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Effect of aberrations on the beam quality factor of Hermite-Gauss beams

Nokwazi Mphuthi, Teboho Bell, and Chemist M. Mabena

Open Access Open Access

Abstract

The effect of aberrations on the beam quality factor (M2) of Hermite-Gauss (HG) beams is examined. Using the method of moments, we derive closed-form analytical expressions of M2 due to astigmatism and spherical aberration. Our analysis reveals that the radius of the HG beams plays a significant role in determining the effect of the aberrations on M2. For each aberration, we establish a critical width that separates the region where M2 changes infinitesimally from the region where it changes sharply. The analytical results are validated through numerical simulations.

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

1. Introduction

The beam quality factor is a useful parameter for assessing the quality of a laser beam. It gives information about the propagation dynamics of a laser beam such as how much it will diverge upon propagation [1–3]. It is invariant in media with linear and quadratic transfer functions, as well as in ideal optical systems. However, physical optical elements or systems are not always free from aberrations, which can be caused by various factors such as variations in propagation media, fabrication errors and general imperfections of optical elements. Aberrations can be described mathematically by a complex transfer function that perturbs the wavefront of a laser beam. The effect of aberrations on laser beams has been extensively studied, resulting in a substantial body of knowledge [2,4–13].

In recent years, significant advancements have been made in the generation of structured light, which refers to light with tailored spatial properties such as amplitude, phase, and polarization. Structured light, such as Hermite-Gauss (HG) beams, hold promise for various applications, including nonlinear optics [14], electron acceleration [15–17], atom trapping [18–20] and free-space optical communications [21–29]. The beam quality of Hermite-Gauss beams has been studied both in free space and through atmospheric turbulence [30–33]. While turbulence has been shown to have a detrimental effect on the beam quality of Hermite-Gaussian beams, however, to date, to the best of our knowledge, the effect of individual aberrations on the beam quality factor of Hermite-Gauss beams has not been studied. These beams can be generated intra-cavity through specially designed lasers [34–41] and extra-cavity using phase-shifting diffractive optical elements [42–49].

Astigmatism and spherical aberration are two of the most common types of aberrations that occur in optical elements. Furthermore, these aberrations have been shown to have detrimental effects in applications where Hermite-Gauss beams are used. In optical trapping, for example, spherical aberration has been shown to decrease the trap stiffness in oil immersion objectives [50]. Astigmatism, on the other hand, has been shown to be the main effect for transverse stiffness degradation in optical traps [51]. In electron acceleration, it was demonstrated how laser inhomogeneities and phase defects can lead to lower electron charges and energies, thereby severely decreasing the number of emitted photons [52]. In order to inform the design and development of laser systems where astigmatism and spherical aberration occur, it is useful and important to quantify the amount of degradation that these aberrations can cause.

In this work, therefore, and to the best of our knowledge, for the first time, we derive closed-form analytical expressions of the beam quality factor of Hermite-Gauss beams due to the individual aberrations mentioned above. Our approach uses the generating function for the Hermite-Gauss beams [53,54] and is based on the method of moments [7]. Through these expressions, we are able to quantitatively assess how these aberrations impact the beam quality factor. A key finding of our analysis is the pivotal role of the radius of the Hermite-Gauss beams in determining the effect of each aberration on the beam quality factor. To this extent, we establish a critical width that serves as a boundary, distinguishing between regions where the beam quality factor changes infinitesimally and regions where it undergoes drastic and sharp changes for each aberration. Furthermore, we note an important difference between the behavior of Hermite-Gauss beams in comparison to Laguerre-Gauss beams [10]. In the case of Laguerre-Gauss beams, the critical width is independent of direction and thus is the same along the $x$ and $y$ directions. For Hermite-Gauss beams, however, we found that in determining the critical width, the $x$ direction behaves differently from the $y$ direction. Therefore, to determine the beam size where the laser beam is unaffected by a particular aberration, at a particular aberration strength, it is important to consider the minimum of the two critical widths. To validate the accuracy of our derived analytical expressions, we complement our theoretical work with numerical simulations. This work provides a tool for conveniently performing parametric studies and obtaining quantitative results expeditiously in cases where the beam quality factor is of concern.

2. Method of moments

The electric field of a laser beam can be expressed as follows [55],

$$E(x, y) = u(x, y)e^{{-}i \frac{2 \pi}{\lambda} \phi(x, y)},$$
where $u(x, y)$ represents the amplitude distribution of the laser beam, and $\phi (x, y)$ denotes the phase of the beam. In this work, the phase term, $\phi (x, y)$, specifically accounts for the aberrations present in an optical system. The aberrations considered here are shown in Table 1.

Tables Icon

Table 1. Algebraic expressions for some primary aberrations.

View Table

It is common practice in experiments to determine the beam quality factor along, $x$ and $y$, independently. Therefore, here, we also calculate the beam quality factor separately in the $x$ and $y$ directions using the following expressions [7],

$$M^2_{x} = 4 \pi \sqrt{ \langle x^2 \rangle \langle \theta^2_x \rangle - \langle x \theta_x \rangle^2 },$$
and
$$M^2_{y} = 4 \pi \sqrt{ \langle y^2 \rangle \langle \theta^2_y \rangle - \langle y \theta_y \rangle^2 }.$$

Here, $\langle x^2 \rangle$ and $\langle y^2 \rangle$ represent the second-order spatial moments along the $x$ and $y$ directions, respectively. Similarly, $\langle \theta ^2_x \rangle$ and $\langle \theta ^2_y \rangle$ denotes the second-order angular moments in each direction. Lastly, $\langle x\theta _x \rangle$ and $\langle y\theta _y \rangle$ represent the first spatial-angular moments. The calculations for these terms are given by the following equations,

$$\langle x^2 \rangle = \int^{\infty}_{-\infty} \int^{\infty}_{-\infty} x^2 u^2\left(x, y\right)dxdy,$$
$$\langle y^2 \rangle = \int^{\infty}_{-\infty} \int^{\infty}_{-\infty} y^2 u^2\left(x, y\right)dx dy,$$
$$\begin{aligned} \langle \theta^2_x \rangle &= \frac{1}{4 \pi^2} \int^{\infty}_{-\infty} \int^{\infty}_{-\infty}\left[ \left( \frac{\partial u}{\partial x} \right)^2 + \left( u \frac{\partial \phi}{\partial x} \right)^2 \right] dx dy \\ & - \frac{1}{4 \pi^2}\left( \int^{\infty}_{-\infty} \int^{\infty}_{-\infty} u^2 \frac{\partial \phi}{\partial x} dx dy\right)^2, \end{aligned}$$
$$\begin{aligned} \langle \theta^2_y \rangle &= \frac{1}{4 \pi^2} \int^{\infty}_{-\infty} \int^{\infty}_{-\infty}\left[ \left( \frac{\partial u}{\partial y} \right)^2 + \left( u \frac{\partial \phi}{\partial y} \right)^2 \right] dx dy \\ & - \frac{1}{4 \pi^2}\left( \int^{\infty}_{-\infty} \int^{\infty}_{-\infty} u^2 \frac{\partial \phi}{\partial y} dx dy\right)^2, \end{aligned}$$
$$\langle x \theta_x \rangle = \frac{1}{2 \pi } \int^{\infty}_{-\infty} \int^{\infty}_{-\infty} u^2 x \frac{\partial \phi}{\partial x} dxdy,$$
$$\langle y \theta_y \rangle = \frac{1}{2 \pi } \int^{\infty}_{-\infty} \int^{\infty}_{-\infty} u^2 y \frac{\partial \phi}{\partial y} dxdy.$$

3. Aberrated Hermite-Gauss beams

Hermite-Gauss beams are solutions of the paraxial wave equation in Cartesian coordinates and have a complex amplitude that is given as follows [55],

$$\begin{aligned}H\left( x, y, z\right) &= E_0 \frac{\omega_0}{\omega\left( z \right)}H_{n}\left( \frac{\sqrt{2}x}{\omega\left( z \right)} \right)H_{m}\left( \frac{\sqrt{2}y}{\omega\left( z \right)} \right) \exp\left( -\frac{x^2 + y^2}{\omega^2\left( z \right)} \right)\\ & \times \exp\left({-}i\frac{\pi \left[ x^2 + y^2 \right]}{\lambda R\left( z \right)} \right) \exp\left( i\psi\left( z \right) \right)\exp\left({-}i\frac{2\pi z}{\lambda} \right), \end{aligned}$$
where $E_0$ is a normalization constant, $\omega _0$ is the beam radius at the waist, $H_{n}\left ( \cdot \right )$ and $H_{m}\left ( \cdot \right )$ represent the Hermite polynomials of order $n$ and $m$, respectively. The beam radius at some arbitrary axial position $z$ is given as,
$$\omega\left( z\right) = \omega_0\sqrt{1 + \left(\frac{ z}{z_R} \right)^2 },$$
the radius of curvature is expressed as,
$$R\left( z \right) = z\left[ 1 + \left( \frac{z_R}{z} \right)^2 \right].$$

