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Twisted elliptical multi-Gaussian Schell-model beams and their propagation properties: reply

Haiyun Wang, Xiaofeng Peng, Lin Liu, Fei Wang, and Yangjian Cai

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Abstract

The correction pointed out by Nemes [J. Opt. Soc. Am. A 39, 667 (2022) [CrossRef]  ] to our paper [J. Opt. Soc. Am. A 37, 89 (2020) [CrossRef]  ] is welcome. The author claims that the beam propagation factors $M_x^2$, $M_y^2$ are inadequate to characterize twisted elliptical multi-Gaussian Schell-model (TEMGSM) beams, which are introduced in our paper. Further, the author points out that there are two pairs of independent, normalized invariants to classify the TEMGSM beams. In this reply, we present the numerical example of the normalized intrinsic propagation invariants of the TEMGSM beams.

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In our previous paper [1], we introduced a new kind of partially coherent source named twisted elliptical multi-Gaussian Schell-model (TEMGSM) sources. The general ${{4}} \times {{4}}$ beam matrix of the second-order moments is obtained to analyze the beam propagation ratios $M_x^2$ and $M_y^2$. In the comment on our paper, the author points out that only the beams with separable properties in $x$ and $y$ axes can be characterized by $M_x^2$ and $M_y^2$ such as a stigmatic (ST) beam and an aligned simple astigmatic (ASA) beam [2,3]. According to Eqs. (19)–(23) in [1], the ${{4}} \times {{4}}$ beam matrix ${\textbf{P}}$ for the TEMGSM beams can be represented as

$${\textbf P}= \left[{\begin{array}{*{20}{c}}{\langle {x^2}\rangle}&\,\,\,0&\,\,\,0&\,\,\,{\langle x{\theta _y}\rangle}\\0&\,\,\,{\langle {y^2}\rangle}&\,\,\,{\langle y{\theta _x}\rangle}&\,\,\,0\\0&\,\,\,{\langle y{\theta _x}\rangle}&\,\,\,{\langle \theta _x^2\rangle}&\,\,\,0\\{\langle x{\theta _y}\rangle}&\,\,\,0&\,\,\,0&\,\,\,{\langle \theta _y^2\rangle}\end{array}} \right] = \left[{\begin{array}{*{20}{c}}W&\,\,\,M\\{{M^T}}&\,\,\,U\end{array}} \right],$$
where ${{W}}$, ${{M}}$, and ${{U}}$ are three ${{2}} \times {{2}}$ submatrices. The superscript $T$ denotes the transpose of the matrix. Note that the terms $\langle xy\rangle$, $\langle x{\theta _x}\rangle$, $\langle y{\theta _y}\rangle$, and $\langle {\theta _x}{\theta _y}\rangle$ are all zero. Following from Eq. (1), the matrix ${\textbf{P}}$ has six independent elements, but it cannot be separated in $x$ and $y$ directions due to its nonzero $\langle x{\theta _y}\rangle$ and $\langle y{\theta _x}\rangle$ elements. Therefore, the TEMGSM beams belong to so-called general astigmatic (GA) beams in classification [2], which means that it is not enough to fully characterize the beam propagation properties using $M_x^2$ and $M_y^2$.

In [2], the author suggests the use of two pairs of independent and normalized invariants, i.e., ($M_{{\rm eff}}^4$, $a$) and ($a$, ${a_M}$), to characterize the TEMGSM beams, given by

$$M_{{\rm eff}}^4 = 4{k^2}{\Delta ^{1/2}},$$
$$a = 2{k^2}\left({T - 2{\Delta ^{1/2}}} \right) \ge 0,$$
$${a_M} = 2{k^2}\left({{T_M} - 2{\Delta ^{1/2}}} \right) = \left({1/2} \right){\left({M_{{\rm eff}}^4 - 1} \right)^2} \ge a,$$
where $\Delta = {\rm Det}[{\textbf P}], T = {\rm{Tr}}[WU - {M^2}]$ and ${T_M} = 4{k^2}\Delta + 1/(4{k^2})\cdot k = 2\pi /\lambda$ is the wavenumber, with $\lambda$ being the wavelength. Det and ${\rm{Tr}}$ denote the determinant and trace of the matrix, respectively. On substituting Eq. (1) into Eqs. (2)–(4), we obtain the normalized invariants ($M_{{\rm eff}}^4$, $a$) and ($a$, ${a_M}$) for the TEMGSM beams, i.e.,
$$\Delta = \left({\langle {y^2}\rangle \langle \theta _x^2\rangle - {{\langle y{\theta _x}\rangle}^2}} \right)\left({\langle {x^2}\rangle \langle \theta _y^2\rangle - {{\langle x{\theta _y}\rangle}^2}} \right),$$
$$T = \langle {x^2}\rangle \langle \theta _x^2\rangle + \langle {y^2}\rangle \langle \theta _y^2\rangle - 2\langle x{\theta _y}\rangle \langle y{\theta _x}\rangle ,$$
$${T_M} = 4{k^2}\Delta + 1/\left({4{k^2}} \right).$$

