laser resonators,” in Opt. Resonators, vol. 1224 (SPIE, 1990), pp. 2–14.

13. A. E. Siegman, “Analysis of laser beam quality degradation caused by quartic phase aberrations,” Appl. Opt. 32(30), 5893–5901 (1993). [CrossRef]  

14. T. Johnston, “M2 concept characterizes beam quality,” Laser Focus World 26, 173–183 (1990).

15. J. A. Ruff and A. Siegman, “Measurement of beam quality degradation due to spherical aberration in a simple lens,” Opt. Quantum Electron. 26(6), 629–632 (1994). [CrossRef]  

16. T. Jeong and J. Lee, “Accurate determination of the beam quality factor of an aberratedhigh-power laser pulse,” J. Korean Phys. Soc. 55(2), 488–494 (2009). [CrossRef]  

17. C. Mafusire and A. Forbes, “Generalized beam quality factor of aberrated truncated gaussian laser beams,” J. Opt. Soc. Am. A 28(7), 1372–1378 (2011). [CrossRef]  

18. J. George, R. Seihgal, S. Oak, and S. Mehendale, “Beam quality degradation of a higher order transverse mode beam due to spherical aberration of a lens,” Appl. Opt. 48(32), 6202–6206 (2009). [CrossRef]  

19. J. Harrison, W. T. Buono, A. Forbes, and D. Naidoo, “Aberration-induced vortex splitting in amplified orbital angular momentum beams,” Opt. Express 31(11), 17593–17608 (2023). [CrossRef]  

20. S. Turtaev, I. T. Leite, K. J. Mitchell, M. J. Padgett, D. B. Phillips, and T. Čižmár, “Comparison of nematic liquid-crystal and dmd based spatial light modulation in complex photonics,” Opt. Express 25(24), 29874–29884 (2017). [CrossRef]  

21. G. Lazarev, P.-J. Chen, J. Strauss, N. Fontaine, and A. Forbes, “Beyond the display: phase-only liquid crystal on silicon devices and their applications in photonics,” Opt. Express 27(11), 16206–16249 (2019). [CrossRef]  

22. A. Rubano, F. Cardano, B. Piccirillo, and L. Marrucci, “Q-plate technology: a progress review,” J. Opt. Soc. Am. B 36(5), D70–D87 (2019). [CrossRef]  

23. R. C. Devlin, A. Ambrosio, D. Wintz, S. L. Oscurato, A. Y. Zhu, M. Khorasaninejad, J. Oh, P. Maddalena, and F. Capasso, “Spin-to-orbital angular momentum conversion in dielectric metasurfaces,” Opt. Express 25(1), 377–393 (2017). [CrossRef]  

24. A. J. Caley, M. J. Thomson, J. Liu, A. J. Waddie, and M. R. Taghizadeh, “Diffractive optical elements for high gain lasers with arbitrary output beam profiles,” Opt. Express 15(17), 10699–10704 (2007). [CrossRef]  

25. D. Naidoo, F. S. Roux, A. Dudley, I. Litvin, B. Piccirillo, L. Marrucci, and A. Forbes, “Controlled generation of higher-order poincaré sphere beams from a laser,” Nat. Photonics 10(5), 327–332 (2016). [CrossRef]  

26. D. Naidoo, I. A. Litvin, and A. Forbes, “Brightness enhancement in a solid-state laser by mode transformation,” Optica 5(7), 836–843 (2018). [CrossRef]  

27. S. Ngcobo, I. Litvin, L. Burger, and A. Forbes, “A digital laser for on-demand laser modes,” Nat. Commun. 4(1), 2289 (2013). [CrossRef]  

28. S. Ngcobo, T. Bell, I. A. Litvin, D. Naidoo, K. Ait-Ameur, and A. Forbes, “Selective excitation and detection of higher-order doughnut laser modes as an incoherent superposition of two petals modes in a digital laser resonator,” in Laser Beam Shaping XVI, vol. 9581 (SPIE, 2015), pp. 15–22.