The Gouy phase is given as,

$$\psi\left( z \right) = \left( n + m + 1 \right) \text{arctan} \left( \frac{ z }{z_R} \right),$$
and,
$$z_R = \frac{1}{\left( m + n + 1 \right)}\frac{\pi \omega_0^2}{ \lambda},$$
is the Rayleigh range. At the waist, the Hermite-Gauss beam expression can be expressed in terms of a generating function [53,54] as follows,
$$\begin{aligned}\mathcal{H} &= \exp\left[ \frac{ 2\sqrt{2 }x \mu }{\omega_0} + \frac{ 2\sqrt{2 }y \eta }{\omega_0} - \left(\mu^2 + \eta^2\right) \right]\\ & \times \exp\left[ -\frac{\left(x^2 + y^2\right)}{\omega^2_0} \right], \end{aligned}$$
where $\mu$ and $\eta$ are the generating parameters for the Hermite polynomials along $x$ and $y$, respectively. A specific Hermite-Gauss beam is obtained by the following operation,
$$\text{HG}_{n, m} = {\mathcal{N}_\text{HG}}\left[ \frac{\partial^m}{\partial \eta^m}\frac{\partial^{n}}{\partial \mu^{n}} \mathcal{H}\right]_{\mu, \eta = 0},$$
where,
$$\mathcal{N}_\text{HG} = \sqrt{\frac{1}{\pi 2^{n + m - 1}n!m! }} \frac{1}{\omega_0}.$$

Examples of the intensity profiles of selected Hermite-Gauss beams, at the waist, with their respective phases are shown in Fig. 1.

 figure: Fig. 1.

Fig. 1. Intensity profiles of selected Hermite-Gauss beams with indices $n= 0, 1,2,3$ and $m=0, 1, 2$ at the waist plane. The insets represent the phase of the beams.

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Upon using the generating function for the Hermite-Gauss beams and substituting into Eq. (4), the second-order spatial moment along the $x$ direction is calculated as follows,

$$\begin{aligned} \langle {x}^2 \rangle_{\mu,\eta} &= \mathcal{N}_\text{HG}^2 \int^{-\infty}_{\infty}\int^{-\infty}_{\infty} x^2 \exp\{\frac{ 2 \sqrt{2} \left[ \left(\mu_1 + \mu_2\right) x + y\left( \eta_1 + \eta_2\right) \right] }{\omega_0} \}\\ & \times \exp\{ \frac{\left[-\eta_1^2 - \eta_2^2 - \mu_1^2 - \mu_2^2\right] \omega_0^2 - 2\left[x^2 + y^2\right]}{\omega_0^2} \} dx dy\\ &= \frac{ \mathcal{N}_\text{HG}^2 \omega_0^4 \pi \left[ 2\left( \mu_1 + \mu_2\right)^2 + 1\right] \exp\left\{2\left[\mu_1 \mu_2 + \eta_1\eta_2 \right]\right\}}{8} , \end{aligned}$$
where $\mu _1, \eta _1$ are the generating parameters for the complex electric field and $\mu _2, \eta _2$ are the generating parameters for its complex conjugate. Using the same approach, the results for the second-order spatial moments along the $y$ after substituting into Eq. (5) becomes,
$$\begin{aligned} \langle y^2 \rangle_{\mu,\eta} &= \frac{\mathcal{N}_\text{HG}^2 \omega_0^4 \pi \exp\left(2\left[\mu_1 \mu_2 + \eta_1\eta_2 \right]\right)}{8} \\ & \times \left[ 2\left( \eta_1 + \eta_2\right)^2 + 1\right]. \end{aligned}$$

For convenience, we break down the expression of the second-order angular moments [Eq. (7), Eq. (8)] expression into three parts. Here, we show the breakdown only for Eq. (7),

$$\theta_{1,x} = \frac{1}{4 \pi^2} \int^{\infty}_{-\infty} \int^{\infty}_{-\infty} \left( \frac{\partial u}{\partial x} \right)^2 dxdy,$$
$$\theta_{2,x} = \frac{1}{4 \pi^2} \int^{\infty}_{-\infty} \int^{\infty}_{-\infty} \left( u \frac{\partial \phi}{\partial x} \right)^2dx dy,$$
$$\theta_{3,x} = \frac{1}{2 \pi} \int^{\infty}_{-\infty} \int^{\infty}_{-\infty} u^2 \frac{\partial \phi}{\partial x} dx dy,$$
where the last term is the square root of the second term in Eq. (7). That is, the $x$ second-order angular moment expression can be rewritten as follows,
$$\langle \theta^2_x \rangle = \theta_{1,x} + \theta_{2,x} - \theta_{3,x}^2,$$
and the $y$ second-order angular moment expression can be rewritten, in the same way, as follows,
$$\langle \theta^2_y \rangle = \theta_{1,y} + \theta_{2,y} - \theta_{3,y}^2.$$

The terms that do not depend on the phase aberration, Eq. (20) and its $y$ variant, are given below in terms of the generating parameters, as follows,

$$\begin{aligned} \theta_{1,x} &= \frac{ \mathcal{N}_\text{HG}^2 \exp\left(2\left[\mu_1 \mu_2 + \eta_1\eta_2 \right]\right)}{8 \pi} \\ & \times \left[ 2\left( \mu_1 - \mu_2\right)^2 - 1\right] , \end{aligned}$$
and,
$$\begin{aligned} \theta_{1,y} &= \frac{\mathcal{N}_\text{HG}^2 \exp\left(2\left[\mu_1 \mu_2 + \eta_1\eta_2 \right]\right)}{8 \pi} \\ & \times \left[ 2\left( \eta_1 - \eta_2\right)^2 - 1\right]. \end{aligned}$$

3.1 Beam quality factor of aberrated Hermite-Gauss beams

3.1.1 $45^\circ$ astigmatism

Using the expression for $45^\circ$ astigmatism in Table 1, we obtain the aberration-dependent terms below. First, we generate the expressions for the second-order angular moment, Eq. (21) and Eq. (22), for both $x$ and $y$,

$$\begin{aligned} \theta^{\text{ast45}}_{2,x} &= \frac{D_{\text{ast45}}^2 \mathcal{N}_\text{HG}^2 \omega_0^4 \pi \exp\left(2\left[\mu_1 \mu_2 + \eta_1\eta_2 \right]\right)}{8 \lambda^2} \\ & \times \left[ 2\left( \eta_1 + \eta_2\right)^2 + 1\right], \end{aligned}$$
and,
$$\begin{aligned} \theta^{\text{ast45}}_{2,y} &= \frac{D_{\text{ast45}}^2 \mathcal{N}_\text{HG}^2 \omega_0^4 \pi \exp\left(2\left[\mu_1 \mu_2 + \eta_1\eta_2 \right]\right)}{8 \lambda^2 } \\ & \times \left[ 2\left( \mu_1 + \mu_2\right)^2 + 1\right], \end{aligned}$$
and
$$\begin{aligned} \theta^{\text{ast45}}_{3,x} &= \frac{D_{\text{ast45}} \mathcal{N}^2_\text{HG}\omega_0^3 \pi \exp\left(2\left[\mu_1 \mu_2 + \eta_1\eta_2 \right]\right)}{ 2\sqrt{2} \lambda} \\ & \times \left[ \eta_1 + \eta_2 \right], \end{aligned}$$
and
$$\begin{aligned} \theta^{\text{ast45}}_{3,y} &= \frac{D_{\text{ast45}} \mathcal{N}^2_\text{HG}\omega_0^3 \pi \exp\left(2\left[ \mu_1 \mu_2 + \eta_1\eta_2 \right]\right)}{ 2\sqrt{2} \lambda } \\ & \times \left[ \mu_1 + \mu_2 \right]. \end{aligned}$$

The first-order spatial-angular momentum, in the $x$ direction, is given as follows,

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

The first-order spatial-angular momentum, in the $y$ direction, works out to be the same expression as for the $x$ direction,

$$\langle y \theta_y \rangle^{\text{ast45}} = \langle x \theta_x\rangle^{\text{ast45}}.$$

