# Propagation Characteristics of Hermite–Gaussian Beam under Pointing Error in Free Space

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## Abstract

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## 1. Introduction

_{mn}mode Gaussian beam, can be generated using higher-order solutions of the paraxial equation with Hermite polynomials (CO

_{2}laser) in rectangular coordinates [5]. A higher-order HG beam can form a multiple-spot pattern for the irradiance rather than a single spot as generated by lowest-order Gaussian beam, and furthermore, the HG beam has unique orthogonal spatial modes and well-preserved irradiance distribution in free-space propagation [5,6,7,8,9,10]. The definition of the HG beam’s spot size is different from that of the Gaussian beam which was proposed by Carter [11]. For FSOC system application, the HG beam has garnered attention due to its potential to improve the capacity using mode-division multiplexing (MDM) [12,13,14,15]. Meanwhile, the HG beam can be also used in FSOC with a single-input multiple-output (SIMO) system owing to its propagation characteristics. Therefore, research on the propagation characteristics and expressions of HG beams, which include average irradiance, spot size and location of the local extreme value in the average irradiance for the HG beam, is critical for HG beam applications in FSOC.

## 2. Modeling of the Average Irradiance under Pointing Error

_{mn}HG beam can be treated as a generalized eigenfunction of the optical field equation in free space, which is obtained as follows [5,8]:

_{00}Gaussian beam. Additionally, W

_{0}denotes the TEM

_{00}beam waist at the transmitter, and W denotes the TEM

_{00}spot size at the receiver, which is $W={W}_{0}\sqrt{1+{\mathsf{\Lambda}}_{0}^{2}}$, where ${\mathsf{\Lambda}}_{0}=2L/(k{W}_{0}^{2})$, $k=2\pi /\lambda $ denotes the optical wave number and $\lambda $ is the wavelength. Owing to the orthogonal spatial modes of the HG beam, its irradiance in Equation (1) allows for splitting into the horizontal (x-coordinate) and vertical (y-coordinate) direction components [5,6,7,8,9,10], and we observe the following: ${I}_{m}(x,L)=\frac{{W}_{0}}{W}{H}_{m}^{2}\left(\frac{\sqrt{2}x}{W}\right)\mathrm{exp}\left[-\frac{2{x}^{2}}{{W}^{2}}\right]$ and ${I}_{n}(y,L)=\frac{{W}_{0}}{W}{H}_{n}^{2}\left(\frac{\sqrt{2}y}{W}\right)\mathrm{exp}\left[-\frac{2{y}^{2}}{{W}^{2}}\right]$.

_{11}HG beam was selected to illustrate this process. The term S denotes the center of the HG beam at the transmitter, and O denotes the center of the receiver plane. The term SO denotes the propagation axis without pointing error (Figure 1a). The center of the HG beam moves to position O

_{j}under the pointing error angle θ

_{j}at time Δt

_{j}, and the cases of j = 1, 2 and 3 are shown in Figure 1b. The term SO

_{j}(j = 1, 2, 3, …) denotes the propagation direction under the pointing error, and θ

_{j}denotes the pointing error angle between SO

_{j}and SO. The average irradiance under the pointing error can be obtained by statistically averaging multiple HG beams at different positions (Figure 1c). Because the probability of appearance at a specific position is determined by the probability density function (PDF) of the pointing error angle, the statistical averaging results can be formulated through position probability weighting for the HG beam irradiance at different positions (see Figure 1c).

_{21}HG beam under the pointing error is equal to the product of ${\langle {I}_{2}(x,L)\rangle}_{PE}$ and ${\langle {I}_{1}(y,L)\rangle}_{PE}$ as follows:

## 3. Numerical Simulation Results and Verification

_{0}= 0.05 m, wavelength λ = 850 nm and propagation distance L changes from 1000 to 3000 m. According to Equations (2) and (3), 10,000 groups of random displacements along the horizontal and vertical axes were generated, and the standard variance of the pointing error angle was set as 0, 5, 10 and 20 μrad. The simulation parameters are set in Table 2; they are often used in practical FSOC systems.

_{0}= 0.05, 0.1, and 0.2 m, respectively. Figure 5a–c shows the cases of p = 1, 2 and 3, respectively. We found that a larger beam waist at the transmitter could mitigate the profile change of the average irradiance of the HG beam under an increasing pointing error. Therefore, the original profile of the HG beam can be maintained by properly increasing the beam waist and controlling the standard variance of pointing error.