29. T. Bell, M. Kgomo, and S. Ngcobo, “Digital laser for on-demand intracavity selective excitation of second harmonic higher-order modes,” Opt. Express 28(11), 16907–16923 (2020). [CrossRef]  

30. G. Foo, D. M. Palacios, and G. A. Swartzlander, “Optical vortex coronagraph,” Opt. Lett. 30(24), 3308–3310 (2005). [CrossRef]  

31. H. He, N. Heckenberg, and H. Rubinsztein-Dunlop, “Optical particle trapping with higher-order doughnut beams produced using high-efficiency computer-generated holograms,” J. Mod. Opt. 42(1), 217–223 (1995). [CrossRef]  

32. X.-L. Wang, X.-D. Cai, Z.-E. Su, M.-C. Chen, D. Wu, L. Li, N.-L. Liu, C.-Y. Lu, and J.-W. Pan, “Quantum teleportation of multiple degrees of freedom of a single photon,” Nature 518(7540), 516–519 (2015). [CrossRef]  

33. M. Mirhosseini, O. S. Maga na-Loaiza, M. N. O’Sullivan, B. Rodenburg, M. Malik, M. P. Lavery, M. J. Padgett, D. J. Gauthier, and R. W. Boyd, “High-dimensional quantum cryptography with twisted light,” New J. Phys. 17(3), 033033 (2015). [CrossRef]  

34. D. L. Andrews, Structured light and its applications: An introduction to phase-structured beams and nanoscale optical forces (Academic, 2011).

35. C. Martinez and P.-A. Belanger, “Characterizing the aberrations of a laser beam according to the second-moment method,” in Third International Workshop on Laser Beam and Optics Characterization, vol. 2870 (SPIE, 1996), pp. 62–71.

36. F. S. Roux, T. Wellens, and V. N. Shatokhin, “Entanglement evolution of twisted photons in strong atmospheric turbulence,” Phys. Rev. A 92(1), 012326 (2015). [CrossRef]  

37. A. Wada, H. Ohminato, T. Yonemura, Y. Miyamoto, and M. Takeda, “Effect of comatic aberration on the propagation characteristics of laguerre–gaussian beams,” Opt. Rev. 12(6), 451–455 (2005). [CrossRef]  

Data availability

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

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Figures (8)
Fig. 1.
Fig. 1. Intensity cross section for selected LG modes with indices $p=0,1,2,3$ and $l=0, 1, 2$. The insets illustrate the corresponding phase of the respective mode.

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Fig. 2.
Fig. 2. Phase profiles of the aberrations in Tab. 1. a) $0^{\circ }$ astigmatism, b) $45^{\circ }$ astigmatism, c) $x-$triangular astigmatism, and d) Spherical aberration.

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Fig. 3.
Fig. 3. Beam quality factor due to $45^{\circ }$ astigmatism as a function of beam width. The solid lines represent the beam quality factor due to an aberration coefficient of $0.0005~\text {cm}^{-1}$ and the dashed lines represent an aberration coefficient of $0.05~\text {cm}^{-1}$. The same color is used for both lines to represent the same optical beam.

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Fig. 4.
Fig. 4. Beam quality factor as a function of the $45^o$ astigmatism for various LG optical beams. The plots show results for the LG modes with (a) $\omega _0 = 0.05~\text {cm}$; (b) $\omega _0 = 2~\text {cm}$. The solid lines represent the analytical prediction of the beam quality factor. The discrete markers represent the results of the numerical simulation

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Fig. 5.
Fig. 5. Beam quality factor due to $x-$triangular astigmatism as a function of beam width. The solid lines represent the beam quality factor due to an aberration coefficient of $0.0005~\text {cm}^{-2}$ and the dashed lines represent an aberration coefficient of $0.05~\text {cm}^{-2}$. The same color is used for both lines to represent the same optical beam.

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Fig. 6.
Fig. 6. Beam quality factor as a function of the $x-$ triangular astigmatism for various LG modes. The plots show results for the LG modes with (a) $\omega _0 = 0.05~\text {cm}$; (b) $\omega _0 = 2~\text {cm}$. The solid lines represent the analytical prediction of the beam quality factor. The discrete markers represent the results of the numerical simulation.

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Fig. 7.
Fig. 7. Beam quality factor due to spherical aberration as a function of beam width. The solid lines represent the beam quality factor due to an aberration coefficient of $0.0005~\text {cm}^{-3}$ and the dashed lines represent an aberration coefficient of $0.05~\text {cm}^{-3}$. The same color is used for both lines to represent the same optical beam.

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Fig. 8.
Fig. 8. Beam quality factor as a function of spherical aberration for various LG modes. The plots show results for the LG modes with (a) $\omega _0 = 0.05~\text {cm}$; (b) $\omega _0 = 2~\text {cm}$. The solid lines represent the analytical prediction of the beam quality factor. The discrete markers represent the results of the numerical simulation.

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Tables (1)

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Table 1. Algebraic expressions for some primary aberrations.