The value of $M^4_x$ for a particular Hermite-Gauss beam can be obtained by the following operation,

$$\begin{aligned} \mathcal{M}^4_x &= 16\pi^2\left\{ \left[ \frac{\partial^m}{\partial \eta_1^m}\frac{\partial^m}{\partial \eta_2^m}\frac{\partial^{n}}{\partial \mu_1^{n} }\frac{\partial^{n} \langle x^2 \rangle_{\mu,\eta} }{\partial \mu_2^{n} } \right]_{\eta_1,\eta_2,\mu_1,\mu_2 = 0} \left( \left[ \frac{\partial^m}{\partial \eta_1^m}\frac{\partial^m}{\partial \eta_2^m}\frac{\partial^{n}}{\partial \mu_1^{n} }\frac{\partial^{n} \theta^\text{ast45}_{1,x} }{\partial \mu_2^{n} } \right]_{\eta_1,\eta_2,\mu_1,\mu_2 = 0} \right. \right.\\ & \left. \left. + \left[ \frac{\partial^m}{\partial \eta_1^m}\frac{\partial^m}{\partial \eta_2^m}\frac{\partial^{n}}{\partial \mu_1^{n} }\frac{\partial^{n} \theta^\text{ast45}_{2,x} }{\partial \mu_2^{n} } \right]_{\eta_1,\eta_2,\mu_1,\mu_2 = 0} - \left[ \frac{\partial^m}{\partial \eta_1^m}\frac{\partial^m}{\partial \eta_2^m}\frac{\partial^{n}}{\partial \mu_1^{n} }\frac{\partial^{n} \theta^\text{ast45}_{3,x} }{\partial \mu_2^{n} } \right]_{\eta_1,\eta_2,\mu_1,\mu_2 = 0} ^2 \right) \right.\\ & \left. - \left[ \frac{\partial^m}{\partial \eta_1^m}\frac{\partial^m}{\partial \eta_2^m}\frac{\partial^{n}}{\partial \mu_1^{n} }\frac{\partial^{n} \langle x\theta_x \rangle^\text{ast45} }{\partial \mu_2^{n} } \right]_{\eta_1,\eta_2,\mu_1,\mu_2 = 0}^2 \right\}. \end{aligned}$$

Substitution of relevant expressions into Eq. (33) above, and performing some long calculations, the final general expression for the beam quality factor of Hermite-Gauss beams aberrated with $45^\circ$ astigmatism becomes,

$$\begin{aligned} M^4_x &= \left(2n + 1 \right)\left[\frac{ \pi^2 D^2_{\text{ast45}}\omega^4_0}{\lambda^2 } \left(2m + 1 \right) \right.\\ & \left. + \left(2n + 1 \right) \right]. \end{aligned}$$

Using the same approach as above, $M^4_y$ for a particular Hermite-Gauss beam is given as,

$$\begin{aligned} \mathcal{M}^4_y &= 16\pi^2\left\{ \left[ \frac{\partial^m}{\partial \eta_1^m}\frac{\partial^m}{\partial \eta_2^m}\frac{\partial^{n}}{\partial \mu_1^{n} }\frac{\partial^{n} \langle y^2 \rangle_{\mu,\eta} }{\partial \mu_2^{n} } \right]_{\eta_1,\eta_2,\mu_1,\mu_2 = 0} \left( \left[ \frac{\partial^m}{\partial \eta_1^m}\frac{\partial^m}{\partial \eta_2^m}\frac{\partial^{n}}{\partial \mu_1^{n} }\frac{\partial^{n} \theta^\text{ast45}_{1,y} }{\partial \mu_2^{n} } \right]_{\eta_1,\eta_2,\mu_1,\mu_2 = 0} \right. \right.\\ & \left. \left. + \left[ \frac{\partial^m}{\partial \eta_1^m}\frac{\partial^m}{\partial \eta_2^m}\frac{\partial^{n}}{\partial \mu_1^{n} }\frac{\partial^{n} \theta^\text{ast45}_{2,y} }{\partial \mu_2^{n} } \right]_{\eta_1,\eta_2,\mu_1,\mu_2 = 0} - \left[ \frac{\partial^m}{\partial \eta_1^m}\frac{\partial^m}{\partial \eta_2^m}\frac{\partial^{n}}{\partial \mu_1^{n} }\frac{\partial^{n} \theta^\text{ast45}_{3,y} }{\partial \mu_2^{n} } \right]_{\eta_1,\eta_2,\mu_1,\mu_2 = 0} ^2 \right) \right.\\ & \left. - \left[ \frac{\partial^m}{\partial \eta_1^m}\frac{\partial^m}{\partial \eta_2^m}\frac{\partial^{n}}{\partial \mu_1^{n} }\frac{\partial^{n} \langle y\theta_y \rangle^\text{ast45} }{\partial \mu_2^{n} } \right]_{\eta_1,\eta_2,\mu_1,\mu_2 = 0}^2 \right\}, \end{aligned}$$
and the general result becomes,
$$\begin{aligned} M^4_y &= \left(2m + 1 \right)\left[\frac{ \pi^2 D^2_{\text{ast45}}\omega^4_0}{\lambda^2 } \left(2n + 1 \right) \right.\\ & \left. + \left(2m + 1 \right) \right]. \end{aligned}$$

Figure 2 shows the beam quality factor of selected Hermite-Gauss beams that are aberrated with $45^{\circ}$ astigmatism as a function of beam radius. The aberration strength is represented by the coefficient of the aberration. The solid lines represent the beam quality factor due to an astigmatism coefficient of 0.1 $\text {cm}^{-1}$ and the dashed lines represent an astigmatism coefficient of 10 $\text {cm}^{-1}$. The beam quality factor along the $x$, $M^2_x$, is shown in Fig. 2(a), and the beam quality factor along the $y$, $M^2_y$, is shown in Fig. 2(b). The general trend is the same in both cases. That is, the beam quality factor is minimally affected by astigmatism up until a certain beam radius is reached. For values of the beam radius that are much larger than the said radius, the beam quality factor starts changing sharply. The radius that separates the region where the beam quality factor is shown to vary infinitesimally and the region where it changes sharply is called the critical width. The expressions for the critical width is determined by rewriting Eq. (34) as follows,

$$M^4_x = \left(2n + 1 \right)^2\left[\frac{ \pi^2 D^2_{\text{ast45}}\omega^4_0 \left(2m + 1 \right)}{\lambda^2 \left(2n + 1 \right) } + 1\right].$$
We then define the critical width as the beam radius where $M^4_x$ is twice its unaberrated value. Working this out gives the following result for the critical width along the $x$,
$$\omega_\text{c,x} = \left[ \frac{\lambda^2 \left( 2n + 1 \right) }{\pi^2 D^2_\text{ast45} \left( 2m + 1 \right)} \right]^{\frac{1}{4}}.$$

 figure: Fig. 2.

Fig. 2. Beam quality factor due to $45^{\circ}$ astigmatism as a function of beam radius along the $x$ and $y$ direction for various $\text {HG}_{nm}$ beams. The solid lines illustrate an astigmatism coefficient of 0.1 $\text {cm}^{-1}$ while dotted lines illustrate an astigmatism coefficient of 10 $\text {cm}^{-1}$.

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Similarly, for the beam radius along the $y$, we obtain,

$$\omega_\text{c,y} = \left[ \frac{\lambda^2 \left( 2m + 1 \right) }{\pi^2 D^2_\text{ast45} \left( 2n + 1 \right)} \right]^{\frac{1}{4}}.$$

As can be seen, the critical width is a function of the wavelength, aberration strength, and Hermite-Gauss beam indices $n,m$. When $n = m$, the critical width along $x$ and $y$ is the same. However, for other combinations of indices, the two critical widths are not necessarily the same. As an example, it can be seen, in Fig. 2, that for the Hermite-Gauss beam with indices $n = 0, m = 1$, $M^2_y$ deviates before $M^2_x$. For the Hermite-Gauss beam with indices $n = 1, m = 0$, the opposite is true. From Eq. (38) and Eq. (39), it is easy to see that the ratio of the aberration-free beam quality factors along the $x$ and $y$ directions determines the critical width. The effect of the aberration strength is to shift the critical width for all the beams. Visual inspection of Fig. 2 shows that changing the aberration strength translates all the Hermite-Gauss beams to a different region along the $x-$axis. The overall behavior for all the Hermite-Gauss beams, however, remains unchanged.

The beam quality factor for radius values that are significantly larger than the critical width, $\omega _c \ll \omega _0$, changes sharply. In this region, Eq. (34) and Eq. (36) simplify to the following expressions,

$$M^2_{x,y} = \frac{ \pi \omega^2_0 D_{\text{ast45}}}{\lambda } \sqrt{\left(2n + 1 \right)\left(2m + 1 \right)}.$$

This shows that the beam quality factors along $x$ and $y$ for a particular Hermite-Gauss beam are the same in this region. Furthermore, since Fig. 2 is a log-log plot, Eq. (40) appears as a linear relationship between the beam quality factor and the beam width. The slope of the line is given by the exponent of the beam radius. A linear plot of the beam quality factor in this region would be a quadratic function of the beam radius.