## 4. Discussion

#### 4.1. Effective Spot Size

_{00}Gaussian beam, the HG beam forms a pattern of spots rather than a single spot of light; the conventional spot size W of a Gaussian beam is not suitable for an HG beam, and a new definition of the spot size of an HG beam is needed.

_{0}equal to 0.05, 0.1 and 0.2 m, respectively. We found that the effective spot size of the higher-order HG beam experiences less broadening under the pointing error than that of the lower-order HG beam. Meanwhile, the pointing error-induced HG beam spot size broadening decreased with an increase in the beam waist at the transmitter.

#### 4.2. Location of the Local Extreme Value in the Average Irradiance under Pointing Error

#### 4.3. Average Received Power and SNR Loss

_{11}, TEM

_{22}and TEM

_{33}, are selected for the average received power investigation. Each array receiving the aperture center is located at the location of the local maxima. The TEM

_{11}HG beam irradiance without pointing error had four local maximum points, and their peak irradiances were equal. Therefore, the receiving aperture D

_{0}is arranged at four locations of the local maxima. The TEM

_{22}HG beam without pointing error has nine local maximum points, which can be divided into three groups according to their peak irradiance. Therefore, the receiving apertures D

_{1}, D

_{2}and D

_{3}were arranged at nine locations of the local maxima (see Figure 7b). Similarly, the TEM

_{33}HG beam without pointing error has 16 local maxima points, and it can also be divided into three groups according to their peak irradiance: the receiving apertures D

_{1}’, D

_{2}’ and D

_{3}’ are arranged at 16 locations of local maxima (see Figure 7c). Taking the TEM

_{33}HG beam as an example, the receiving apertures are the same but are divided into three groups according to the peak irradiance (red, yellow and orange), and the average received power values of similar color receiving apertures are equal.

_{f}, y

_{f}), Equation (16) can be approximated as follows:

_{11}, TEM

_{22}and TEM

_{33}are as follows:

_{11}HG beam, the average received power on the receiving aperture D

_{0}decreases with an increase in the pointing error. As shown in Figure 8b, for the TEM

_{22}HG beam, the average received power on D

_{1}and D

_{2}decreased with an increase in the pointing error, and the average received power on D

_{1}decreased faster than the average received power on D

_{2}. The average received power on D

_{3}fluctuated with an increase in the pointing error. Notably, the average received powers on D

_{1}, D

_{2}and D

_{3}are equal when ${\sigma}_{r}$ = 12.67 μrad; we define it as the “equal power point”. As shown in Figure 8c, for the TEM

_{33}HG beam, the variation trend of the average received power on D

_{1}’, D

_{2}’ and D

_{3}’ with an increase in the pointing error was the same as that of the TEM

_{22}HG beam. The “equal power point” appears at ${\sigma}_{r}$ = 14.05 μrad.

_{22}HG beam at the “equal power point” ${\sigma}_{r}$ = 12.67 μrad exhibits a flat-topped profile. As shown in Figure 10, the average irradiance of the TEM

_{33}HG beam at the “equal power point” ${\sigma}_{r}$ = 14.05 μrad exhibits a slightly hollow shape. The average irradiance of the HG beam appears to have a nearly flat-topped shape at the “equal power point”.

_{33}HG beam is taken as an example to analyze the received power loss ratio (also SNR loss ratio) under the influence of pointing error in the corresponding receiving aperture. The numerical simulation parameters are the same as above, and the transmission distance is 3000 m.

_{2}’ and the average SNR decaying speed of receiving aperture D

_{1}’ is faster than that of receiving aperture D

_{2}’. For receiving aperture D

_{3}’, the average SNR decreases and then increases with the increase in pointing error, and the increase in average SNR is 0.87 dB when ${\sigma}_{r}$ = 16.97 μrad. This shows that the transmitted power is homogenized at the receiver under the influence of pointing error. In order to achieve better reception efficiency for the HG beam, we should control the transmission parameters. It can be seen that when the propagation parameter is set as shown in Table 2, the standard variance of pointing error should be less than 7.5 μrad for the TEM

_{33}HG beam so that the SNR loss at each receiving aperture is less than 3.00 dB.

