## Abstract

Instead of Zernike polynomials, ellipse Gaussian model is proposed to represent localized wave-front deformation in researching pointing and tracking errors in inter-satellite laser communication links, which can simplify the calculation. It is shown that both pointing and tracking errors depend on the center deepness *h*, the radiuses *a* and *b*, and the distance *d* of the Gaussian distortion and change regularly as they increase. The maximum peak values of pointing and tracking errors always appear around *h*=0.2*λ*. The influence of localized deformation is up to 0.7*µ*rad for pointing error, and 0.5*µ*rad for tracking error. To reduce the impact of localized deformation on pointing and tracking errors, the machining precision of optical devices, which should be more greater than 0.2*λ*, is proposed. The principle of choosing the optical devices with localized deformation is presented, and the method that adjusts the pointing direction to compensate pointing and tracking errors is given. We hope the results can be used in the design of inter-satellite lasercom systems.

©2008 Optical Society of America

## 1. Introduction

Comparing to microwave communications, inter-satellite laser communication (lasercom) has many advantages, such as smaller size and weight of the terminal, less power consumption, greater immunity to interference, larger data rate, and denser satellite orbit population, consequently it provides an attractive alternate to microwave communications for both commercial and military applications. [1–5] Inter-satellite lasercom relates to laser beam transmission which has recently been extensively studied. [6, 7] Due to the small beam divergence and the ultra-long distance of the communication links, wave-front aberrations strongly affect the spatial pointing and tracking of laser beams.

There are two major reasons which cause the wave-front aberrations. The first reason is the space environment which includes space radiation, contamination, and especially temperature variation. Temperature variation causes local changes in the optical properties of the devices, such as variation of the reflective index, variation of the curvature of the lens surface, variation of the thickness of the lens, and variation in the gap between lenses. The second reason is the processing technic. It is difficult for the optical devices, especially for that with large aperture, to be processed to the precision of 0.01*λ* and remain unchanged for long time, consequently localized distortions is almost inevitable. Both of the two reasons are equivalent to the deformation of the optical devices. When the beam transmits the optical devices with deformation, its wave-front will change locally.

Toyoshima *et al.* have studied mutual alignment errors in circle region using Zernike polynomials expressing wave-front aberrations. [10] Furthermore, Sun *et al.* developed the research to annular region. [11] Due to the orthogonality of Zernike polynomials, almost all the wave-front aberrations in the optical system can be represented by them. [12–14] However, it generally needs too many items of Zernike polynomials to express localized distortion, which complicates the calculation. To simplify the analysis, we proposed ellipse Gaussian model to represent localized deformation, which is proved simple in the calculation by comparison with Zernike polynomials. Based on ellipse Gaussian model, the effects of localized wave-front deformation on pointing and tracking errors are researched. The purpose of the research is to estimate how much the influence of localized wave-front deformation on pointing and tracking errors, and try to provide the evidence of processing precision for the optical devices used in lasercom.

This paper has the following outline. In Section 2 the ellipse Gaussian Model is introduced to describe local distortion. In Section 3 pointing and tracking errors are defined. Section 4 is devoted to numerical analysis. Section 5 summarizes our results.

## 2. Ellipse Gaussian model

Due to the limitation of processing technic and the effects of space environment, the localized distortion is extremely likely to appear in satellite optical system, especially in the primary mirror of transmitter antenna due to the large aperture. To simplify the analysis, we propose ellipse Gaussian model to express them, which is shown in Fig. 1 and can be written as

where *A* is the center value of the ellipse Gaussian function (the center deepness *h*=*A*(1-1/e)), *a* and *b* are the radiuses of the localized distortion, (*x*
_{0},*y*
_{0}) is the coordinate of the center, *d* is the distance from (*0*,*0*) to (*x*
_{0},*y*
_{0}), which can be represented as

Assuming that there is localized deformation in the primary mirror of reflection-style antenna, when the beam is reflected by it, localized wave-front deformation is generated. The forming process is shown in Fig. 2. The wave-front deformation can be written as

$$\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}=\psi \mathrm{exp}\left\{-\left[\frac{{\left(x-{x}_{0}\right)}^{2}}{{a}^{2}}+\frac{{\left(y-{y}_{0}\right)}^{2}}{{b}^{2}}\right]\right\}-\frac{\psi}{e},$$

where *ψ* denotes the center amplitude of ellipse Gaussian function, which is considered to be 4*Aπ*/*λ*. Equation (3) is composed of two parts, Φ_{1} and Φ_{2}. Φ_{1} is ellipse Gaussian function, and Φ_{2} is a constant.

The optical field of the beam reflected by mirrors can be shown in the form

where *H*(*x*,*y*) is the optical field before the optical device, *exp*(*j*Φ) is called aberration term caused by the localized distortion.

Root mean square (rms) is a conventional factor to evaluate the degree of wave-front aberrations, for the ellipse Gaussian function Φ_{1}, which can be expressed as

where *S* denotes the deformation area which is an ellipse with major axis radius *a* and minor axis radius *b*. From Eq. (5) we can find that rms proportionably depends on the center deepness *h*, but has no relation to the radiuses *a* and *b*.