3.1.2 Triangular astigmatism

A similar approach to that used for $45^\circ$ astigmatism is used for triangular astigmatism. The aberration-dependent terms of the second-order angular moment are calculated below using the expression for $x-$triangular astigmatism in Table 1,

$$\begin{aligned} \theta^{\text{trix}}_{2,x} &={-}\frac{9 D_{\text{trix}}^2 \mathcal{N}^2_{HG} \omega_0^6 \pi \exp\left(2\left[\mu_1 \mu_2 + \eta_1\eta_2 \right]\right)}{ 8 \lambda^2 } \\ & \times \left[ \left(\mu_1 + \mu_2\right)^2 + \left(\eta_1 + \eta_2 \right)^2 + 1\right.\\ & \left. + 2\eta_1 \mu_1 + 2 \eta_1 \mu_2 + 2 \eta_2 \mu_1 + 2 \eta_2 \mu_2 \right]\\ & \times \left[ \left(\mu_1 + \mu_2\right)^2 + \left(\eta_1 + \eta_2 \right)^2 + 1\right.\\ & \left. - \left(2\eta_1 \mu_1 + 2 \eta_1 \mu_2 + 2 \eta_2 \mu_1 + 2 \eta_2 \mu_2 \right) \right], \end{aligned}$$
and
$$\begin{aligned} \theta^{\text{trix}}_{2,y} &={-}\frac{9 D_{\text{trix}}^2 \mathcal{N}^2_{HG} \omega_0^6 \pi \exp\left(2\left[\mu_1 \mu_2 + \eta_1\eta_2 \right]\right)}{ 8 \lambda^2 } \\ & \times \left[ 2\left(\mu_1 + \mu_2\right)^2 + 1\right] \left[ 2\left(\eta_1 + \eta_2 \right)^2 + 1 \right], \end{aligned}$$
and
$$\begin{aligned} \theta^{\text{trix}}_{3,x} &={-}\frac{3 D_{\text{trix}} \mathcal{N}^2_{HG} \omega_0^4 \pi \exp\left(2\left[\mu_1 \mu_2 + \eta_1\eta_2 \right]\right) }{4 \lambda}\\ & \times \left[ \mu_1 + \eta_1 + \mu_2 + \eta_2\right]\left[ \mu_1 - \eta_1 + \mu_2 - \eta_2\right] , \end{aligned}$$
and
$$\begin{aligned} \theta^{\text{trix}}_{3,y} &= \frac{3 D_{\text{trix}} \mathcal{N}^2_{HG} \omega_0^4 \pi \exp\left(2\left[\mu_1 \mu_2 + \eta_1\eta_2 \right]\right) }{2 \lambda}\\ & \times \left[ \mu_1 + \mu_2\right]\left[ \eta_1 + \eta_2\right]. \end{aligned}$$

The first-order angular-spatial moment, for both directions, is given as follows,

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

Substitution of the above expressions, together with the expression for the second-order spatial moment into Eq. (2) and Eq. (3), the final expressions for the beam quality factor of Hermite-Gauss beams aberrated with $x-$triangular astigmatism become,

$$\begin{aligned} M^4_x &= \left(2n + 1 \right)\left[ \frac{9 \pi^2 D^2_{\text{trix}}\omega^6_0}{2 \lambda^2 } \right.\\ & \times \left. \left( n^2 + m^2 + n + m + 2 \right) \right.\\ & \left. + \left(2n + 1\right) \right], \end{aligned}$$
and
$$M^4_y = \left(2m + 1 \right)^2\left[ \frac{9 \pi^2 D^2_{\text{trix}}\omega^6_0}{ \lambda^2} \left(2n + 1\right) + 1 \right].$$

The beam quality factor for Hermite-Gauss beams that are aberrated with $y-$triangular astigmatism has a form similar to that of $x-$triangular astigmatism, except that the results for $M^4_x$ and $M^4_y$ are swapped around, with the appropriate replacements of $n$ and $m$. Therefore, we do not give the explicit expressions for the $y-$triangular astigmatism because they can be simply obtained from the above results.

Figure 3 shows the beam quality factor of selected Hermite-Gauss beams that are aberrated with $x$-triangular astigmatism as a function of beam radius. The solid lines represent the beam quality factor due to a coefficient of 0.1 $\text {cm}^{-2}$ and the dashed lines represent an aberration coefficient of 10 $\text {cm}^{-2}$. The critical widths are given below as follows,

$$\omega_{c,x} = \left[ \sqrt{\frac{2 \left(2n + 1\right)}{9}} \frac{ \lambda }{ \pi D_\text{trix} \Pi_x } \right]^\frac{1}{3},$$
and,
$$\omega_{c,y} = \left[ \sqrt{\frac{1}{9 \left(2n + 1\right)}} \frac{ \lambda }{ \pi D_\text{trix} } \right]^\frac{1}{3},$$
where,
$$\Pi^2_x = n^2 + m^2 + n + m + 2.$$

 figure: Fig. 3.

Fig. 3. Beam quality factor due to $x$-triangular astigmatism as a function of beam radius along the $x$ and $y$ direction for various $\text {HG}_{nm}$ modes. The solid lines illustrate an aberration coefficient of 0.1 $\text {cm}^{-2}$ while dotted lines illustrate a strength of 10 $\text {cm}^{-2}$.

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Here, unlike the case for $45^\circ$ astigmatism, the expressions do not have any symmetry. This implies that the relationship between the two critical widths is not as simple. While in the case of $45^\circ$ astigmatism, the two widths were the same for Hermite-Gauss beams with indices that are equal, $n = m$, this is not necessarily true for $x-$triangular astigmatism. However, the role of the aberration strength is still the same. The aberration strength translates the plots to a different region along the $x-$axis. It does not change the general behavior of the different Hermite-Gauss beams. Here, we also find that the beam quality factor, for beam radii that are significantly larger than the critical width, is not the same for $x$ and $y$, rather the beam quality factor reduces to the following expressions,

$$M^2_x =\sqrt{\frac{9 \left(2n + 1\right) }{2}} \frac{ \pi \omega^3_0 D_\text{trix} \Pi_x}{ \lambda},$$
and,
$$M^2_y = \frac{ \sqrt{9 \left(2n + 1\right) } \pi \omega^3_0 D_\text{trix} \left(2m+ 1\right)}{ \lambda}.$$

3.1.3 Spherical aberration

The effect of spherical aberration, here, is evaluated by considering a quartic phase. The expression for the quartic phase is shown in Table 1. The generating expressions for the aberration-dependent terms are very complicated and we do not give them here. However, it is not a complicated task to generate them by following the same approach as above for the different types of astigmatism. The final expressions for the beam quality factor when the Hermite-Gauss beam is aberrated with a quartic phase are given as follows,

$$\begin{aligned} M^4_{x} &= \frac{2 \pi^2 D^2_{\text{sp}}\omega^8_0}{\lambda^2 }\left[ 2n^4 + 4n^3 \right.\\ & \left. + \left( 4m^2 + 4m + 21 \right)n^2 \right.\\ & \left. + \left( 4m^2 + 4m + 23\right)n \right.\\ & \left. + \left( m^2 + m + 4\right) \right]\\ & + \left( 2n + 1\right)^2, \end{aligned}$$
$$\begin{aligned} M^4_{y} &= \frac{2 \pi^2 D^2_{\text{sp}}\omega^8_0}{\lambda^2 }\left[ 2m^4 + 4m^3 \right.\\ & \left. + \left( 4n^2 + 4n + 23 \right)m^2 \right.\\ & \left. + \left( 4n^2 + 4n + 21\right)m \right.\\ & \left. + \left( n^2 + n + 4\right) \right]\\ & + \left( 2m + 1\right)^2. \end{aligned}$$

Figure 4 shows the beam quality factor of selected Hermite-Gauss beams with spherical aberration as a function of beam radius. The solid lines represent the beam quality factor due to an aberration coefficient of 0.1 $\text {cm}^{-3}$ and the dashed lines represent an aberration coefficient of 10 $\text {cm}^{-3}$. The critical width for the spherical aberration is given below as,

$$\omega_{c,x} = \left[ \frac{\lambda \left( 2n + 1 \right) }{\sqrt{2}\pi D_\text{sp} \Lambda_x } \right]^\frac{1}{4},$$
and
$$\omega_{c,y} = \left[ \frac{\lambda \left( 2m + 1 \right) }{\sqrt{2}\pi D_\text{sp} \Lambda_y } \right]^\frac{1}{4},$$
where,
$$\begin{aligned}\Lambda^2_x &= 2n^4 + 4n^3 + (4m^2 + 4m + 21)n^2\\ & + (4m^2 + 4m + 23)n + m^2 + m + 4, \end{aligned}$$
and
$$\begin{aligned}\Lambda^2_y &= 2m^4 + 4m^3 + (4n^2 + 4n + 21)m^2\\ & + (4n^2 + 4n + 23)m + n^2 + n + 4. \end{aligned}$$

The expression for the beam quality factor in the region of drastic growth is given as,

$$M^2_{x} = \frac{\sqrt{2} \pi \omega^4_0 D_{\text{sp}} \Lambda_x }{\lambda },$$
and
$$M^2_{y} = \frac{\sqrt{2} \pi \omega^4_0 D_{\text{sp}} \Lambda_y }{\lambda }.$$

 figure: Fig. 4.

