## Abstract

The analytical propagation formulae of Hermite-Gaussian (H-G) beams propagating in Kerr media are derived by using the variational approach. The analytical formulae of the self-focusing critical power, the Rayleigh range and the beam quality factor of H-G beams propagating in Kerr media are also derived. It is demonstrated that the ABCD law is valid if a new complex parameter is introduced, which presents a simple method to study the propagation of H-G beams through an optical system in Kerr media. It is shown that, as the beam order increases, the self-focusing critical power of H-G beams increase and H-G beams are less affected by the Kerr nonlinearity. Finally, it is found that the focus point of H-G beams never appears in self-focusing media, and the focal length is not suitable for characterizing the self-focusing effect of H-G beams.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

It is important to study the propagation of laser beams through nonlinear media. In 1964, Chiao et al. proposed a self-trapping model, which successfully explained the filamentation phenomenon when a laser beam propagates in the glass material [1]. In 1965, Kelley and Talanov presented the nonlinear Schrödinger equation, and obtained the critical power and the focal length of the steady-state self-focusing [2,3]. After that, the theory of nonlinear optics was studied widely and deeply [4,5]. It is noted that these studies only concerned about the nonlinearity of Gaussian (GS) laser beams [1–5]. On the other hand, Cornolti et al. examined the self-focusing of elliptic Gaussian (EGS) laser beams in nonlinear media, and showed that the critical power increases with ellipcity [6]. Singh et al. investigated the dynamics of self-focusing and self-phase modulation of EGS laser beam in Kerr media, and showed that EGS laser beam exhibits no stationary self-trapping [7]. In addition, very recently, our group studied the propagation characteristics of partially coherent laser beams in Kerr media, and our results can reduce to those of GS laser beams in Kerr media when the beam spatial coherence width is large enough [8].

Until now, a few studies involved the self-focusing effect of Hermite-Gaussian (H-G) laser beams in nonlinear media. The self-focusing effect of H-G laser beams in plasma was studied [9–11]. It is found that H-G beams show an oscillatory self-focusing and defocusing behavior with the propagation distance [11]. Based on the Snyder-Mitchell model, exact analytical H-G solutions in strongly nonlocal nonlinear media were obtained [12]. The propagation of scalar and vector H-G solitons in strongly nonlocal media with exponential-decay response was investigated, and the evolution equations were obtained [13]. However, these studies mentioned above concerned about H-G laser beams in plasma or strongly nonlocal nonlinear media [9–13].

In this paper, the propagation characteristics of H-G laser beams in local Kerr media is studied in detail. It is known that the variational approach is useful to study nonlinear effects (e.g., Kerr nonlinearities [7], thermal optical nonlinearities [14] and high-order optical nonlinearities [15]). By using the variational approach, we derive the analytical formulae (i.e., the beam width, the curvature radius, the phase, the self-focusing critical power, the Rayleigh range and the beam quality factor) of H-G beams propagating in Kerr media. Furthermore, the analytical formulae obtained in this paper can reduce to those of GS beams in Kerr media, and to those of H-G beams in linear media. It is known that the ABCD law is valid when H-G beams propagate in linear media. In this paper, we examine if the ABCD law is still valid when H-G beams propagate through optical systems in Kerr media. In addition, the features of Kerr nonlinearity of H-G beams are also investigated in this paper.

## 2. Propagation formulae and characteristics

Under the standard paraxial approximation, the key features of the beam evolution (diffraction and Kerr nonlinearity) in local Kerr media can be described by the nonlinear Schrödinger (NLS) equation, i.e [16],

*E*is the complex amplitude envelope of the electric field, $k=2\pi {n}_{0}/\lambda $ is the wave-number in linear media, $\lambda $ is the wave length,

*n*

_{2}and

*n*

_{0}are the nonlinear and linear refractive index respectively.

It is known that the beam profile will be retained when a H-G beam propagates in linear media. In this paper, our numerical results demonstrate that the beam profile for a situation without filamentation is close to H-G one when a H-G beam propagates in Kerr media, and then we may assume the solution of Eq. (1) is in the form as

*P*

_{0}is the beam power, H

*and H*

_{m}*denote*

_{n}*m*

^{th}and

*n*

^{th}order Hermite polynomials in

*x*and

*y*directions respectively. A H-G beam reduces to a GS beam when

*m*=

*n*= 0,

*w*(

_{x}*z*) and

*w*(

_{y}*z*) are the GS beam width in

*x*and

*y*directions respectively, and

*w*

_{0}is the GS beam width at the plane

*z*= 0.

