1. Introduction

In 1993, Simon and Mukunda firstly introduced the twisted Gaussian Schell-model (TGSM) beams in theory [1] based on the anisotropic Gaussian Schell-model (AGSM) beams presented by Li and Wolf [2]. And then, Simon et al. showed a comprehensive normal-mode decomposition analysis for TGSM beams introduced by Simon and Mukunda [3]. In 1994, Ambrosini et al. indicated that TGSM beams can also be obtained by an incoherent superposition of ordinary Gaussian beams [4]. On the other hand, in 1994, Friberg et al. firstly presented experimental demonstrations of TGSM beams from AGSM beams based on the theory presented by Simon and Mukunda [5]. Furthermore, in 2019, Wang et al. experimentally realized TGSM beams by converting an AGSM beam into a TGSM beam with a set of just three cylindrical lenses [6].

Until now, many studies have been carried out concerning the propagation characteristics of TGSM beams in vacuum or through atmospheric turbulence [7–15]. It was shown that the TGSM beams have critical advantages in practice. For example, Sundar et al. indicated that the twist phase of TGSM beams lifts the degeneracy in the eigenvalue spectrum [7]. Cai et al. showed that the ghost image disappears gradually and its visibility increases as the twist parameter increases [11]. Tong and Korotkova showed that the TGSM beam could serve as illumination that might produce images with a resolution overcoming the Rayleigh limit [12]. Wang et al. indicated that the second-order moments of TGSM beams are less affected by atmospheric turbulence than a GSM beam without twist phase [13], and Cui et al. showed this result is true in non-Kolmogorov turbulence [14]. Furthermore, Wang et al. indicated that the twist of TGSM beams can actively reduce turbulence-induced scintillation in weak turbulent atmosphere [15]. However, these studies mentioned above concerned about TGSM laser beams only in linear media [7–15], but not including in nonlinear media.

A powerful laser beam propagation in nonlinear media is affected by the Kerr nonlinearity [16,17]. Rubenchik et al. indicated that in the atmosphere the self-focusing (i.e., Kerr effect) can noticeably decrease the laser intensity on the debris target [18]. Our group proved that the uniform irradiation on the space debris may be generated by self-focusing effect in the inhomogeneous atmosphere [19], and presented the quadratic approximation of the nonlinear phase shift to study analytically the characteristics of powerful laser beams propagating upwards in the turbulent atmosphere [20]. Furthermore, some works have been carried out concerning the influence of the nonlinear self-focusing, self-defocusing and optical field collapse on characteristics of different types of beams (e.g., Gaussian–Schell model (GSM) beam, vector beam and hybridly polarized beam, etc.). For example, Li et al. demonstrated that hybridly polarized vector fields with axial symmetry broken polarization could achieve controllable and robust multiple filamentation [21]. Wang et al. revealed that the synergy of optical anisotropy and polarization structure is indeed a very effective means for controlling the optical field collapse [22]. Gu et al. showed that the isotropic optical nonlinearity can manipulate the spatial self-phase modulation intensity pattern, the distribution of state of polarization, and the spin angular momentum flux of a hybridly polarized vector beam [23]. Wen et al. indicated that the optical nonlinearity enhances the polarization rotation of the focused fundamental Poincaré beams during propagation and the anisotropy of optical nonlinearity causes the symmetry breaking of the polarization distribution [24]. Our group recently studied the propagation characteristics of partially coherent GSM beams and partially coherent light pulses (PCLPs) in nonlinear media [25,26]. We indicated an optimal beam will necessarily involve a trade-off between collimation and spatial coherence when a GSM beam propagates in self-focusing media [25], and demonstrated that a PCLP has more advantage to avoid the optical damage of materials than a fully one [26]. However, the propagation characteristics of TGSM beams through nonlinear Kerr media has not been examined until now.

In this paper, the analytical propagation formulae of TGSM beams propagating through nonlinear Kerr media are derive, and the propagation characteristics are studied in detail. In addition, beam quality of TGSM beams and focusing characteristics of focused TGSM beams are also investigated. The results obtained in this paper are not only theoretical interest, but also useful for laser ablation propulsion’s applications in space. The main results obtained in this paper are explained in physics.