Fig. 4. Beam quality factor due to spherical aberration as a function of beam radius along the $x$ and $y$ direction for various $\text {HG}_{nm}$ beams. The solid lines illustrate an aberration coefficient of $0.1 \text {cm}^{-3}$ while the dashed lines illustrate an aberration coefficient of $10 \text {cm}^{-3}$.

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A linear plot of Eq. (60) and Eq. (61) would give a quartic relationship between the beam quality factor and the beam radius. However, Fig. 4 is a loglog plot, therefore, the relationship between $M^2_{x,y}$ and $\omega _0$ in this region is linear, with the slope of the line given by the exponent of the $\omega _0$ in Eq. (60) and Eq. (61). From visual inspection, it can be seen that the slope of the lines in Fig. 4 is steeper than that of the lines in Fig. 2 and Fig. 3, as is expected based on the exponents of $\omega _0$ in the respective equations for $M^2_x$ [ Eq. (40), Eq. (52), Eq. (60) ], for instance.

4. Numerical simulations

4.1 Numerical method

The beam propagation simulation in this study utilizes the angular spectrum method, a widely used technique for analyzing the propagation behavior of optical beams. The simulation process involves several steps, beginning with the computation of the Fourier transform of the input function, which is represented by the equation,

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

Here, $u(x,y)$ represents the input function that characterizes the optical field. The variables $a$ and $b$ correspond to the spatial frequency components in the Fourier domain.

Next, the angular spectrum obtained from the Fourier transform is multiplied by the propagation phase factor given by,

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

In this equation, $z$ represents the propagation distance, and $\lambda$ denotes the wavelength of the light. To obtain the optical field at a specific position along the propagation axis ($z$), an inverse Fourier transform is performed on the product of the angular spectrum and the Fourier transform,

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

To assess the quality factor of the beam, the method proposed by Siegman [1] is employed. This approach measures the radius of the beam along the propagation axis and compares the results with a quadratic equation. The equation used for comparison is given as follows,

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

Here, $\omega ^2(z)$ represents the beam radius squared at a specific position $z$ along the propagation axis. The parameters $M^2$, $\omega _0$, $z_0$, and $\lambda$ correspond to the beam quality factor, beam waist radius, beam waist position, and wavelength, respectively. The data obtained from the simulations is then fitted with a quadratic polynomial of the form,

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

Finally, the beam quality factor ($M^2$) is calculated using the formula,

$$M^2 = \frac{\pi}{\lambda}\sqrt{AC - \frac{B^2}{4}},$$
where, $A$, $B$, and $C$ can be obtained by comparing Eq. (65) and Eq. (66).

4.2 Numerical validation

The analytical expressions derived in Sec. 3 are validated in this section using the numerical method of Sec 4. All the results are generated using $\lambda = 680 \text {nm}$, two beam radii $1\text {cm}$ and $3\text {cm}$. The solid lines represent the beam quality as per analytical expression in Sec. 3 and the discrete markers represent the beam quality obtained through numerical simulation. For validation, we only consider the beam quality factor along the $x$ direction.

Figure 5 shows the beam quality factor of selected Hermite-Gauss (HG) optical beams which are aberrated with $45^\circ$ astigmatism. There is excellent agreement between the analytical and numerical results for both beam radii. In Fig. 5(a), the change in the beam quality factor is not sharp. However, the situation is different in Fig. 5(b). The change in the beam quality factor with increasing astigmatism strength is steep. For instance, the beam quality factor for the Hermite-Gauss mode with indices $n = 1, m = 0$ starts at $M^2 = 1$ when there is no astigmatism and becomes just below $40$ when the astigmatism strength is $D_\text {ast} = 5 \text {cm}^{-1}$. The increase in the beam quality factor is seen to be proportional to the un-aberrated beam quality factor. The most drastic increase at $D_\text {ast} = 5 \text {cm}^{-1}$ corresponds to the Hermite-Gauss beam with indices $n = 3, m = 2$.

 figure: Fig. 5.

Fig. 5. The graph depicts how the beam quality factor changes with 45$^{\circ}$ astigmatism for different HG beams. The plots show the results for selected HG beams with (a) a beam waist of 1 cm and (b) a beam waist of 3 cm. The solid lines illustrate the predicted beam quality factor using analytical methods, while the discrete markers indicate the results obtained from numerical simulations.

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Figure 6 illustrates the beam quality factor due to $x$-triangular astigmatism for various Hermite-Gauss beams. In Fig. 6(a), we observe the impact of $x$-triangular astigmatism when the Hermite-Gauss beams have a beam radius of 1 $\text {cm}$. In Fig. 6(a), we observe the impact of $x$-triangular astigmatism when the Hermite-Gauss beams have a beam radius of 3 $\text {cm}$. Apart from some differences, particularly quantitative differences, the overall quantitative behaviour is comparable to the case of $45^\circ$ astigmatism. That is, the beam quality factor change is not sharp for the beams with radius 1 $\text {cm}$, but the change becomes very drastic at the beam radius given by 3 $\text {cm}$.

 figure: Fig. 6.

Fig. 6. The graph depicts how the beam quality factor changes with x-triangular astigmatism for different HG beams. The plots show the results for selected HG beams with (a) a beam waist of 1 cm and (b) a beam waist of 3 cm. The solid lines illustrate the predicted beam quality factor using analytical methods, while the discrete markers indicate the results obtained from numerical simulations.

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The effect of spherical aberration on the beam quality factor of Hermite-Gauss beams is shown in Fig. 7. For Hermite-Gauss beams with a radius of 1 $\text {cm}$, there is no effect on the beam quality as can be seen in Fig. 7(a). However, the same is not true for the Hermite-Gauss beams with a radius of 3 $\text {cm}$. The beam quality factor is affected sharply and the amount by which it changes is dependent on the particular Hermite-Gauss beam.

 figure: Fig. 7.

Fig. 7. The graph depicts how the beam quality factor changes with spherical aberration for different HG beams. The plots show the results for selected HG beams with (a) a beam waist of 1 cm and (b) a beam waist of 3 cm. The solid lines illustrate the predicted beam quality factor using analytical methods, while the discrete markers indicate the results obtained from numerical simulations.

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5. Summary

In this work, we explored the concept of beam quality factor and its relationship to aberrations in optical beams. Particularly, we focused on different types of astigmatism: $0^{\circ}$ and $45^{\circ}$, triangular astigmatism, and spherical aberration. We found that $0^{\circ}$ astigmatism has no effect on the beam quality factor of Hermite-Gauss beams. Triangular and $45^{\circ}$ astigmatism have an effect on the beam quality factor of the Hermite-Gauss beams. In the case of triangular astigmatism, we found that the effect of $x-$triangular astigmatism on the beam quality factor along the $x$ direction is the same as the effect of $y-$triangular astigmatism on the beam quality factor along the $y$ direction, and vice-versa. We also found that spherical aberration has an effect on the beam quality factor of Hermite-Gauss beams. In all cases, we found that the beam quality factor has the same qualitative behavior. The aberrations have a negligible effect up to some beam radius, where after the beam quality factor starts to deviate sharply from the aberration-free case. Based on this observation, we further derived a radius that can be used as a boundary to separate the region of minimal effect and the region of sharp increase. We call the derived beam radius the critical width. We found that the critical width depends on the strength of the aberrations, and also on the wavelength and the indices of the Hermite-Gauss beam. Furthermore, given that the Hermite-Gauss beams are not circularly symmetric, we found that the critical width is in general different for the beam quality along the $x$ and the beam quality factor along the $y$ direction. Overall, the findings of this article highlight the importance of considering aberrations in optical beams and their effects on the beam quality factor. By understanding and quantifying these effects, it becomes possible to optimize optical systems and ensure the delivery of high-quality beams for various applications.