## 5. Conclusions

_{33}HG beam, the standard variance of pointing error should be within 7.5 μrad so that the SNR loss at each receiving aperture is less than 3.00 dB with the transmission distance of 3000 m. The fundamental theoretical expressions of average irradiance for an HG beam under pointing error have provided effective guidance for analyzing the propagation characteristics and link performance.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**Forming process of average irradiance of HG beam under pointing error. (

**a**) Propagation axis without pointing error, (

**b**) Pointing error induced random deviation at different times, (

**c**) Average irradiance under pointing error.

**Figure 2.**Average irradiance of HG beam under pointing error with the increase in propagation distance in the case of p = 1.

**Figure 3.**Average irradiance of HG beam under pointing error with the increase in propagation distance in the case of p = 2.

**Figure 4.**Average irradiance of HG beam under pointing error with the increase in propagation distance in the case of p = 3.

**Figure 5.**Average irradiance of HG beam under pointing error with different beam waists at transmitter. (

**a**) p = 1, (

**b**) p = 2 and (

**c**) p = 3.

**Figure 8.**Average received power at the corresponding receiving aperture under the influence of pointing error. (

**a**) TEM

_{11}, (

**b**) TEM

_{22}and (

**c**) TEM

_{33}.

**Figure 9.**Average irradiance of TEM22 HG beam when ${\sigma}_{r}$ = 12.67 μrad. (

**a**) A 3D diagram, (

**b**) a vertical diagram and (

**c**) a cross-section along the x axis.

**Figure 10.**Average irradiance of TEM33 HG beam when ${\sigma}_{r}$ = 14.05 μrad. (

**a**) A 3D diagram, (

**b**) a vertical diagram and (

**c**) a cross-section along the x axis.

${\langle {I}_{p}(s,L)\rangle}_{PE}$ | p |

$\frac{8{W}_{0}}{{({W}^{2}+4{\sigma}_{r}^{2})}^{5/2}}\mathrm{exp}\left(-\frac{2{s}^{2}}{{W}^{2}+4{\sigma}_{r}^{2}}\right)\left[{W}^{2}({s}^{2}+{\sigma}_{r}^{2})+4{\sigma}_{r}^{2}\right]$ | 1 |

$\begin{array}{l}\frac{4{W}_{0}}{{({W}^{2}+4{\sigma}_{r}^{2})}^{9/2}}\mathrm{exp}\left(-\frac{2{s}^{2}}{{W}^{2}+4{\sigma}_{r}^{2}}\right)[{W}^{8}+8{W}^{6}(-{s}^{2}+{\sigma}_{r}^{2})\\ +16{W}^{4}({s}^{4}+2{s}^{2}{\sigma}_{r}^{2}+3{\sigma}_{r}^{4})+256{W}^{2}{\sigma}_{r}^{4}({s}^{2}+{\sigma}_{r}^{2})+512{\sigma}_{r}^{8}]\end{array}$ | 2 |

$\begin{array}{l}\frac{32{W}_{0}}{{({W}^{2}+4{\sigma}_{r}^{2})}^{13/2}}\mathrm{exp}\left(-\frac{2{s}^{2}}{{W}^{2}+4{\sigma}_{r}^{2}}\right)[9{W}^{10}({s}^{2}+{\sigma}_{r}^{2})+12{W}^{8}(-2{s}^{4}+9{\sigma}_{r}^{4})\\ +16{W}^{6}({s}^{6}+3{s}^{4}{\sigma}_{r}^{2}-9{s}^{2}{\sigma}_{r}^{4}+33{\sigma}_{r}^{6})+576{W}^{4}{\sigma}_{r}^{4}({s}^{4}+2{s}^{2}{\sigma}_{r}^{2}+3{\sigma}_{r}^{4})+4608{W}^{2}{\sigma}_{r}^{8}({s}^{2}+{\sigma}_{r}^{2})+6144{\sigma}_{r}^{12}]\end{array}$ | 3 |

Parameters | Value |
---|---|

Optical beam waist (W_{0}) | 0.05 m |

Wavelength (λ) | 850 nm |

Propagation distance (L) | 1000, 2000 and 3000 m |

Standard variance of pointing error angle (${\sigma}_{r}$) | 0, 5, 10 and 20 μrad |