2. Derivation of propagation formulae of TGSM beams in nonlinear Kerr media

The principal features of diffraction and Kerr nonlinearity of TGSM beams propagating in nonlinear Kerr media can be described by the nonlinear Schrödinger (NLS) equation. Under the standard paraxial approximation, the NLS equation is expressed as [20]

It was shown that the Gaussian profile will be retained when a TGSM beam propagates in linear media [9]. Our numerical simulation results indicates that, when a TGSM beam propagates in nonlinear Kerr media, the beam profile for a situation without filamentation is also close to a Gaussian one. And then, we suppose the solution of Eq. (1) is in the form as

And setting $\nabla _ \bot ^2W({{{\boldsymbol{\mathrm{\rho}} }_1},{{\boldsymbol{\mathrm{\rho}} }_2},z} )$ in Eq. (2), we obtain

Under the paraxial approximation (i.e., ${w^2}(z) \gg {\rho ^2}$), and applying Taylor series expansion, we have $\textrm{exp} [{{{ - {\rho^2}} / {{w^2}(z )}}} ]\approx 1 - {{{\rho ^2}} / {{w^2}(z )}}.$ Thus, from Eq. (2) we have

Substituting Eqs. (3–5) into Eq. (1), separating the real part and the imaginary part, we can derive the analytical expressions of u(z), w(z), δ(z) and R(z) when the boundary conditions u(z=0)=u₀, w(z=0)=w₀, δ(z=0)=δ₀ and R(z=0)=R₀ are adopted, i.e.,

When u₀=0 and α→∞, Eq. (10) reduces to that of Gaussian beams propagating in nonlinear Kerr media (i.e., ${\eta _{\textrm{GS}}} = {{{I_0}{k^2}{n_2}w_0^2} / {{n_0}}}$ [25]). It is noted that the formulae obtained in this paper [i.e., Eqs. (6–9)] are more general, which reduce to those of a TGSM beam being incident in nonlinear Kerr media at the waist position when R₀→∞, to those of a TGSM beam propagating in linear media when n₂ = 0, and to those of a GSM beam propagating in nonlinear Kerr media when u₀=0 (which are in agreement with those in [9,25]).

In self-focusing media and R₀→∞, when η=1, the Kerr nonlinearity is completely balanced by the diffraction caused by spatial coherence and twist, i.e., the self-trapping of TGSM beams occurs. Letting η=1 in Eq. (10), we obtain the self-focusing critical power of TGSM beams in nonlinear Kerr media, i.e.,

From Eq. (11), together with $\alpha _{\textrm{eff}}^2 = {{4{\alpha ^4}} / {({4{\alpha^2} + {\zeta^2}} )}}$ and $\zeta=k u_{0} \delta_{0}^{2}$, one can see that P_cr increases as u₀ increases. The physical reason is that the twist reduces the effective degree of coherence and enhances the beam divergence [1]. It is noted that Eq. (11) is more general, which can reduce to that of GSM beams when u₀=0 (i.e., ${P_{\textrm{cr - GSM}}} = {{{\varepsilon _0}cn_0^2\pi ({1 + {\alpha^{ - 2}}} )} / {({{k^2}{n_2}} )}}$), and to that of Gaussian beams when u₀=0 and α→∞ (i.e., ${P_{\textrm{cr - GS}}} = {{{\varepsilon _0}cn_0^2\pi } / {({{k^2}{n_2}} )}}$). Obviously, it is ${P_{\textrm{cr}}} > {P_{\textrm{cr - GSM}}} > {P_{\textrm{cr - GS}}}$. Namely, a TGSM beam is less sensitive to Kerr nonlinearity than a GSM beam and a Gaussian beam.

It is noted that a TGSM beam propagating in self-focusing media may spread or focus, which can be controlled by the twist parameter u₀. By letting η=1 in Eq. (10), the initial twist parameter of TGSM beams u_0cr should satisfy

In self-focusing media, a TGSM beam will spread when u₀>u_0cr (i.e., η<1), while will focus when u₀<u_0cr (i.e., η>1). In addition, Eq. (12) shows that u_0cr increases as P increases.