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Equations (50)

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E ( x , y ) = u ( x , y ) e − i 2 π λ ϕ ( x , y ) ,
M x 2 = 4 π ⟨ x 2 ⟩ ⟨ θ x 2 ⟩ − ⟨ x θ x ⟩ 2 ,
M y 2 = 4 π ⟨ y 2 ⟩ ⟨ θ y 2 ⟩ − ⟨ y θ y ⟩ 2 ,
⟨ x 2 ⟩ = ∫ − ∞ ∞ ∫ − ∞ ∞ x 2 u 2 ( x , y ) d x d y ,
⟨ y 2 ⟩ = ∫ − ∞ ∞ ∫ − ∞ ∞ y 2 u 2 ( x , y ) d x d y ,
⟨ θ x 2 ⟩ = 1 4 π 2 ∫ − ∞ ∞ ∫ − ∞ ∞ [ ( ∂ u ∂ x ) 2 + ( u ∂ ϕ ∂ x ) 2 ] d x d y − 1 4 π 2 ( ∫ − ∞ ∞ ∫ − ∞ ∞ u 2 ∂ ϕ ∂ x d x d y ) 2 ,
⟨ θ y 2 ⟩ = 1 4 π 2 ∫ − ∞ ∞ ∫ − ∞ ∞ [ ( ∂ u ∂ y ) 2 + ( u ∂ ϕ ∂ y ) 2 ] d x d y − 1 4 π 2 ( ∫ − ∞ ∞ ∫ − ∞ ∞ u 2 ∂ ϕ ∂ y d x d y ) 2 ,
⟨ x θ x ⟩ = 1 2 π ∫ − ∞ ∞ ∫ − ∞ ∞ u 2 x ∂ ϕ ∂ x d x d y ,
⟨ y θ y ⟩ = 1 2 π ∫ − ∞ ∞ ∫ − ∞ ∞ u 2 y ∂ ϕ ∂ y d x d y .
G = 1 1 − η exp ⁡ [ ( x + i T y ) μ ω 0 ( 1 − η ) − ( x 2 + y 2 ) ( 1 + η ) ω 0 2 ( 1 − η ) ] ,
L G ℓ , p = N LG p ! [ ∂ p ∂ η p ∂ | ℓ | ∂ μ | ℓ | G ] η , μ = 0 ,
N LG = [ 2 | ℓ | + 1 p ! π ω 0 2 ( p + | ℓ | ) ! ] 1 / 2 .
M 2 = 2 p + | ℓ | + 1.
⟨ x 2 ⟩ LG = N L G 2 p ! 2 π ω 0 4 e − μ 1 μ 2 2 ( η 1 η 2 − 1 ) 32 ( η 1 η 2 − 1 ) 3 [ 2 μ 1 μ 2 + μ 2 2 + ( 2 η 2 μ 1 μ 2 + 4 η 2 2 − 2 μ 1 μ 2 − 2 μ 2 2 − 4 ) η 1 + η 2 2 μ 1 2 + ( − 2 μ 1 2 − 2 μ 1 μ 2 − 4 ) η 2 + μ 1 2 + ( μ 2 2 + 4 η 2 − 4 η 2 2 ) η 1 2 + 4 ] .
θ 1 = 1 4 π 2 ∫ − ∞ ∞ ∫ − ∞ ∞ ( ∂ u ∂ x ) 2 d x d y ,
θ 2 = 1 4 π 2 ∫ − ∞ ∞ ∫ − ∞ ∞ ( u ∂ ϕ ∂ x ) 2 d x d y ,
θ 3 = 1 2 π ∫ − ∞ ∞ ∫ − ∞ ∞ u 2 ∂ ϕ ∂ x d x d y ,
⟨ θ x 2 ⟩ = θ 1 + θ 2 − θ 3 2 .
θ 1 = N L G 2 exp ⁡ ( μ 1 μ 2 2 η 1 η 2 − 2 ) 32 π ( η 1 η 2 − 1 ) 3 × [ ( 4 η 2 2 + μ 2 2 + 4 η 2 ) η 1 2 + ( − 2 η 2 μ 1 μ 2 + 4 η 2 2 − 2 μ 1 μ 2 + 2 μ 2 2 − 4 ) η 1 + η 2 2 μ 1 2 + ( 2 μ 1 2 − 2 μ 1 μ 2 − 4 ) η 2 + ( μ 1 − μ 2 + 2 ) ( μ 1 − μ 2 − 2 ) ] .
M x 2 = 4 π ⟨ x 2 ⟩ θ 1 ,