Grid points (N × N) | 1024 × 1024 |

Simulated numbers (M) | 10,000 |

Locations of the local maxima | p |

$\pm \frac{\sqrt{{W}^{4}+2{W}^{2}{\sigma}_{r}^{2}-8{\sigma}_{r}^{4}}}{\sqrt{2}W}$; $\pm \frac{W}{\sqrt{2}}$ (${\sigma}_{r}=0$). | 1 |

0, $\pm \sqrt{\frac{3{W}^{2}}{4}+{\sigma}_{r}^{2}-\frac{16{\sigma}_{r}^{4}-\sqrt{{W}^{8}+4{W}^{6}{\sigma}_{r}^{2}-8{W}^{4}{\sigma}_{r}^{4}+128{\sigma}_{r}^{8}}}{2{W}^{2}}}$; 0, $\pm \frac{\sqrt{5}W}{2}$ (${\sigma}_{r}=0$). | 2 |

$\pm \sqrt{\frac{5{W}^{16}+16{W}^{14}{\sigma}_{r}^{2}-64{W}^{12}{\sigma}_{r}^{4}+4{W}^{6}{\sigma}_{r}^{2}{q}^{1/3}-48{W}^{4}{\sigma}_{r}^{4}{q}^{1/3}+{q}^{2/3}+4{W}^{8}(192{\sigma}_{r}^{8}+{q}^{1/3})}{4{W}^{6}{q}^{1/3}}}$, $\pm \sqrt{\frac{\left[5{W}^{16}-16{W}^{14}{\sigma}_{r}^{2}+64{W}^{12}{\sigma}_{r}^{4}+8{W}^{6}{\sigma}_{r}^{2}\right]{q}^{1/3}-96{W}^{4}{\sigma}_{r}^{4}{q}^{1/3}-{q}^{2/3}+8{W}^{8}\left[96{\sigma}_{r}^{8}+{q}^{1/3}\right]}{8{W}^{6}{q}^{1/3}}}$; $\pm \frac{\sqrt{3}W}{2}$, $\pm \frac{\sqrt{9{W}^{2}+\sqrt{57}{W}^{2}}}{2\sqrt{2}}$ (${\sigma}_{r}=0$). | 3 |

Locations of the local minima | p |

0 | 1 |

$\pm \sqrt{\frac{3{W}^{2}}{4}+{\sigma}_{r}^{2}-\frac{16{\sigma}_{r}^{4}+\sqrt{{W}^{8}+4{W}^{6}{\sigma}_{r}^{2}-8{W}^{4}{\sigma}_{r}^{4}+128{\sigma}_{r}^{8}}}{2{W}^{2}}}$; $\pm \frac{W}{2}$ (${\sigma}_{r}=0$). | 2 |

0,$\pm \sqrt{\frac{\left[-5{W}^{16}-16{W}^{14}{\sigma}_{r}^{2}+64{W}^{12}{\sigma}_{r}^{4}+8{W}^{6}{\sigma}_{r}^{2}\right]{q}^{1/3}-96{W}^{4}{\sigma}_{r}^{4}{q}^{1/3}-{q}^{2/3}-8{W}^{8}\left[96{\sigma}_{r}^{8}+{q}^{1/3}\right]}{8{W}^{6}{q}^{1/3}}}$; 0, $\pm \frac{\sqrt{9{W}^{2}-\sqrt{57}{W}^{2}}}{2\sqrt{2}}$ (${\sigma}_{r}=0$). | 3 |

