In the past decade, relying on the wide application from material processing and nonlinear optics to free-space optical communication and other fields, the high-power laser beams^[1–4] have increasingly drawn attention, on the other hand, an increase in the laser output power inevitably results in the gain non-uniformity, gain saturation, nonlinear optical effects, and thermal effect further triggering beam distortions on both amplitude and phase which are able to impact the propagation properties of laser beams.^[5–11] Therefore, it is practical important to study the propagation properties of laser beams with beam distortion, including amplitude modulations and phase fluctuations. In order to analyze propagation properties of laser beams with beam distortions theoretically, Manes and Simmons proposed statistical optics model of high-power laser beams for the first time.^[12] Further studies^[12,13] have shown that statistical optics methods can provide reliable estimates with experimental results and simulations. Besides, the beam propagation quality factor M² introduced by Siegman^[14] is a very useful beam parameter to feature propagation properties of laser beams in various applications.^[15–18]

Generally, the larger the value of the factor is, the poorer the beam quality is. Meanwhile, the practical optical systems mostly are equipped with limited apertures which usually cause the laser beams propagating through these systems truncated. Thus, based on the mathematics model proposed by Manes and Simmons, a series of works concerning the propagation properties have been accomplished,^[19–25] including M² factor of truncated high-power laser beams passing through turbulent atmospheric and passing through optical system with an aperture. The results showed that the propagation properties of high-power laser beams are related to both amplitude modulations and phase fluctuations when passing through optical system with an aperture. In addition, it is also shown that when lasers propagate through turbulent atmosphere, the decrement of the turbulence effect on laser beams is caused by amplitude modulations and phase fluctuations. In 2004, by expanding the window function into complex Gaussian function sum, an approximate analytical recurrence formula specified for laser beams with amplitude modulations and phase fluctuations propagating through multi-apertured ABCD system was deduced by Lü et al.,^[26] which provided a fast algorithm for simulating the beam propagation through complex optical trains with a series of apertures. This analytical expression is in satisfactory agreement with direct diffraction integration. To the best of our knowledge, as the valuable theoretical reference for the design and asses of high-power laser optical system, the generalized M² factor of high-power laser beams passing through multi-apertured ABCD system has not been reported yet. With the truncated second-order moments method and expanding the hard-edged aperture function into a finite sum of complex Gaussian functions, in this paper, we deduce an approximate analytical expression for the generalized M² factor of high-power laser beams passing through the multi-apertured ABCD optical system, and discuss the influence of beam distortions on M² factor of laser beams. The approximate analytical propagation expression deduced in Ref. [26] is applied and the generalized M² factor of high-power laser beams passing through an aperture-spatial filter system is calculated numerically in this paper.

In the space–time domain, the high-power laser beams with AMs and PFs introduced by Manes and Simmons^[12] is featured with the mutual intensity J₀ (, , z = 0) in two-dimensional case are expressed as

Letting

The high-power laser beams is assumed to propagate through the paraxial ABCD optical system consisting of m A_iB_iC_iD_i subsystems with m hard-edged apertures in the b_i (i = 1, 2, …, m) width, as shown in Fig. 1, the herein refers to the ray transfer matrix.

Fig. 1.

	Figure Option ViewDownloadNew Window
	Fig. 1. A multi-apertured ABCD optical system.

The window function of a rectangular aperture is expressed as T(x). According to Wen and Breazeale,^[22] T(x) can be expanded into a finite sum of complex-valued Gaussian functions

The relevant parameters and expressions in this paper are subject to the corresponding parameters and expressions in Ref. [26]. On the basis of the mutual intensity expression J_m(x₁, x₂, z) at the output plane RP_m in Ref. [26], the total power of high-power laser beams, passing through the m-th hard-edged rectangular aperture with width of b_m at the m-th aperture plane z can be deduced as

The truncated second-order moments in the spatial domain of high-power laser beams through the m-th hard-edged rectangular aperture at the m-th aperture plane z is given by^[28]

The corresponding cross second-order moment can be obtained as

Supposing that σ₀ → ∞, , γ → ∞, the approximate analytical expression (13) will be reduced to the generalized M² factor of Gaussian beams passing through a multi-apertured optical ABCD system. Furthermore, supposing that m = 1, the approximate analytical expression (13) will be reduced to the generalized M² factor of hard-edged apertured diffracted Gaussian beams, which is consistent with Eq. (25) in Ref. [29].

In this section, some numerical calculations have been involved and accomplished with the above deduced analytical expression.