Acknowledgments

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

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

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

4. W. T. Buono, C. Peters, J. Tau, I. Nape, and A. Forbes, “Eigenmodes of aberrated systems: the tilted lens,” J. Opt. 24(12), 125602 (2022). [CrossRef]  

5. J. Trevino, V. Coello, A. Jaimes-Nájera, C. Garcia-Ortiz, S. Chávez-Cerda, and J. Gómez-Correa, “Direct observation of longitudinal aberrated wavefields,” Photonics Res. 11(6), 1015–1020 (2023). [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. 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]  

8. 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]  

9. 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]  

10. C. M. Mabena, T. Bell, N. Mphuthi, J. Harrison, and D. Naidoo, “Beam quality factor of aberrated laguerre-gaussian optical beams,” Opt. Express 31(16), 26435–26450 (2023). [CrossRef]  

11. C. M. Mabena, N. Mphuthi, L. Bell, and D. Naidoo, “Effect of astigmatism on the beam quality factor of hermite-gauss laser beams,” Proc. SPIE 12664, 126640L (2023). [CrossRef]  

12. Y. Qiu, L. Huang, M. Gong, L. Qiang, P. Yan, and H. Zhang, “Method to evaluate beam quality of gaussian beams with aberrations,” Appl. Opt. 51(27), 6539–6543 (2012). [CrossRef]  

13. C. M. Mabena, “Analytical analysis of the beam propagation factor of elegant hermite-gaussian and elegant laguerre-gaussian beams with astigmatism,” Opt. Commun. 550, 129933 (2024). [CrossRef]  

14. S. Zhang and L. Yi, “Two-dimensional hermite–gaussian solitons in strongly nonlocal nonlinear medium with rectangular boundaries,” Opt. Commun. 282(8), 1654–1658 (2009). [CrossRef]  

15. P. Wang, Y. Ho, C. X. Tang, and W. Wang, “Field structure and electron acceleration in a laser beam of a high-order hermite-gaussian mode,” J. Appl. Phys. 101(8), 083113 (2007). [CrossRef]  

16. H. Zhang, S. Liu, and H. Guo, “Direct acceleration of electrons by intense crossed hermite–gaussian laser beams,” Phys. Lett. A 367(4-5), 402–407 (2007). [CrossRef]  

17. Z. Zhao, D. Yang, and B. Lü, “Electron acceleration by a tightly focused hermite-gaussian beam: higher-order corrections,” J. Opt. Soc. Am. B 25(3), 434–442 (2008). [CrossRef]  

18. T. Meyrath, F. Schreck, J. Hanssen, C.-S. Chuu, and M. Raizen, “A high frequency optical trap for atoms using hermite-gaussian beams,” Opt. Express 13(8), 2843–2851 (2005). [CrossRef]  

19. A. Porfirev and R. Skidanov, “Optical trapping and manipulation of light-absorbing particles by means of a hermite–gaussian laser beam,” J. Opt. Technol. 82(9), 587–591 (2015). [CrossRef]  

20. J. Su, N. Li, J. Mou, Y. Liu, X. Chen, and H. Hu, “Simultaneous trapping of two types of particles with focused elegant third-order hermite–gaussian beams,” Micromachines 12(7), 769 (2021). [CrossRef]  

21. C. Y. Young, Y. V. Gilchrest, and B. Macon, “Turbulence induced beam spreading of higher order mode optical waves,” Opt. Eng. 41(5), 1097–1103 (2002). [CrossRef]  

22. A. Amphawan, S. Chaudhary, and T.-K. Neo, “Hermite-gaussian mode division multiplexing for free-space optical interconnects,” Adv. Sci. Lett. 21(10), 3050–3053 (2015). [CrossRef]  

23. B. Ndagano, N. Mphuthi, G. Milione, and A. Forbes, “Comparing mode-crosstalk and mode-dependent loss of laterally displaced orbital angular momentum and hermite–gaussian modes for free-space optical communication,” Opt. Lett. 42(20), 4175–4178 (2017). [CrossRef]  

24. K. Kiasaleh, “Spatial beam tracking for hermite–gaussian-based free-space optical communications,” Opt. Eng. 56(7), 076106 (2017). [CrossRef]  

25. M. A. Cox, L. Maqondo, R. Kara, G. Milione, L. Cheng, and A. Forbes, “The resilience of hermite–and laguerre–gaussian modes in turbulence,” J. Lightwave Technol. 37(16), 3911–3917 (2019). [CrossRef]  

26. M. Singh and J. Malhotra, “2× 10 gbit/s–10 ghz radio over free space optics transmission system incorporating mode division multiplexing of hermite gaussian modes,” J. Opt. Commun. 44(4), 495–503 (2023). [CrossRef]  

27. J. Ugalde-Ontiveros, A. Jaimes-Nájera, S. Luo, J. Gómez-Correa, J. Pu, and S. Chávez-Cerda, “What are the traveling waves composing the hermite-gauss beams that make them structured wavefields?” Opt. Express 29(18), 29068–29081 (2021). [CrossRef]  

28. K. Y. Bliokh, E. Karimi, M. J. Padgett, et al., “Roadmap on structured waves,” J. Opt. 25(10), 103001 (2023). [CrossRef]  

29. A. Jaimes-Nájera, J. E. Gomez-Correa, J. Ugalde-Ontiveros, H. Mendez-Dzul, M. Iturbe-Castillo, and S. Chávez-Cerda, “On the physical limitations of structured paraxial beams with orbital angular momentum,” J. Opt. 24(10), 104004 (2022). [CrossRef]  

30. S. Saghafi and C. Sheppard, “The beam propagation factor for higher order gaussian beams,” Opt. Commun. 153(4-6), 207–210 (1998). [CrossRef]  

31. G. Yang, L. Liu, Z. Jiang, J. Guo, and T. Wang, “The effect of beam quality factor for the laser beam propagation through turbulence,” Optik 156, 148–154 (2018). [CrossRef]  

32. Y. Yuan, Y. Cai, J. Qu, H. T. Eyyuboğlu, and Y. Baykal, “Propagation factors of hermite–gaussian beams in turbulent atmosphere,” Opt. Laser Technol. 42(8), 1344–1348 (2010). [CrossRef]  

33. Y.-P. Huang and A.-P. Zeng, “Propagation properties of hermite-gaussian beams in non-kolmogorov turbulence,” Acta Photonica Sinica 41, 818–823 (2012). [CrossRef]  

34. Y.-F. Chen, T. Huang, C. Kao, C. Wang, and S. Wang, “Generation of hermite-gaussian modes in fiber-coupled laser-diode end-pumped lasers,” IEEE J. Quantum Electron. 33(6), 1025–1031 (1997). [CrossRef]  

35. S.-C. Chu, Y.-T. Chen, K.-F. Tsai, and K. Otsuka, “Generation of high-order hermite-gaussian modes in end-pumped solid-state lasers for square vortex array laser beam generation,” Opt. Express 20(7), 7128–7141 (2012). [CrossRef]  

36. W. Kong, A. Sugita, and T. Taira, “Generation of hermite–gaussian modes and vortex arrays based on two-dimensional gain distribution controlled microchip laser,” Opt. Lett. 37(13), 2661–2663 (2012). [CrossRef]  

37. M. L. Lukowski, J. T. Meyer, C. Hessenius, E. M. Wright, and M. Fallahi, “High-power higher order hermite–gaussian and laguerre–gaussian beams from vertical external cavity surface emitting lasers,” IEEE J. Sel. Top. Quantum Electron. 25(6), 1–6 (2019). [CrossRef]  

38. F. Schepers, T. Bexter, T. Hellwig, and C. Fallnich, “Selective hermite–gaussian mode excitation in a laser cavity by external pump beam shaping,” Appl. Phys. B 125(5), 75 (2019). [CrossRef]  

39. J. Ugalde-Ontiveros, A. Jaimes-Nájera, J. Gómez-Correa, and S. Chávez-Cerda, “Siegman’s elegant laser resonator modes,” Opt. Laser Technol. 143, 107340 (2021). [CrossRef]  

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

41. 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]  

42. M. W. Beijersbergen, L. Allen, H. Van der Veen, and J. Woerdman, “Astigmatic laser mode converters and transfer of orbital angular momentum,” Opt. Commun. 96(1-3), 123–132 (1993). [CrossRef]  

43. A. Alekseev, K. Alekseev, O. Borodavka, A. Volyar, and Y. A. Fridman, “Conversion of hermite-gaussian and laguerre-gaussian beams in an astigmatic optical system. 1. experiment,” Tech. Phys. Lett. 24(9), 694–696 (1998). [CrossRef]  

44. Y.-X. Ren, Z.-X. Fang, L. Gong, K. Huang, Y. Chen, and R.-D. Lu, “Digital generation and control of hermite–gaussian modes with an amplitude digital micromirror device,” J. Opt. 17(12), 125604 (2015). [CrossRef]  

45. A. S. Larkin, D. V. Pushkarev, S. A. Degtyarev, S. N. Khonina, and A. B. Savel’ev, “Generation of hermite–gaussian modes of high-power femtosecond laser radiation using binary-phase diffractive optical elements,” Quantum Electron. 46(8), 733 (2016). [CrossRef]  