In the presence of pointing error | |

$\left(0,-\frac{\sqrt{{W}^{4}+2{W}^{2}{\sigma}_{r}^{2}-8{\sigma}_{r}^{4}}}{\sqrt{2}W}\right)$, $\left(0,\frac{\sqrt{{W}^{4}+2{W}^{2}{\sigma}_{r}^{2}-8{\sigma}_{r}^{4}}}{\sqrt{2}W}\right)$, $\left(-\sqrt{\frac{3{W}^{2}}{4}+{\sigma}_{r}^{2}-\frac{16{\sigma}_{r}^{4}-\sqrt{{W}^{8}+4{W}^{6}{\sigma}_{r}^{2}-8{W}^{4}{\sigma}_{r}^{4}+128{\sigma}_{r}^{8}}}{2{W}^{2}}},-\frac{\sqrt{{W}^{4}+2{W}^{2}{\sigma}_{r}^{2}-8{\sigma}_{r}^{4}}}{\sqrt{2}W}\right)$, $\left(\sqrt{\frac{3{W}^{2}}{4}+{\sigma}_{r}^{2}-\frac{16{\sigma}_{r}^{4}-\sqrt{{W}^{8}+4{W}^{6}{\sigma}_{r}^{2}-8{W}^{4}{\sigma}_{r}^{4}+128{\sigma}_{r}^{8}}}{2{W}^{2}}},-\frac{\sqrt{{W}^{4}+2{W}^{2}{\sigma}_{r}^{2}-8{\sigma}_{r}^{4}}}{\sqrt{2}W}\right)$, $\left(-\sqrt{\frac{3{W}^{2}}{4}+{\sigma}_{r}^{2}-\frac{16{\sigma}_{r}^{4}-\sqrt{{W}^{8}+4{W}^{6}{\sigma}_{r}^{2}-8{W}^{4}{\sigma}_{r}^{4}+128{\sigma}_{r}^{8}}}{2{W}^{2}}},\frac{\sqrt{{W}^{4}+2{W}^{2}{\sigma}_{r}^{2}-8{\sigma}_{r}^{4}}}{\sqrt{2}W}\right)$, $\left(\sqrt{\frac{3{W}^{2}}{4}+{\sigma}_{r}^{2}-\frac{16{\sigma}_{r}^{4}-\sqrt{{W}^{8}+4{W}^{6}{\sigma}_{r}^{2}-8{W}^{4}{\sigma}_{r}^{4}+128{\sigma}_{r}^{8}}}{2{W}^{2}}},\frac{\sqrt{{W}^{4}+2{W}^{2}{\sigma}_{r}^{2}-8{\sigma}_{r}^{4}}}{\sqrt{2}W}\right)$. | Coordinates of the local maximum irradiance |

$\left(x,0\right)$, $\left(-\sqrt{\frac{3{W}^{2}}{4}+{\sigma}_{r}^{2}-\frac{16{\sigma}_{r}^{4}+\sqrt{{W}^{8}+4{W}^{6}{\sigma}_{r}^{2}-8{W}^{4}{\sigma}_{r}^{4}+128{\sigma}_{r}^{8}}}{2{W}^{2}}},y\right)$, $\left(\sqrt{\frac{3{W}^{2}}{4}+{\sigma}_{r}^{2}-\frac{16{\sigma}_{r}^{4}+\sqrt{{W}^{8}+4{W}^{6}{\sigma}_{r}^{2}-8{W}^{4}{\sigma}_{r}^{4}+128{\sigma}_{r}^{8}}}{2{W}^{2}}},y\right)$. | Coordinates of the local minimum irradiance |

In the absence of pointing error | |

$\left(-\frac{\sqrt{5}W}{2},-\frac{W}{\sqrt{2}}\right)$, $\left(0,-\frac{W}{\sqrt{2}}\right)$, $\left(\frac{\sqrt{5}W}{2},-\frac{W}{\sqrt{2}}\right)$, $\left(-\frac{\sqrt{5}W}{2},\frac{W}{\sqrt{2}}\right)$, $\left(0,\frac{W}{\sqrt{2}}\right)$, $\left(\frac{\sqrt{5}W}{2},\frac{W}{\sqrt{2}}\right)$. | Coordinates of the local maximum irradiance |

$\left(x,0\right)$, $\left(-\frac{W}{2},y\right)$, $\left(\frac{W}{2},y\right)$. | Coordinates of the local minimum irradiance |

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**MDPI and ACS Style**

Liu, X.; Jiang, D.; Zhang, Y.; Kong, L.; Zeng, Q.; Qin, K.
Propagation Characteristics of Hermite–Gaussian Beam under Pointing Error in Free Space. *Photonics* **2022**, *9*, 478.
https://doi.org/10.3390/photonics9070478

**AMA Style**

Liu X, Jiang D, Zhang Y, Kong L, Zeng Q, Qin K.
Propagation Characteristics of Hermite–Gaussian Beam under Pointing Error in Free Space. *Photonics*. 2022; 9(7):478.
https://doi.org/10.3390/photonics9070478

**Chicago/Turabian Style**

Liu, Xin, Dagang Jiang, Yu Zhang, Lingzhao Kong, Qinyong Zeng, and Kaiyu Qin.
2022. "Propagation Characteristics of Hermite–Gaussian Beam under Pointing Error in Free Space" *Photonics* 9, no. 7: 478.
https://doi.org/10.3390/photonics9070478