As a numerical example, Fig. 1 shows a variation of the propagation invariants ($M_{{\rm eff}}^4$, $a$) and ($a$, ${a_M}$) of the TEMGSM beams with different beam indices $M$ as a function of the twist factor $|u|$. The beam parameters are the same as those used in Fig. 8 in [1]. It can be seen from Fig. 1 that the effective beam propagation ratio $M_{{\rm eff}}^4$ increases with the increase in twist factor and beam index $M$, implying that the beam with larger $u$ or $M$ has poor beam quality. The variation of $M_{{\rm eff}}^4$ with the twist factor is almost the same as that of the propagation factor $M_x^2$($M_y^2$) with the twist factor reported in Fig. 8 in [1]. Therefore, the results obtained in [1] are also consistent with the variation of $M_{{\rm eff}}^4$.

 figure: Fig. 1.

Fig. 1. Variation of the propagation invariants of a TEMGSM beam for different values of twist factor $u$ with different beam indices $M$.

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Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

REFERENCES

1. H. Wang, X. Peng, L. Liu, F. Wang, and Y. Cai, “Twisted elliptical multi-Gaussian Schell-model beams and their propagation properties,” J. Opt. Soc. Am. A 37, 89–97 (2020). [CrossRef]  

2. G. Nemes, “Twisted elliptical multi-Gaussian Schell-model beams and their propagation properties: comment,” J. Opt. Soc. Am. A 39,667–671 (2022) [CrossRef]  .

3. ISO, “’Lasers and laser-related equipment—Test methods for laser beam widths, divergence angles and beam propagation ratios’—part 1: stigmatic and simple astigmatic beams,” ISO 11146-1:2005, Part 2: “General astigmatic beams”; ISO/TR 11146-3:2004, Part 3: “Intrinsic and geometrical laser beam classification, propagation, and details of test methods”; ISO/TR 11146-3:2004/Cor 1 (2005).

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Figures (1)
Fig. 1.
Fig. 1. Variation of the propagation invariants of a TEMGSM beam for different values of twist factor $u$ with different beam indices $M$.

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Equations (7)

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P = [ ⟨ x 2 ⟩ 0 0 ⟨ x θ y ⟩ 0 ⟨ y 2 ⟩ ⟨ y θ x ⟩ 0 0 ⟨ y θ x ⟩ ⟨ θ x 2 ⟩ 0 ⟨ x θ y ⟩ 0 0 ⟨ θ y 2 ⟩ ] = [ W M M T U ] ,
M e f f 4 = 4 k 2 Δ 1 / 2 ,
a = 2 k 2 ( T − 2 Δ 1 / 2 ) ≥ 0 ,
a M = 2 k 2 ( T M − 2 Δ 1 / 2 ) = ( 1 / 2 ) ( M e f f 4 − 1 ) 2 ≥ a ,
Δ = ( ⟨ y 2 ⟩ ⟨ θ x 2 ⟩ − ⟨ y θ x ⟩ 2 ) ( ⟨ x 2 ⟩ ⟨ θ y 2 ⟩ − ⟨ x θ y ⟩ 2 ) ,
T = ⟨ x 2 ⟩ ⟨ θ x 2 ⟩ + ⟨ y 2 ⟩ ⟨ θ y 2 ⟩ − 2 ⟨ x θ y ⟩ ⟨ y θ x ⟩ ,
T M = 4 k 2 Δ + 1 / ( 4 k 2 ) .
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