*R*(

_{mx}*z*) and

*R*(

_{ny}*z*) are the curvature radius in

*x*and

*y*directions respectively, and ${\phi}_{mn}(z)$ is the phase function.

In this paper, we apply the variational approach to solve the NLS equation, namely, the NLS equation is expressed as a variational equation, and then the variational equation is solved by using the assumed solution (i.e., Eq. (2)) of the NLS equation.

The NLS Eq. (1) can be restated as a variational problem corresponding to the Lagrange density equation given by

Letting

*g*(

*m*) or

*g*(

*n*) increases as

*m*or

*n*increases, e.g., we have

*g*(0) = 1,

*g*(1) = 3 and

*g*(2) = 41.

According to the variational principle Eq. (4), and using Eq. (5), we arrive at a reduced variational problem with the following Euler-Lagrange equation, i.e.,

*S*(

*z*) denotes the beam parameters, e.g.,

*w*(

_{x}*z*),

*R*(

_{mx}*z*),

*w*(

_{y}*z*),

*R*(

_{ny}*z*) and ${\phi}_{mn}(z)$.

Considering the beam parameters denoted by *S*(*z*), and substituting Eq. (6) into Eq. (7), we obtain

*w*(

_{y}*z*) and

*R*(

_{ny}*z*) should satisfy can be obtained from Eqs. (8) and (9) if

*m*and

*n*,

*x*and

*y*are interchanged respectively.

From Eqs. (8)-(10), it can be seen that there exists coupling between the H-G beam parameters in *x* and *y* directions because of the Kerr nonlinearity, which is quite different from the behavior that in linear media.

It is mentioned that Eqs. (8)-(10) can be solved analytically only when *n* = *m* (i.e., *w _{x}*(

*z*) =

*w*(

_{y}*z*) =

*w*(

*z*),

*R*(

_{mx}*z*) =

*R*(

_{ny}*z*) =

*R*(

_{m}*z*) and ${\phi}_{mn}(z)={\phi}_{m}(z)$). The following derivations are all under this condition. According to Eqs. (8) and (9), we can obtain

Based on Eqs. (11), (8), (10), and the relationship *w _{m}*(

*z*) = (2

*m*+ 1)

^{1/2}

*w*(

*z*) (it is noted that

*w*(

_{m}*z*) is the H-G beam width), we can derive the analytical propagation formulae of

*w*(

_{m}*z*),

*R*(

_{m}*z*) and ${\phi}_{m}(z)$ after very tedious calculations, which are given by

*P*

_{cr}is the self-focusing critical power of H-G beams, which can reduce to that of GS beams when

*m*= 0 (i.e.,

*P*

_{cr GS}= 2

*n*

_{0}

*π*/(

*k*

^{2}

*n*

_{2})). It is mentioned that the analytical formulae (i.e., Eqs. (12)-(14)) are general, which can reduce to those of GS beams propagating in Kerr media when

*m*= 0, and to those of H-G beams propagating in linear media when $\eta =1$ (i.e., ${n}_{2}=0$). In addition, when $\eta <1$ (i.e., ${n}_{2}>0$) and $\eta >1$ (i.e., ${n}_{2}<0$), Eqs. (12)-(14) reduce to those of H-G beams propagating in self-focusing and self-defocusing media, respectively.

The validity of the analytical expression *w _{m}*(

*z*) (i.e., Eq. (12)) is confirmed by using numerical simulations (see Fig. 1). Equation (1) can also be solved numerically by using the multi-phase screen method, which is shown as follows. Let

*E*(

*x, y*,

*z*) be the solution of Eq. (1) at

_{j}*z*plane, and the solution of Eq. (1) at

_{j}*z*

_{j}_{+1}=

*z*+ Δ

_{j}*z*plane can be expressed as [17]

*z*nonlinear propagation. From Eq. (15), one can see that the propagating field over a distance Δ

*z*in Kerr media consists of three steps, i.e., a linear propagation over a distance Δ

*z*/2, then an incrementing of the phase caused by the Kerr effect within Δ

*z*nonlinear propagation, and a linear propagation over a distance Δ

*z*/2 again. Thus, we can design a computer code of the beam propagation in Kerr media by using the multi-phase screen method.