46. Y. Zhou, J. Zhao, Z. Shi, S. M. H. Rafsanjani, M. Mirhosseini, Z. Zhu, A. E. Willner, and R. W. Boyd, “Hermite–gaussian mode sorter,” Opt. Lett. 43(21), 5263–5266 (2018). [CrossRef]  

47. G. Wang, Y. Li, X. Shan, Y. Miao, and X. Gao, “Hermite–gaussian beams with sinusoidal vortex phase modulation,” Chin. Opt. Lett. 18, 042601 (2020). [CrossRef]  

48. M. Yan and L. Ma, “Generation of higher-order hermite–gaussian modes via cascaded phase-only spatial light modulators,” Mathematics 10(10), 1631 (2022). [CrossRef]  

49. V. Arrizón, U. Ruiz, R. Carrada, and L. A. González, “Pixelated phase computer holograms for the accurate encoding of scalar complex fields,” J. Opt. Soc. Am. A 24(11), 3500–3507 (2007). [CrossRef]  

50. K. C. Vermeulen, G. J. Wuite, G. J. Stienen, and C. F. Schmidt, “Optical trap stiffness in the presence and absence of spherical aberrations,” Appl. Opt. 45(8), 1812–1819 (2006). [CrossRef]  

51. N. Viana, M. Rocha, O. Mesquita, A. Mazolli, P. M. Neto, and H. Nussenzveig, “Towards absolute calibration of optical tweezers,” Phys. Rev. E 75(2), 021914 (2007). [CrossRef]  

52. J. Ferri, X. Davoine, S. Fourmaux, J. Kieffer, S. Corde, K. Ta Phuoc, and A. Lifschitz, “Effect of experimental laser imperfections on laser wakefield acceleration and betatron source,” Sci. Rep. 6(1), 27846 (2016). [CrossRef]  

53. M. J. Bastiaans and T. Alieva, “Generating function for hermite–gaussian modes propagating through first-order optical systems,” J. Phys. A: Math. Gen. 38(6), L73 (2005). [CrossRef]  

54. A. Wünsche, “Hermite and laguerre 2d polynomials,” J. Comput. Appl. Math. 133(1-2), 665–678 (2001). [CrossRef]  

55. N. Hodgson and H. Weber, Laser Resonators and Beam Propagation: Fundamentals, Advanced Concepts, Applications, vol. 108 (Springer, 2005).

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Figures (7)
Fig. 1.
Fig. 1. Intensity profiles of selected Hermite-Gauss beams with indices $n= 0, 1,2,3$ and $m=0, 1, 2$ at the waist plane. The insets represent the phase of the beams.

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Fig. 2.
Fig. 2. Beam quality factor due to $45^{\circ}$ astigmatism as a function of beam radius along the $x$ and $y$ direction for various $\text {HG}_{nm}$ beams. The solid lines illustrate an astigmatism coefficient of 0.1 $\text {cm}^{-1}$ while dotted lines illustrate an astigmatism coefficient of 10 $\text {cm}^{-1}$.

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Fig. 3.
Fig. 3. Beam quality factor due to $x$-triangular astigmatism as a function of beam radius along the $x$ and $y$ direction for various $\text {HG}_{nm}$ modes. The solid lines illustrate an aberration coefficient of 0.1 $\text {cm}^{-2}$ while dotted lines illustrate a strength of 10 $\text {cm}^{-2}$.

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Fig. 4.
Fig. 4. Beam quality factor due to spherical aberration as a function of beam radius along the $x$ and $y$ direction for various $\text {HG}_{nm}$ beams. The solid lines illustrate an aberration coefficient of $0.1 \text {cm}^{-3}$ while the dashed lines illustrate an aberration coefficient of $10 \text {cm}^{-3}$.

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Fig. 5.
Fig. 5. The graph depicts how the beam quality factor changes with 45$^{\circ}$ astigmatism for different HG beams. The plots show the results for selected HG beams with (a) a beam waist of 1 cm and (b) a beam waist of 3 cm. The solid lines illustrate the predicted beam quality factor using analytical methods, while the discrete markers indicate the results obtained from numerical simulations.

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Fig. 6.
Fig. 6. The graph depicts how the beam quality factor changes with x-triangular astigmatism for different HG beams. The plots show the results for selected HG beams with (a) a beam waist of 1 cm and (b) a beam waist of 3 cm. The solid lines illustrate the predicted beam quality factor using analytical methods, while the discrete markers indicate the results obtained from numerical simulations.

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Fig. 7.
Fig. 7. The graph depicts how the beam quality factor changes with spherical aberration for different HG beams. The plots show the results for selected HG beams with (a) a beam waist of 1 cm and (b) a beam waist of 3 cm. The solid lines illustrate the predicted beam quality factor using analytical methods, while the discrete markers indicate the results obtained from numerical simulations.

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Tables (1)

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Table 1. Algebraic expressions for some primary aberrations.

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Equations (67)