Based on the definition of ${M^2}\textrm{ - factor}$ [27], we derive the ${M^2}\textrm{ - factor}$ of TGSM beams propagating in nonlinear Kerr media by using Eq. (7), i.e.,

In Eq. (13), η≤1 should be satisfied. The ${M^2}\textrm{ - factor}$ defined by Siegman is only valid for beam spreading case [27]. When η>1, Eq. (13) is not valid because a TGSM beam will focus.

The ABCD law is one of important methods to study the laser beam propagation. Until now, the ABCD law of TGSM beams propagating through an optical system in nonlinear Kerr media has not been made. In this paper, we introduce a new complex parameter q of TGSM beams propagating in nonlinear Kerr media, i.e.,

Based on the new complex parameter q defined in this paper (i.e., Eq. (14)), together with the transformation of a TGSM beam by a thin lens and by Kerr media (i.e., Eqs. (7) and (9)), we derive the ABCD law of TGSM beams propagating through an optical system in nonlinear Kerr media, i.e.,

3. Propagation characteristics of TGSM beams in nonlinear Kerr media

In the numerical calculation examples of this paper, it is assumed that a TGSM beam waist is at the initial plane z=0 (i.e., R₀→∞ at the initial plane z=0), and the initial beam width is taken as w₀=0.4mm. For the numerical calculation examples in Sections 3 and 4, a TGSM beam propagates through nonlinear Kerr media without optical systems. On the other hand, for the numerical calculation example in Section 5, a TGSM beam is focused by a lens located at the initial plane z=0, and then propagates through self-focusing media. Furthermore, in this paper, the value of the beam power P is given by the dimensionless parameter η_GS, the value of the initial twist parameter u₀ is given by the dimensionless parameter ζ, and the value of the initial spatial coherence width δ₀ is given by the dimensionless parameter α. It is known that ${\eta _{\textrm{GS}}} = {{P{n_2}{k^2}} / {({\pi {\varepsilon_0}cn_0^2} )}}$, and $\zeta \textrm{ = }k{u_0}\delta _0^2$, namely, P increases as |η_GS| increases, and u₀ increases as ζ increases, respectively.

In the numerical calculation examples of this paper, the calculation parameters λ=1.06µm, n₀=2.4 and ${n_2} = 2 \times {10^{ - 13}}{{\textrm{c}{\textrm{m}^2}} / \textrm{W}}$ are adopted for the self-focusing media (i.e., the As₂S₃ glass); λ=0.532µm, n₀=1.56 and ${n_2} ={-} 1.5 \times {10^{ - 13}}{{\textrm{c}{\textrm{m}^2}} / \textrm{W}}$ are adopted for the self-defocusing media (i.e., the synthesized soluble polyoxadiazoles containing 3,4-dialkoxythiophenes). In addition, the beam spreading case (i.e., η<1) is considered in Figs. 1–5, Fig. 6(b) and Figs. 8,9.

 

Fig. 1. Twist parameter u versus the propagation distance z. α=0.3. (a) self-focusing media; (b) self-defocusing media.

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Fig. 2. Relative twist parameter u/u_lin versus the propagation distance z, α=0.3. Solid curves: in Self-focusing media; dashed curves: in self-defocusing media. (a) for different values of η_GS; (b) for different values of ζ.

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Fig. 3. In self-focusing media, (a) twist parameter u and (b) relative twist parameter u/u_lin versus the propagation distance z, η_GS=4.

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Fig. 4. In self-focusing media, (a) beam width w and (b) relative beam width w/w_lin versus the propagation distance z, α=0.3. Solid curves: ζ=0.5; dashed curves: ζ=0.8.

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Fig. 5. (a) Spatial coherence width δ and (b) relative spatial coherence width δ/δ_lin versus the propagation distance z, α=0.3. Solid curves: in self-focusing medium, η_GS=5; dashed curves: in self-defocusing medium, η_GS=−8.

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Fig. 6. In self-focusing media, curvature radius R versus the propagation distance z, η_GS=20, α=0.3. (a) beam focusing; (b) beam divergence.

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Fig. 7. In self-focusing media (a) M² -factor and (b) M²/${M^{2}_{\textrm{lin}}}$ versus the normalized twist parameter ζ and the degree of global coherence α.

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Fig. 8. ${{{M^2}} / {M_{\textrm{GSM - lin}}^2}}$ versus the twist parameter u₀, α=0.3.