Figure 2 is a special case of Fig. 1, i.e., the aperture-spatial filter system. Among calculation parameters, we take w₀ = 1 mm, λ = 1.06 μm, f_q1 = f₂ = 1 m, and d₁ = d₂ = 1 m, f₁ and f₂ refer to the focal lengths of the two lens and d₁ and d₂ refer to the separation of the first aperture to the first lens and the second aperture to the second lens. The generalized M² factor of different beams (Gaussian beam and high-power laser beams) passing through one aperture-lens optical system versus truncation parameter b₁/w₀ is shown in Fig. 3. It is worth noting that the width of the high-power laser beams is assumed to 2w₀, in this circumstance, the diffraction effect can be negligible if the aperture width is larger than 2w₀. From Figs. 3(a)–3(b), it is seen that the M² factor decreases with increasing σ₀ or decreasing for a certain b₁/w₀. The M² factor of truncated Gaussian beam and truncated high-power laser beams with only PFs () decreases with increasing b₁/w₀ till it respectively reaches the valve 1, 1.421 corresponding to the non-truncated case, whereas M² factor varies non-monotonically with b₁/w₀ for the truncated high-power laser beams with In the beginning, the M² factor decreases with increasing b₁/w₀ till it reaches minimum (, π₀ = 0.5 mm), 1.606 (, π₀ = 1 mm), 1.316 (, π₀ = ∞), and 1.734 (, π₀ = 1 mm) respectively, thereafter, it increases with increasing b₁/w₀. With b₁/w₀ further increasing, the M² factor reaches a certain value corresponding to the non-truncated case, for instance, M² = 2.969 (, π₀ = 0.5 mm), 2.033 (, π₀ = 1 mm), 1.605 (, π₀ = ∞), and 2.328 (, π₀ = 1 mm) respectively. Furthermore, regarding to the strong truncated case, the M² factor converges to 2.313 of truncated plane waves which coincides with the conclusion in Ref. [29]. It demonstrates that when high-power laser beams are passing through an aperture-lens optical system, AMs will change the dependence of Gaussian beams on the truncation parameters b₁/w₀, moreover, both AMs and PFs will degrade the beam propagation quality significantly compared with Gaussian beams. It is also suggested in Fig. 3 that when the aperture width b₁ is within a suitable range, the optical system with an aperture-lens will result in the decreasing of M² factor of high-power laser beams with both AMs and PFs.

Fig. 2.

	Figure Option ViewDownloadNew Window
	Fig. 2. An aperture-spatial filter system.

Fig. 3.

	Figure Option ViewDownloadNew Window
	Fig. 3. M2 factor versus b1/w0.

The generalized M² factor of high-power laser beams passing through the aperture-spatial filter system versus b₂/w₀ for different π₀ and with b₁/w₀ = 1 is shown in Fig. 4. From Figs. 4(a) and 4(b), it turns out that the influence of π₀ and on M² factor is negligible when b₂/w₀ is small, nevertheless, when b₂/w₀ is large, the M² factor will increase with the increase of or decrease of π₀ for a certain b₂/w₀. Regarding to high-power laser beams passing through the aperture-spatial filter system and the aperture-lens optical system, the dependences of their M² factor on b₂/w₀ and on b₁/w₀ are parallel. In other words, there is an optimal aperture width b₂ resulting in M² minimal with preset beam parameters and l-th aperture-lens parameters. Respectively, the is 1.87, 1.39, 1.1 corresponding to π₀ = 0.5 mm, 1 mm, ∞ with , as shown in Fig. 4(a). The is 1.35, 1.39, 1.41 respectively corresponding to , 0.1, 0.2 with π₀ = 1 mm, as shown in Fig. 4(b). Regarding the high-power laser beams passing through the aperture-spatial filter system, its M² will be reduced similarly by setting the appropriate aperture width b₂ with the preset aperture width b₁.

Fig. 4.

	Figure Option ViewDownloadNew Window
	Fig. 4. M2 factor versus b2/w0.

Figures 5(a) and 5(b) show versus π₀ and versus when high-power laser beams passing through the aperture-spatial filter system, which is suggested that will decrease to an asymptotic value when π₀ increases with preset b₁/w₀. Moreover, when π₀ is determined, the larger the b₁/w₀ value is, the larger will be. As shown in Fig. 5(b), it can be seen will not affect the value of for b₁/w₀ = 0.2, 0.5, nevertheless when b₁/w₀ = 1, will increase with the increase of . In other words, the aperture-spatial filter system with smaller aperture width b₁ can be applied to upgrade the beam propagation quality of high-power laser beams passing through the aperture-spatial filter system.

Fig. 5.

	Figure Option ViewDownloadNew Window
	Fig. 5. versus π0 or .

Figure 6 shows M² factor versus b₂/w₀ for different wavelengths λ with b₁/w₀ = 1, π₀ = 1 mm, when high-power laser beams passing through the aperture-spatial filter system. It is clear that the variation rule of M² factor with b₂/w₀ will not be impacted by the variation of wavelengths. Nevertheless, when b₂/w₀ is small, the variation of the wavelengths will affect the value of M² factor, and when b₂/w₀ is large, corresponding to one aperture-lens case, M² factor will not vary with the variation of wavelengths. It is concluded that the M² factor is independent of the wavelengths λ when high-power laser beams passing through one-apertured ABCD optical system.

Fig. 6.

	Figure Option ViewDownloadNew Window
	Fig. 6. M2 factor versus b2/w0.

It is worth noting that further numerical calculations on the M² factor of high-power laser beams passing through three or more aperture-lens optical system have as well been fulfilled. The results show that when the number of aperture-lens is equal to or larger than three, M² factor with preset high-power laser beams parameters can be also minimized by optimal aperture parameters and the influence of AMs and PFs on the M² factor is negligible with small width of the front apertures, the relevant detailed statements is omitted.

In this paper, the approximate propagation expression of beam propagation factor M² of high-power laser beams passing through a multi-apertured optical ABCD system is derived with the truncated second-order moments method and by expanding the rectangular function into a finite sum of complex Gaussian functions, it can be applied to evaluate the beam propagation quality of high-power laser beams passing through a multi-apertured optical ABCD system. Numerical calculation results indicate the influence of the multi-apertures on the beam propagation quality of high-power laser beams. It is suggested that the M² factor of high-power laser beams will be decreased with optimal beam parameters and aperture parameters regardless passing through one or two apertures, and the phase fluctuations have greater influence on M² factor than amplitude modulations. The conclusions reported in this paper provides a reference for design and engineering applications of high-power laser spatial filter systems.