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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 .
H ( x , y , z ) = E 0 ω 0 ω ( z ) H n ( 2 x ω ( z ) ) H m ( 2 y ω ( z ) ) exp ⁡ ( − x 2 + y 2 ω 2 ( z ) ) × exp ⁡ ( − i π [ x 2 + y 2 ] λ R ( z ) ) exp ⁡ ( i ψ ( z ) ) exp ⁡ ( − i 2 π z λ ) ,
ω ( z ) = ω 0 1 + ( z z R ) 2 ,
R ( z ) = z [ 1 + ( z R z ) 2 ] .
ψ ( z ) = ( n + m + 1 ) arctan ( z z R ) ,
z R = 1 ( m + n + 1 ) π ω 0 2 λ ,
H = exp ⁡ [ 2 2 x μ ω 0 + 2 2 y η ω 0 − ( μ 2 + η 2 ) ] × exp ⁡ [ − ( x 2 + y 2 ) ω 0 2 ] ,
HG n , m = N HG [ ∂ m ∂ η m ∂ n ∂ μ n H ] μ , η = 0 ,
N HG = 1 π 2 n + m − 1 n ! m ! 1 ω 0 .
⟨ x 2 ⟩ μ , η = N HG 2 ∫ ∞ − ∞ ∫ ∞ − ∞ x 2 exp ⁡ { 2 2 [ ( μ 1 + μ 2 ) x + y ( η 1 + η 2 ) ] ω 0 } × exp ⁡ { [ − η 1 2 − η 2 2 − μ 1 2 − μ 2 2 ] ω 0 2 − 2 [ x 2 + y 2 ] ω 0 2 } d x d y = N HG 2 ω 0 4 π [ 2 ( μ 1 + μ 2 ) 2 + 1 ] exp ⁡ { 2 [ μ 1 μ 2 + η 1 η 2 ] } 8 ,
⟨ y 2 ⟩ μ , η = N HG 2 ω 0 4 π exp ⁡ ( 2 [ μ 1 μ 2 + η 1 η 2 ] ) 8 × [ 2 ( η 1 + η 2 ) 2 + 1 ] .
θ 1 , x = 1 4 π 2 ∫ − ∞ ∞ ∫ − ∞ ∞ ( ∂ u ∂ x ) 2 d x d y ,
θ 2 , x = 1 4 π 2 ∫ − ∞ ∞ ∫ − ∞ ∞ ( u ∂ ϕ ∂ x ) 2 d x d y ,
θ 3 , x = 1 2 π ∫ − ∞ ∞ ∫ − ∞ ∞ u 2 ∂ ϕ ∂ x d x d y ,
⟨ θ x 2 ⟩ = θ 1 , x + θ 2 , x − θ 3 , x 2 ,
⟨ θ y 2 ⟩ = θ 1 , y + θ 2 , y − θ 3 , y 2 .
θ 1 , x = N HG 2 exp ⁡ ( 2 [ μ 1 μ 2 + η 1 η 2 ] ) 8 π × [ 2 ( μ 1 − μ 2 ) 2 − 1 ] ,
θ 1 , y = N HG 2 exp ⁡ ( 2 [ μ 1 μ 2 + η 1 η 2 ] ) 8 π × [ 2 ( η 1 − η 2 ) 2 − 1 ] .
θ 2 , x ast45 = D ast45 2 N HG 2 ω 0 4 π exp ⁡ ( 2 [ μ 1 μ 2 + η 1 η 2 ] ) 8 λ 2 × [ 2 ( η 1 + η 2 ) 2 + 1 ] ,
θ 2 , y ast45 = D ast45 2 N HG 2 ω 0 4 π exp ⁡ ( 2 [ μ 1 μ 2 + η 1 η 2 ] ) 8 λ 2 × [ 2 ( μ 1 + μ 2 ) 2 + 1 ] ,
θ 3 , x ast45 = D ast45 N HG 2 ω 0 3 π exp ⁡ ( 2 [ μ 1 μ 2 + η 1 η 2 ] ) 2 2 λ × [ η 1 + η 2 ] ,
θ 3 , y ast45 = D ast45 N HG 2 ω 0 3 π exp ⁡ ( 2 [ μ 1 μ 2 + η 1 η 2 ] ) 2 2 λ × [ μ 1 + μ 2 ] .
⟨ x θ x ⟩ ast45 = D ast45 N HG 2 ω 0 4 π exp ⁡ ( 2 [ μ 1 μ 2 + η 1 η 2 ] ) 4 λ × [ η 1 + η 2 ] [ μ 1 + μ 2 ] .
⟨ y θ y ⟩ ast45 = ⟨ x θ x ⟩ ast45 .
M x 4 = 16 π 2 { [ ∂ m ∂ η 1 m ∂ m ∂ η 2 m ∂ n ∂ μ 1 n ∂ n ⟨ x 2 ⟩ μ , η ∂ μ 2 n ] η 1 , η 2 , μ 1 , μ 2 = 0 ( [ ∂ m ∂ η 1 m ∂ m ∂ η 2 m ∂ n ∂ μ 1 n ∂ n θ 1 , x ast45 ∂ μ 2 n ] η 1 , η 2 , μ 1 , μ 2 = 0 + [ ∂ m ∂ η 1 m ∂ m ∂ η 2 m ∂ n ∂ μ 1 n ∂ n θ 2 , x ast45 ∂ μ 2 n ] η 1 , η 2 , μ 1 , μ 2 = 0 − [ ∂ m ∂ η 1 m ∂ m ∂ η 2 m ∂ n ∂ μ 1 n ∂ n θ 3 , x ast45 ∂ μ 2 n ] η 1 , η 2 , μ 1 , μ 2 = 0 2 ) − [ ∂ m ∂ η 1 m ∂ m ∂ η 2 m ∂ n ∂ μ 1 n ∂ n ⟨ x θ x ⟩ ast45 ∂ μ 2 n ] η 1 , η 2 , μ 1 , μ 2 = 0 2 } .
M x 4 = ( 2 n + 1 ) [ π 2 D ast45 2 ω 0 4 λ 2 ( 2 m + 1 ) + ( 2 n + 1 ) ] .
M y 4 = 16 π 2 { [ ∂ m ∂ η 1 m ∂ m ∂ η 2 m ∂ n ∂ μ 1 n ∂ n ⟨ y 2 ⟩ μ , η ∂ μ 2 n ] η 1 , η 2 , μ 1 , μ 2 = 0 ( [ ∂ m ∂ η 1 m ∂ m ∂ η 2 m ∂ n ∂ μ 1 n ∂ n θ 1 , y ast45 ∂ μ 2 n ] η 1 , η 2 , μ 1 , μ 2 = 0 + [ ∂ m ∂ η 1 m ∂ m ∂ η 2 m ∂ n ∂ μ 1 n ∂ n θ 2 , y ast45 ∂ μ 2 n ] η 1 , η 2 , μ 1 , μ 2 = 0 − [ ∂ m ∂ η 1 m ∂ m ∂ η 2 m ∂ n ∂ μ 1 n ∂ n θ 3 , y ast45 ∂ μ 2 n ] η 1 , η 2 , μ 1 , μ 2 = 0 2 ) − [ ∂ m ∂ η 1 m ∂ m ∂ η 2 m ∂ n ∂ μ 1 n ∂ n ⟨ y θ y ⟩ ast45 ∂ μ 2 n ] η 1 , η 2 , μ 1 , μ 2 = 0 2 } ,
M y 4 = ( 2 m + 1 ) [ π 2 D ast45 2 ω 0 4 λ 2 ( 2 n + 1 ) + ( 2 m + 1 ) ] .
M x 4 = ( 2 n + 1 ) 2 [ π 2 D ast45 2 ω 0 4 ( 2 m + 1 ) λ 2 ( 2 n + 1 ) + 1 ] .
ω c,x = [ λ 2 ( 2 n + 1 ) π 2 D ast45 2 ( 2 m + 1 ) ] 1 4 .
ω c,y = [ λ 2 ( 2 m + 1 ) π 2 D ast45 2 ( 2 n + 1 ) ] 1 4 .
M x , y 2 = π ω 0 2 D ast45 λ ( 2 n + 1 ) ( 2 m + 1 ) .
θ 2 , x trix = − 9 D trix 2 N H G 2 ω 0 6 π exp ⁡ ( 2 [ μ 1 μ 2 + η 1 η 2 ] ) 8 λ 2 × [ ( μ 1 + μ 2 ) 2 + ( η 1 + η 2 ) 2 + 1 + 2 η 1 μ 1 + 2 η 1 μ 2 + 2 η 2 μ 1 + 2 η 2 μ 2 ] × [ ( μ 1 + μ 2 ) 2 + ( η 1 + η 2 ) 2 + 1 − ( 2 η 1 μ 1 + 2 η 1 μ 2 + 2 η 2 μ 1 + 2 η 2 μ 2 ) ] ,
θ 2 , y trix = − 9 D trix 2 N H G 2 ω 0 6 π exp ⁡ ( 2 [ μ 1 μ 2 + η 1 η 2 ] ) 8 λ 2 × [ 2 ( μ 1 + μ 2 ) 2 + 1 ] [ 2 ( η 1 + η 2 ) 2 + 1 ] ,
θ 3 , x trix = − 3 D trix N H G 2 ω 0 4 π exp ⁡ ( 2 [ μ 1 μ 2 + η 1 η 2 ] ) 4 λ × [ μ 1 + η 1 + μ 2 + η 2 ] [ μ 1 − η 1 + μ 2 − η 2 ] ,
θ 3 , y trix = 3 D trix N H G 2 ω 0 4 π exp ⁡ ( 2 [ μ 1 μ 2 + η 1 η 2 ] ) 2 λ × [ μ 1 + μ 2 ] [ η 1 + η 2 ] .
⟨ x θ x ⟩ trix = − 3 D trix N H G 2 ω 0 5 π exp ⁡ ( 2 [ μ 1 μ 2 + η 1 η 2 ] ) 4 2 λ × [ ( μ 1 + μ 2 ) 2 − ( η 1 + η 2 ) 2 + 1 ] × [ μ 1 + μ 2 ] ,
⟨ y θ y ⟩ trix = 3 D trix N H G 2 ω 0 5 π exp ⁡ ( 2 [ μ 1 μ 2 + η 1 η 2 ] ) 4 2 λ × [ 2 ( η 1 + η 2 ) 2 + 1 ] [ μ 1 + μ 2 ] .
M x 4 = ( 2 n + 1 ) [ 9 π 2 D trix 2 ω 0 6 2 λ 2 × ( n 2 + m 2 + n + m + 2 ) + ( 2 n + 1 ) ] ,
M y 4 = ( 2 m + 1 ) 2 [ 9 π 2 D trix 2 ω 0 6 λ 2 ( 2 n + 1 ) + 1 ] .
ω c , x = [ 2 ( 2 n + 1 ) 9 λ π D trix Π x ] 1 3 ,
ω c , y = [ 1 9 ( 2 n + 1 ) λ π D trix ] 1 3 ,
Π x 2 = n 2 + m 2 + n + m + 2.
M x 2 = 9 ( 2 n + 1 ) 2 π ω 0 3 D trix Π x λ ,
M y 2 = 9 ( 2 n + 1 ) π ω 0 3 D trix ( 2 m + 1 ) λ .
M x 4 = 2 π 2 D sp 2 ω 0 8 λ 2 [ 2 n 4 + 4 n 3 + ( 4 m 2 + 4 m + 21 ) n 2 + ( 4 m 2 + 4 m + 23 ) n + ( m 2 + m + 4 ) ] + ( 2 n + 1 ) 2 ,
M y 4 = 2 π 2 D sp 2 ω 0 8 λ 2 [ 2 m 4 + 4 m 3 + ( 4 n 2 + 4 n + 23 ) m 2 + ( 4 n 2 + 4 n + 21 ) m + ( n 2 + n + 4 ) ] + ( 2 m + 1 ) 2 .
ω c , x = [ λ ( 2 n + 1 ) 2 π D sp Λ x ] 1 4 ,
ω c , y = [ λ ( 2 m + 1 ) 2 π D sp Λ y ] 1 4 ,
Λ x 2 = 2 n 4 + 4 n 3 + ( 4 m 2 + 4 m + 21 ) n 2 + ( 4 m 2 + 4 m + 23 ) n + m 2 + m + 4 ,
Λ y 2 = 2 m 4 + 4 m 3 + ( 4 n 2 + 4 n + 21 ) m 2 + ( 4 n 2 + 4 n + 23 ) m + n 2 + n + 4.
M x 2 = 2 π ω 0 4 D sp Λ x λ ,
M y 2 = 2 π ω 0 4 D sp Λ y λ .
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 ,
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