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Fig. 9. Rayleigh range Z_R versus the normalized twist parameter ζ. α=0.3.

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Changes of the twist parameter u in self-focusing and self-defocusing media versus the propagation distance z are shown in Figs. 1(a) and 1(b), respectively. One can see that u decreases as the propagation distance z increases, which agrees with the result in linear media. As |η_GS| increases (i.e., P increases), u increases in self-focusing medium (see Fig. 1(a)) while decreases in self-defocusing medium (see Fig. 1(b)). It means that the self-focusing effect results in an increase of u, and the self-defocusing effect results in a decrease of u.

It is noted that u₀=u_0cr remains unchanged on propagation when the self-trapping of TGSM beams occurs (i.e., η=1), and u increases as the propagation distance z increases for the beam focusing case (i.e., η>1). The numerical calculation examples are omitted here to save space.

Changes of the relative twist parameter u/u_lin versus the propagation distance z are shown in Fig. 2, where u_lin is the twist parameter in linear media. The u/u_lin is further away from the value of 1 means that the nonlinearity Kerr effect is stronger. It is known that u/u_lin>1 due to self-focusing (see solid curves), and u/u_lin<1 due to self-defocusing (see dashed curves). From Fig. 2(a), one can see that the u/u_lin is further away from the value of 1 as |η_GS| increases because the nonlinearity Kerr effect becomes stronger. Figure 2(b) shows that, as ζ increases (i.e., the initial twist parameter u₀ increases), the u/u_lin is close to the value of 1 in both self-focusing and self-defocusing media. It implies that on propagation, the twist characteristic of TGSM beams with larger value of u₀ are less affected by Kerr nonlinearity.

In self-focusing media, for different values of α, changes of the twist parameter u and the relative twist parameter u/u_lin versus the propagation distance z are shown in Figs. 3(a) and 3(b), respectively. Figure 3(a) shows that u increases as α increases (i.e., the initial spatial coherence width δ₀ increases), which agrees with the result in linear and self-defocusing media. It means that an increase of initial spatial coherence results in an increase of u. Figure 3(b) shows that u/u_lin is further away from the value of 1 as α increases because the self-focusing effect becomes stronger. It means that on propagation, the twist characteristic of TGSM beams with larger value of δ₀ are more affected by Kerr nonlinearity.

In self-focusing media, for different values of η_GS, changes of the beam width w and the relative beam width w/w_lin versus the propagation distance z are shown in Figs. 4(a) and 4(b), respectively. On comparing dashed curves (ζ=0.8) with solid curves (ζ=0.5), one can see both w and w/w_lin increase as ζ increases (i.e., the initial twist parameter u₀ increases). The physical reason is that the twist enhances the beam divergence (this conclusion is the same as that in linear media [1]), and weakens the self-focusing effect. As η_GS increases (i.e., P increases), w and w/w_lin decrease because the self-focusing effect becomes stronger, and this conclusion is the same as that of GSM beams in nonlinear Kerr media [28]. The numerical results in self-defocusing media are omitted here because the conclusions are straightforward.

For different values of ζ, changes of the spatial coherence width δ and the relative spatial coherence width δ/δ_lin versus the propagation distance z are shown in Figs. 5(a) and 5(b) respectively, where δ_lin is the spatial coherence width in linear media. Figure 5(a) shows that δ increases as ζ increases (i.e., u₀ increases), i.e., the twist can result in the improvement of spatial coherence. And the δ/δ_lin is close to the value of 1 as ζ increases (see Fig. 5(b)) because the Kerr nonlinearity effect is weaker as the initial beam twist is stronger. It can be concluded that on propagation, the spatial coherence of TGSM beams with larger value of u₀ are less affected by Kerr nonlinearity.

In self-focusing media, for different values of ζ, changes of the curvature radius R versus the propagation distance z are shown in Fig. 6. One can see that R changes due to the twist. R<0 (see Fig. 6(a)) and R>0 (see Fig. 6(b)) are corresponding to the beam focusing and beam divergence (i.e., η>1 and η<1), respectively. As ζ increases (i.e., u₀ increases), the R changes from R<0 to R>0 (i.e., from focusing to divergence), which is consistent with the result in Eq. (12). Figure 6(b) shows that for a divergent beam, R reaches its minimum at the position of the Rayleigh range, and R increases as z increases beyond the Rayleigh range. Furthermore, the absolute value |R| decreases for R>0 and increases for R<0 as ζ increases. The physical reason is that the twist enhances the beam divergence [1]. In self-defocusing media, it is always R>0 and R decreases as ζ increases, the numerical calculation examples are omitted here to save space.