In this paper, the As_{2}S_{3} glass is adopted as the self-focusing medium (i.e., ${n}_{0}=2.4$, ${n}_{2}=2\times {10}^{-13}{\text{cm}}^{\text{2}}/\text{W}$) [18], the synthesized soluble polyoxadiazoles containing 3,4-dialkoxythiophenes is taken as the self-defocusing medium (i.e., ${n}_{0}=1.56$, ${n}_{2}=-1.5\times {10}^{-13}{\text{cm}}^{\text{2}}/\text{W}$) [19], and the other numerical calculation parameters are taken as ${R}_{0}\to \infty $, ${w}_{0}=0.1\text{mm}$, *λ* = 1.06μm in the self-focusing medium, and *λ* = 0.532μm in the self-defocusing medium. The changes of the beam width with *w _{m}*(

*z*) in the self-focusing medium versus the propagation distance

*z*are shown in Fig. 1. In Fig. 1, for

*m*= 0 and 1 cases,

*w*(

_{m}*z*) decreases as

*z*increases due to ${P}_{0}>{P}_{\text{cr}}$; for

*m*= 2 case,

*w*(

_{m}*z*) increases as

*z*increases due to ${P}_{0}<{P}_{\text{cr}}$. From Fig. 1, one can see that the results obtained by using Eq. (12) are in agreement with those by using numerical simulations. Thus, the analytical formulae obtained in this paper are valid.

The changes of the self-focusing critical power *P*_{cr} versus the beam order *m* are shown in Fig. 2. One can see that *P*_{cr} increases as *m* increases. In particular, the self-focusing critical power of H-G beams is larger than that of GS beams.

The changes of the relative beam width *w _{m}*/

*w*

_{m}_{-lin}in self-focusing and self-defocusing media versus the propagation distance

*z*are shown in Figs. 3(a) and 3(b) respectively, where

*w*

_{m}_{-lin}is the beam width in linear media. It is noted that the value of the

*w*/

_{m}*w*

_{m}_{-lin}is further away from 1 means that the beam width is more affected by the Kerr nonlinearity. In a self-focusing medium we have

*w*/

_{m}*w*

_{m}_{-lin}<1 (see Fig. 3(a)), while in a self-defocusing medium we have

*w*/

_{m}*w*

_{m}_{-lin}>1 (see Fig. 3(b)). However, in both self-focusing and self-defocusing media, the

*w*/

_{m}*w*

_{m}_{-lin}is close to the value of 1 as the beam order

*m*increases. It implies that the beam width is less sensitive to the Kerr nonlinearity as

*m*increases.

The changes of the radius of curvature *R _{m}* versus the propagation distance

*z*are shown in Fig. 4. It is known that in linear media

*R*is independent of the beam order

_{m}*m*. But in Kerr media

*R*depends on

_{m}*m*(see Fig. 4(a) and 4(b)). In a self-focusing medium we have

*R*>0 when ${P}_{0}<{P}_{\text{cr}}$ and

_{m}*R*<0 when ${P}_{0}>{P}_{\text{cr}}$ because the self-focusing effect results in beam convergence(see Fig. 4(a)), while in a self-defocusing medium we always have

_{m}*R*>0 (see Fig. 4(b)). In addition, in a self-focusing medium, the absolute value |

_{m}*R*| decreases as

_{m}*z*increases, and |

*R*| more rapidly approaches to zero as

_{m}*m*decreases because the self-focusing effect becomes stronger.

## 3. Rayleigh range

The Rayleigh range is used in the theory of lasers to characterize the distance over which a beam may be considered effectively non-spreading [20]. The Rayleigh range *Z*_{R} is defined as the propagation distance where the beam cross-sectional area doubles [20]. Based on the definition of the Rayleigh range and Eq. (12) for the *R*_{0}→∞ case, we obtain the *Z*_{R} of H-G beams propagating in Kerr media, i.e.,

*m*= 0, Eq. (16) reduces to the Rayleigh range of GS beams in Kerr media. When $\eta =1$(i.e.,

*n*

_{2}= 0), Eq. (16) reduces to the Rayleigh length of H-G beams in linear media, i.e., ${Z}_{\text{R}-\text{lin}}=k{w}_{0}^{2}/2$. Thus, the relative Rayleigh length of H-G beams can be expressed as ${Z}_{\text{R}}/{Z}_{\text{R}-\text{lin}}={\eta}^{-1/2}$. On the other hand, when the Rayleigh length expression Eq. (16) is adopted, Eqs. (12) and (13) can be simplified as

