M2-factor of high-power laser beams through a multi-apertured ABCD optical system
Zeng Xiangmei1, †, Zhang Meizhi1, Cao Dongmei2, Sun Dingyu1, Zhou Hua1
School of Electronic Engineering, Xi’an University of Posts and Telecommunications, Xi’an 710121, China
School of Physics and Electronic Information, Yanan Univeristiy, Yanan 716000, China

 

† Corresponding author. E-mail: xmzz79@163.com

Project supported by the Science Fund from the Shaanxi Provincial Education Department, China (Grant No. 18JK0723).

Abstract

Based on the generalized truncated second-order moments, an approximate analytical formula of the beam propagation factor M2 of high-power laser beams passing through the optical system with multiple hard-edged apertures is deduced. Numerical examples of the beams passing through an aperture-spatial filter are enclosed, and the influences of amplitude modulations (AMs) and phase fluctuations (PFs) on the beam propagation quality of high-power laser beams passing through the multi-apertured ABCD optical system are considered and discussed. It is shown that PFs are able to degrade the beam propagation quality of laser beams more than AMs when the high-power laser beams passing through the aperture-spatial filter, furthermore, one or two aperture-lens optical systems configured appropriate aperture parameters are both able to upgrade the beam propagation quality of high-power laser beams. The M2 factor of Gaussian beam passing through the multi-aperture optical system is a special case in this paper.

1. Introduction

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[14] 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.[511] 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.[1518]

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,[1925] 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.

2. Basic theory

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

where LA and Lp are the scale of the AMs and PFs respectively, and and measure the strengths of the intensity modulations and the amplitude of the phase error respectively. I presenting Gaussian profile is generally much larger than the noise intensity , i.e.,

where w0 is the waist width.

Letting

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

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

where Fj and Gj denote the expansion coefficients and Gaussian coefficients respectively, which are able to be yielded through optimization computation directly for M = 10 in Table 2 in Ref. [27].

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 Jm(x1, x2, z) at the output plane RPm in Ref. [26], the total power of high-power laser beams, passing through the m-th hard-edged rectangular aperture with width of bm 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]

where

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

where

and k = 2π/λ is the wavenumber with incident wavelength λ. By replacing P2m–3,P2m–2,Sm–1 on the right-hand side of Eqs. (10) with , , respectively, , , can be obtained from Eq. (10).

The corresponding cross second-order moment can be obtained as

By substituting Eqs. (6)–(11) into the definition of M2 factor, we obtain

the generalized M2 factor of high-power laser beams passing through a multi-apertured optical ABCD system can be yielded below:

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].

3. Numerical results and analyses

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, fq1 = 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. An aperture-spatial filter system.
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. 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. 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. 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.

4. Conclusion

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.

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