4. Beam quality of TGSM beams in nonlinear Kerr media

In this section, the ${M^2}\textrm{ - factor}$ and the Rayleigh range Z_R are taken as the characteristic parameters to study the beam quality of TGSM beams propagating in nonlinear Kerr media.

The expression of ${M^2}\textrm{ - factor}$ of TGSM beams propagating in nonlinear Kerr media (i.e., Eq. (13)) can be rewritten as

It is noted that η_GS can also be expressed as η_GS=P/P_crGS, which represents the Kerr nonlinearity effect on ideal Gaussian beams. The second, third and fourth items under the root sign in Eq. (16) denote the contributions to ${M^2}\textrm{ - factor}$ of TGSM beams propagating in nonlinear Kerr media, namely, they are Kerr nonlinearity of ideal beams, spatial coherence of TGSM beams and twist of TGSM beams, respectively. Obviously, Eq. (16) reduces to M² = 1 when an ideal Gaussian beam propagates in linear media.

It can be shown that the self-focusing results in a decrease of M², while both the partially coherence and the twist result in an increase of M². From Eq. (16), we can obtain M² = 1 when the initial beam twist parameter satisfies

Namely, for this case the propagation of TGSM beams in self-focusing media is equivalent to that of ideal Gaussian beams in linear media.

On the other hand, from Eqs. (12) and (16), we can obtain ${M^2}\textrm{ = }0$ when u₀=u_0cr. It is known that in self-focusing media, the ${M^2}\textrm{ - factor}$ measures the goodness of the beam for collimating purposes [29]. Namely, a self-trapping TGSM beam with initial beam twist u_0cr is the best collimated beam.

Changes of the ${M^2}\textrm{ - factor}$ and the relative ${M^2}\textrm{ - factor}$ $\left(M^{2} / M_{\text {lin }}^{2}\right)$ in self-focusing media versus the normalized twist parameter ζ and the global coherence degree α are shown in Figs. 7(a) and 7(b) respectively, where $M_{\text {lin }}^{2}$ is the M²-factor in linear media. One can see that both M² and $M^{2} / M_{\operatorname{lin}}^{2}$ decrease as α or η_GS increases (i.e., δ₀ or P increases). On other hand, both M² and $M^{2} / M_{\operatorname{lin}}^{2}$ increase as ζ increases (i.e., u₀ increases), the reason is that the twist results in the beam divergence and weakens the self-focusing. In addition, from Fig. 7, one can see that M² = 1 and M² = 0 may appear because of self-focusing effect.

Furthermore, when η_GS=0 (i.e., n₂=0) and u₀=0, Eq. (16) reduces to that of GSM beams propagating in linear media (i.e., $M_{\textrm{GSM} - \textrm{lin}}^2 = \sqrt {1 - {\alpha ^{ - 2}}}$). The value of the ratio ${{{M^2}} / {M_{\textrm{GSM - lin}}^2}}$ denotes how the self-focusing and the twist affect the M² -factor. Letting ${{{M^2}} / {M_{\textrm{GSM - lin}}^2}}\textrm{ = }1$ (i.e., $\eta_{\mathrm{GS}}=\left(k u_{0} w_{0}^{2} / 2\right)^{2}$ in Eq. (16)), we can obtain

When u₀=u_0GSM-lin, the Kerr nonlinearity is completely balanced by the diffraction caused by the twist, namely, for this case the propagation of TGSM beams in self-focusing media is equivalent to that of GSM beams in linear media. When u₀<u_0GSM-lin, the self-focusing dominates, and ${{{M^2}} / {M_{\textrm{GSM - lin}}^2}} < 1$ . When u₀>u_0GSM-lin, the twist dominates, and ${{{M^2}} / {M_{\textrm{GSM - lin}}^2}} > 1$. These results are shown clearly in Fig. 8.