The changes of the Rayleigh range *Z*_{R} and the Relative Rayleigh range *Z*_{R}/*Z*_{R-lin} versus the beam order *m* are shown in Figs. 5 and 6, respectively. The *Z*_{R} decreases as *m* increases in self-focusing media (see Fig. 5(a)), but the inverse situation is in self-defocusing media (see Fig. 5(b)). Furthermore, Fig. 5 shows that both in self-focusing media and in self-defocusing media, for a certain value of the beam power, the *Z*_{R} approaches *Z*_{R-lin} when *m* is large enough because the Kerr nonlinearity may be ignored. We have *Z*_{R}/*Z*_{R-lin} >1 due to self-focusing (see Fig. 6(a)), while we have *Z*_{R}/*Z*_{R-lin} <1 due to self-defocusing (see Fig. 6(b)). However, Fig. 6 indicates that both in self-focusing media and in self-defocusing media, for a certain value of the beam power, the Z_{R}/Z_{R-lin} is close to the value of 1 when *m* is large enough. The physical reason is that, for a certain value of the beam power, the Kerr nonlinearity may be ignored when *m* is large enough.

## 4. Beam quality factor (*M*^{2}-factor)

The beam quality factor (*M*^{2}-factor) is a very useful parameter for characterizing various laser beams [21]. The *M*^{2}-factor in self-focusing media measures the goodness of the beam for collimating purposes [22]. For ${R}_{0}\to \infty $ case, we obtain the far-field divergence angle of H-G beams propagating in Kerr media by use of Eq. (12), i.e.,

*M*

^{2}-factor [23], and by using Eqs. (12) and (18), we obtain the

*M*

^{2}-factor of H-G beams in Kerr media, i.e.,When

*m*= 0, Eq. (19) reduces to the

*M*

^{2}-factor of GS beams in Kerr media, i.e., ${M}^{2}={\eta}^{1/2}$. When $\eta =1$ (i.e.,

*n*

_{2}= 0), Eq. (19) reduces to the

*M*

^{2}-factor of H-G beams in linear media, i.e., ${M}_{m-\text{lin}}^{2}=(2m+1)$. Thus, the relative

*M*

^{2}-factor of H-G beams can be expressed as ${M}_{m}^{2}/{M}_{m-\text{lin}}^{2}={\eta}^{1/2}$. Based on Eq. (19), we have ${M}_{m}^{2}=1$ when ${P}_{0}={P}_{\text{cr}}\left[1-1/{(2m+1)}^{2}\right]$, which is equivalent to a GS beam in linear case. According to Eq. (19), we have ${M}_{m}^{2}=0$ when ${P}_{0}={P}_{\text{cr}}$, which is corresponding to the best collimated beam.

The changes of the *M*^{2}-factor and the relative *M*^{2}-factor versus the beam power *P*_{0} for different values of the beam order *m* are shown in Figs. 7 and 8, respectively. The ${M}_{m}^{2}$ decreases as *P*_{0} increases in self-focusing (see Fig. 7(a)), but the inverse situation is in self-defocusing media (see Fig. 7(b)). Furthermore, ${M}_{m}^{2}<1$ and ${M}_{m}^{2}=0$ may appear due to self-focusing (see Fig. 7(a)), but it is always ${M}_{m}^{2}>1$ due to self-defocusing (see Fig. 7(b)).

On the other hand, we have ${M}_{m}^{2}/{M}_{m-\text{lin}}^{2}\le 1$ due to self-focusing effect (see Fig. 8(a)), while we have ${M}_{m}^{2}/{M}_{m-\text{lin}}^{2}\ge 1$ due to self-defocusing effect (see Fig. 8(b)). However, in both self-focusing and self-defocusing media, the ${M}_{m}^{2}/{M}_{m-\text{lin}}^{2}$ is further away from the value of 1 as the beam order *m* decreases. It means that the beam quality is less affected by the Kerr nonlinearity as *m* increases.

## 5. ABCD law

It is known that the ABCD law is very useful to study the laser beam propagation [22]. A real ABCD system may be composed of free propagations and ideal thin lenses. The transfer matrixes of a homogeneous medium and an ideal thin lens with focal length *f* are $\left[\begin{array}{l}1z\\ 01\end{array}\right]$ and $\left[\begin{array}{l}10\\ -1/f1\end{array}\right]$, respectively. When H-G beams propagate through an ideal thin lens, we have

*w*

_{m}_{1}and

*R*

_{m}_{1}, and

*w*

_{m}_{2}and

*R*

_{m}_{2}are the beam width and the curvature radius of H-G beams before and after the ideal thin lens.