θ 2 ast45 = D ast45 2 N L G 2 ω 0 4 π exp ⁡ ( − μ 1 μ 2 2 η 1 η 2 − 2 ) 32 ( η 1 η 2 − 1 ) 3 λ 2 × [ ( η 2 − 1 ) 2 μ 1 2 + ( η 1 − 1 ) 2 μ 2 2 + 4 ( η 2 − 1 ) ( η 1 − 1 ) ( η 1 η 2 − μ 1 μ 2 4 − 1 ) ] ,
θ 3 ast45 = D ast45 N L G 2 ω 0 3 π exp ⁡ ( − μ 1 μ 2 2 η 2 η 1 − 2 ) i 8 ( η 1 η 2 − 1 ) 2 λ × [ ( η 1 − 1 ) μ 2 + ( η 2 − 1 ) μ 1 ] .
⟨ x θ x ⟩ ast45 = D ast45 N L G 2 ω 0 4 π exp ⁡ ( − μ 1 μ 2 2 η 1 η 2 − 2 ) i 32 ( η 1 η 2 − 1 ) 3 λ × [ ( η 2 − 1 ) 2 μ 1 2 − ( η 1 − 1 ) 2 μ 2 2 ] .
M x 4 = ( 2 p + ℓ + 1 ) 2 [ π 2 D ast45 2 ω 0 4 λ 2 + 1 ] ,
θ 2 xtri = − 9 D xtri 2 N L G 2 ω 0 6 π exp ⁡ ( − μ 1 μ 2 2 η 1 η 2 − 2 ) 128 ( η 1 η 2 − 1 ) 5 λ 2 × { 16 ( η 2 − 1 ) 2 [ μ 1 2 μ 2 2 8 − ( η 1 η 2 − 1 ) μ 1 μ 2 + ( η 1 η 2 − 1 ) 2 ] ( η 2 − 1 ) 2 + ( η 2 − 1 ) 4 μ 1 4 + ( η 1 − 1 ) 4 μ 2 4 } ,
θ 3 xtri = 3 D xtri N L G 2 ω 0 4 π exp ⁡ ( − μ 1 μ 2 2 η 1 η 2 − 2 ) 16 ( η 1 η 2 − 1 ) 3 λ × [ ( η 2 − 1 ) 2 μ 1 2 + ( η 1 − 1 ) 2 μ 2 2 ] .
⟨ x θ x ⟩ trix = 3 D xtri N L G 2 ω 0 5 π exp ⁡ ( − μ 1 μ 2 2 η 1 η 2 − 2 ) 64 ( η 1 η 2 − 1 ) 4 λ × [ ( η 2 − 1 ) μ 1 + ( η 1 − 1 ) μ 2 ] × [ ( η 2 − 1 ) 2 μ 1 2 + ( η 1 − 1 ) 2 μ 2 2 − 4 ( η 2 − 1 ) ( η 1 − 1 ) ( η 1 η 2 − 1 ) ] .
M x 4 = ( 2 p + ℓ + 1 ) [ 9 π 2 D xtri 2 ω 0 6 2 λ 2 ( ℓ 2 + 3 [ 2 p + 1 ] ℓ + 6 p [ p + 1 ] + 2 ) + ( 2 p + ℓ + 1 ) ] .
θ 2 sp = − D sp 2 N L G 2 ω 0 8 π exp ⁡ ( − μ 1 μ 2 2 η 1 η 2 − 2 ) ( η 2 − 1 ) 2 ( η 1 − 1 ) 2 32 λ 2 ( η 1 η 2 − 1 ) 7 × [ ( η 2 − 1 ) 2 μ 1 4 μ 2 2 + 2 ( η 2 − 1 ) ( η 1 − 1 ) μ 1 3 μ 2 3 + ( η 1 − 1 ) 2 μ 1 2 μ 2 4 − 16 ( η 2 − 1 ) 2 ( η 1 η 2 − 1 ) μ 1 3 μ 2 − 36 ( η 2 − 1 ) ( η 1 − 1 ) ( η 1 η 2 − 1 ) μ 1 2 μ 2 2 − 16 ( η 1 − 1 ) 2 ( η 1 η 2 − 1 ) μ 1 μ 2 3 + 48 ( η 2 − 1 ) 2 ( η 1 η 2 − 1 ) 2 μ 1 2 + 144 ( η 2 − 1 ) ( η 1 − 1 ) ( η 1 η 2 − 1 ) 2 μ 1 μ 2 + 48 ( η 1 − 1 ) 2 ( η 1 η 2 − 1 ) 2 μ 2 2 − 96 ( η 2 − 1 ) ( η 1 − 1 ) ( η 1 η 2 − 1 ) 3 ] ,