Based on the definition of the Rayleigh range (i.e., the propagation distance at which the beam cross-sectional area doubles) and Eq. (7), for the R₀→∞ case, we obtain the Rayleigh range Z_R of TGSM beams propagating in nonlinear Kerr media, i.e.,

5. Focusing characteristics of focused TGSM beams propagating in self-focusing media

Assume that a TGSM beam is focused by a thin lens located at z=0 plane, and then propagates through self-focusing media. Based on ABCD law and Eq. (7), the beam width of a focused TGSM beam propagating through self-focusing media can be expressed as

A focused TGSM beam propagates in self-focusing media, its beam width will decrease gradually for the beam self-focusing case (i.e., η>1). Ideally, it will extend to the focus point until the value of the beam width reaches zero [30]. Letting w(z) = 0 in Eq. (20), we obtain the focal length z_f of focused TGSM beams in self-focusing media, i.e.,

The focal length should satisfy z_f>0. When (R₀−f)/R₀f<1/f_T, the third item on the right side of Eq. (21) takes the “+” sign and there is only one focus z_f1. When (R₀−f)/R₀f >1/f_T, the third item on the right side of Eq. (21) takes the “±” sign and there are two foci z_f1 and z_f2. It is interesting that one focus or two foci of the TGSM beams can be controlled by the twist. Letting (R₀−f)/R₀f=1/f_T, we obtain

There exists one focus z_f1 when ${u_0} < {u^{\prime}_0}$, and there exist two foci when ${u_0} > {u^{\prime}_0}$. As u₀ increases, z_f1 increases, but z_f2 decreases, namely, both the two foci are close to the focal plane of the thin lens and the distance between the two foci decreases. These results can be shown clearly in Fig. 10. In addition, Fig. 10 shows that, as η_GS increases (i.e., P increases), ${u^{\prime}_0}$ increases and the distance between two foci increases.

 

Fig. 10. In self-focusing media, focal length z_f versus the twist parameter u₀, α=0.3, f=0.2 m.

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6. Conclusions

In this paper, the analytical propagation formulae (e.g., the twist parameter, the beam width, the spatial coherence width, etc.) of TGSM beams in nonlinear Kerr media are derived, and the ABCD law of TGSM beams propagating through an optical system in Kerr media is also derived. Furthermore, analytical expressions of the self-focusing critical power, M ²-factor and Rayleigh range of TGSM beams propagating in Kerr media are derived. The analytical formulae obtained in this paper are more general, which can reduce to those of TGSM beams in linear media when n₂=0, and to those of GSM beams in Kerr media when u₀=0.

It is shown that a TGSM beam may spread or focus in self-focusing media, which can be controlled by the initial twist parameter, and analytical expression of the critical twist parameter u_0cr is also derived. When u_0cr is adopted, the self-trapping of the TGSM beams occurs. On the other hand, the focal length of focused TGSM beams in self-focusing media is also derived. One focus or two foci of focused TGSM beams can be controlled by the initial twist parameter, and the analytical expression of the critical twist parameter ${u^{\prime}_0}$ is also derived. The ${u^{\prime}_0}$ decreases and the distance between two foci increases as the self-focusing effect becomes stronger.

It is found that a TGSM beam is less sensitive to Kerr nonlinearity than a GSM beam. Furthermore, the propagation characteristics of TGSM beams with stronger twist and worse spatial coherence are less affected by Kerr nonlinearity. The atmosphere is one kind of nonlinear self-focusing media. The results obtained in this paper may be useful for laser ablation propulsion’s applications in space (e.g., space-debris removal, launching small probes into Low Earth Orbit, etc.). Based on the results obtained in this paper, we can conclude that a TGSM beam has greater resistance not only to Kerr nonlinearity (i.e., self-focusing in the atmosphere) but also to atmospheric turbulence than a GSM beam or an ideal Gaussian beam. The physical reasons are explained as follows. The self-focusing effect enhances the beam twist, but degrades the beam spatial coherence. On the other hand, a laser beam with twist is less affected by atmospheric turbulence than one without twist [13–15], and a partially coherent beam is less sensitive to atmospheric turbulence than a fully coherent one [31]. In the atmosphere, the self-focusing and turbulence can noticeably decrease the laser intensity on the debris target [18]. Therefore, a TGSM beam may have the potential to be used in laser ablation propulsion’s applications in space.