In Section 2, we derive the propagation formulae of the beam width and the curvature radius of H-G beams propagating through homogeneous Kerr media (i.e., Eqs. (12) and (13)). Furthermore, we find that Eqs. (12), (13) and (20) can be rewritten in a unified form as

*A*,

*B*,

*C*and

*D*are matrix elements of an optical system in Kerr media. In this paper, we introduce a new complex parameter

*q*of H-G beams propagating through Kerr media, i.e.,

_{m}Based on the definition of the new complex parameter *q _{m}* (i.e., Eq. (23)) and Eqs. (21)-(22), we demonstrate that the ABCD law is still valid when H-G beams propagate through an optical system in Kerr media, i.e.,

*m*= 0, and to those of H-G beams in linear media when $\eta =1$ (i.e.,

*n*

_{2}= 0).

## 6. Features of Kerr nonlinearity

It is known that there exist several sub-beams for a H-G beam, and the sub-beams are separated each other by the dark area, which is quite different from that of a GS beam. In this section, the features of Kerr nonlinearity of H-G beams are studied. The Poynting vector in Kerr media can be expressed as [24]

*x*and

*y*respectively, ${\mu}_{0}$ and ${\omega}_{0}$are the magnetic permeability and the angular frequency in vacuum respectively, ${k}_{0}=2\pi /\lambda $ is the wave-number in vacuum, $n={n}_{0}+{n}_{2}{\left|E\right|}^{2}$ is the refractive index in nonlinear media.

By using numerical simulation method, the transverse power flow of H-G beams in self-focusing and self-defocusing media are shown in Figs. 9 and 10 respectively, where *z* = 0.1mm. It can be seen that the self-focusing or the self-defocusing of each sub-beam is independent. The physical reason is that there is no interchange of the power flow between sub-beams because the sub-beams are separated each other by the dark area. Furthermore, the position of the intensity peak of sub-beams only shifts slightly due to the asymmetry of sub-beams for the nonlinear case. Therefore, the focus point of the whole H-G beam never appears in self-focusing media because it is impossible to reach *w _{m}*(

*z*) = 0. It means that the parameter (i.e., focal length) is not suitable for characterizing the self-focusing effect of H-G beams although it is usually adopted to characterizing that of GS beams.

## 7. Conclusions

In this paper, the analytical propagation formulae (i.e., the beam width, the curvature radius and the phase) of H-G beams propagating in Kerr media are derived by using the variational approach, and validity of the formulae are confirmed. Furthermore, the analytical formulae of the self-focusing critical power, the Rayleigh range and the beam quality factor of H-G beams propagating in Kerr media are also derived. It is demonstrated that the ABCD law is valid when H-G beams propagate through an optical system in Kerr media if a new complex parameter is introduced, which presents a simple method to study the propagation of H-G beams through an optical system in Kerr media. It is mentioned that the analytical formulae obtained in this paper are general, which can reduce those of GS beams in Kerr media when the beam order *m* = 0, and to those of H-G beams in linear media when the nonlinear refractive index *n*_{2} = 0. As *m* increases, the self-focusing critical power of H-G beams increases, and H-G beams are less affected by the Kerr nonlinearity. In particular, a H-G beam is less sensitive to the Kerr nonlinearity than a GS one. Finally, it is found that the self-focusing feature of H-G beams is different from that of GS ones, i.e., the focus point of the whole H-G beam never appears in self-focusing media because the self-focusing of each sub-beam is independent. Therefore, the focal length is not suitable for characterizing the self-focusing effect of H-G beams although it is usually adopted to characterizing that of GS ones.

It is known that the air is a Kerr medium. In 2009, Rubenchik *et al*. proposed to exploit a self-focusing effect in the atmosphere to assist delivering powerful laser beams from orbit to the ground, which can greatly relax the requirements for the large orbital optics and the large ground receivers [16]. In this paper, it is proved that the value of the self-focusing critical power of H-G beams is larger than that of GS ones. Therefore, H-G beams can be compressed without filamentation and collapse even if the initial power is much higher, which is useful to assist delivering higher-power laser beams from orbit to the ground. On the other hand, in this paper it is shown that the focus point of the whole H-G beam never appears in Kerr media because the self-focusing of each sub-beam is independent. Thus, it is worth studying further if H-G beams may be applied for the multi all-optical switching.

## Funding

National Natural Science Foundation of China (NSFC) (61775152)

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