θ 3 sp = D sp N L G 2 ω 0 5 π exp ⁡ ( − μ 1 μ 2 2 η 1 η 2 − 2 ) ( η 2 − 1 ) ( η 1 − 1 ) ( 4 η 1 η 2 − μ 1 μ 2 − 4 ) 8 ( η 1 η 2 − 1 ) 4 λ × [ ( η 2 − 1 ) μ 1 + ( η 1 − 1 ) μ 2 ] .
⟨ x θ x ⟩ sp = − D sp N L G 2 ω 0 6 π exp ⁡ ( − μ 1 μ 2 2 η 1 η 2 − 2 ) ( η 2 − 1 ) ( η 1 − 1 ) 16 ( η 1 η 2 − 1 ) 5 λ × [ ( η 1 − 1 ) ( μ 1 2 μ 2 2 − 8 ( η 1 η 2 − 1 ) μ 1 μ 2 + 8 ( η 1 η 2 − 1 ) 2 ) ( η 2 − 1 ) − 3 μ 1 2 ( η 1 η 2 − μ 1 μ 2 6 − 1 ) ( η 2 − 1 ) 2 − 3 μ 2 2 ( η 1 − 1 ) 2 ( η 1 η 2 − μ 1 μ 2 6 − 1 ) ] .
M x 4 = 4 π 2 D sp 2 ω 0 8 λ 2 [ ( 2 p + 1 ) ℓ 3 + ( 3 p 2 + 3 [ p + 1 ] 2 + 1 ) ℓ 2 + ( 2 p + 1 ) ( 2 [ p 2 + 1 ] + 2 [ p + 1 ] 2 + 1 ) ℓ + 2 ( p [ p + 1 ] ( [ p 2 + 4 ] + [ p + 1 ] 2 ) + 1 ) ] + ( 2 p + ℓ + 1 ) 2 ,
F ( a , b ) = ∫ ∫ u ( x , y ) exp ⁡ ( − 2 π i [ a x + b y ] ) d x d y .
Φ ( a , b ) = exp ⁡ ( − 2 π i z a 2 − b 2 − 1 λ 2 ) .
u ( x , y , z ) = ∫ ∫ F ( a , b ) Φ ( a , b ) exp ⁡ ( 2 π i [ a x + b y ] ) d a d b .
ω 2 ( z ) = ( M 2 λ π ω 0 ) 2 z 2 − 2 z 0 ( M 2 λ π ω 0 ) 2 z + ( M 2 λ π ω 0 ) 2 z 0 2 + ω 0 2 .
Y = A z 2 + B z + C .
M 2 = π λ A C − B 2 4 ,
A = ( M 2 λ π ω 0 ) 2 ,
B = − 2 z 0 ( M 2 λ π ω 0 ) 2 ,
C = ( M 2 λ π ω 0 ) 2 z 0 2 + ω 0 2 .
ω c ast45 = λ π D ast .
M x 2 ≃ ( 2 p + | ℓ | + 1 ) [ 1 + π 2 D ast45 2 ω 0 4 2 λ 2 ] .
M x 2 ≃ ( 2 p + | ℓ | + 1 ) π D ast45 ω 0 2 λ .
ω c xtri = [ 2 λ 2 ( 2 p + | ℓ | + 1 ) 9 π 2 D xtri 2 M p ℓ xtri ] 1 / 6 ,
M p ℓ xtri = | ℓ | 2 + 3 ( 2 p + 1 ) | ℓ | + 6 p ( p + 1 ) + 2.
M x 2 ≃ 3 π D xtri [ 2 p + | ℓ | + 1 ] ω 0 3 4 λ M p ℓ xtri .
ω c sp = [ λ ( 2 p + | ℓ | + 1 ) 2 π D sp M p ℓ sp ] 1 / 4 ,
M p ℓ sp = [ ( 2 p + 1 ) | ℓ | 3 + ( 3 p 2 + 3 [ p + 1 ] 2 + 1 ) | ℓ | 2 + ( 2 p + 1 ) ( 2 [ p 2 + 1 ] + 2 [ p + 1 ] 2 + 1 ) | ℓ | + 2 ( p [ p + 1 ] ( [ p 2 + 4 ] + [ p + 1 ] 2 ) + 1 ) ] .
M x 2 ≃ 2 π D xtri ω 0 4 M p ℓ sp λ .
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