## 3. Pointing and tracking errors

Mutual alignment errors are defined in Ref. 10 as the angle between the transmitting and receiving optical axes. We consider that, in fact, mutual alignment errors include two parts: pointing and tracking errors. They are described in the following subsections.

#### 3.1. Pointing error

Pointing error is defined as the angle between the transmitting optical axes with and without wave-front aberrations. Transmitting optical axis is determined by the direction with the peak intensity at a far-field. The definition of the coordinate systems is shown in Fig. 3, which is similar to that in Ref. 10. The transmitter beam is Gaussian beam with localized wave-front aberrations, which can be written as

where *C* is a constant, *F*
_{0} is the radius of curvature at the transmitter, *M*
_{1}(*x*
_{0},*y*
_{0}) is transmitter aperture function which is determined by the transmitter antenna with primary mirror radius *R*
_{1} and secondary mirror radius *R*
_{2}, *ω*
_{0} is waist radius of the Gaussian beam. The intensity distribution *I _{re}*(

*x*,

*y*) in the receiver plane is obtained as the following [15]

where *λ* is the wavelength, *z _{f}* is the distance of the two communication terminals. For transmitter beam free of aberrations, the peak intensity is at the origin. And for the beam with aberrations, it is at the position of

*I*(

_{re}*x*,

*y*) |

_{max}=

*I*(

_{re}*x*,

_{max}*y*). In this case, pointing error

_{max}*θ*can be written in the form

_{P}Related to Eqs. (3) and (8), we can find that pointing error *θ _{P}* caused by localized distortion depends on the following parameters: the center deepness

*h*, the radiuses

*a*and

*b*, and the distance

*d*.

#### 3.2. Tracking error

Tracking error is defined as the angle between the receiving optical axes with and without wave-front deformation. Receiving optical axis is obtained by the gravity center of the received optical power on an optical tracking sensor. Owing to the long distance between the two communication terminals, the received wave can be considered as plane wave. When the plane wave passes through the optical terminal which is equivalent to a lens with focal length *f*, it is focused on the focal plane, and the intensity is given by [16]

where *B* is a constant, *M*
_{2}(*x*,*y*) is receiver aperture function which is determined by the receiver antenna, *r*
_{1} and *r*
_{2} are the primary mirror radius and secondary mirror radius, Φ(*x*,*y*) is wave-front deformation in receiver plane. Similarly, when there is no aberrations in the optical systems, the gravity center of the received optical power in the focus plane is at the origin. However, normally the center of gravity is at (*X*,*Y*) when aberrations exist in the optical systems. By definition tracking error *θ _{T}* can be written as

where *X* and *Y* are given by the following equations [17]

Similar to pointing error *θ _{P}*, tracking error

*θ*also depends on the following parameters: the center deepness

_{T}*h*, the radiuses

*a*and

*b*, and the distance

*d*.

The optical fields for Eqs. (8) and (10) can be shown as the same form (to simplify the analysis, the optical field is considered as one dimension variable.)

where *H*(*x*) denotes Gaussian beam for pointing error, or plane beam for tracking error. *F* is the distance between two satellites for pointing error, or the focal length of receiver optical system for tracking error. And *D*=2*R*
_{1} for pointing error, or *D*=2*r*
_{1} for tracking error. Substituting Eq. (3) into Eq. (14), we can obtain the following equation

$$\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}\phantom{\rule{.2em}{0ex}}+\mathrm{exp}\left(\frac{j\psi}{e}\right){\int}_{{x}_{0}-a}^{{x}_{0}+a}H\left(x\right)\mathrm{exp}\left(j{\Phi}_{1}\right)\mathrm{exp}\left(-\frac{jk}{F}xu\right)dx.$$

Equation (15) shows that the optical field consists of three parts. By definition the first and the second parts don’t cause pointing and tracking errors which are mainly influenced by the third part. Therefore, to simply the analysis, we can only consider the third part which is shown as

From Eq. (16), we can find that pointing and tracking errors are mainly determined by the aberration term *exp*(*j*Φ_{1}). It is known that *exp*(*j*Φ_{1})=*exp*[*j*(Φ_{1}+_{2}
*π*)], namely the aberration term is a periodic function whose period is 2*π*. Therefore, pointing and tracking errors would vary periodically with the change of Φ_{1}. When Φ_{1}=0, pointing and tracking errors are zeros. We know that the wave-front difference for Φ_{1}=0 and Φ_{1}=(2*n*-1)*π* is the maximum. Therefore, the peaks of pointing and tracking errors would appear around Φ_{1}=(2*n*-1)*π* (*n* is positive integer). Due to Φ_{1} being a function of *x* and *y*, the peaks should be around rms=(2*n*-1)*π*. Furthermore, from the integral region we can conclude that it is the localized deformation, not the whole aperture, which determines the pointing and tracking errors, consequently rms should obtained from localized deformation area (See Eq. (5)). Related to Eq. (6), we can conclude that pointing and tracking errors would change periodically as the center deepnees *h* increases. Though the radiuses *a* and *b* don’t contribute to the rms of Φ_{1} according to Eq. (6), it determines the aberration area. When *a* rises, the value of *h*(*u*) increases, namely the influence of wave-front deformation increases too. According to the definitions of pointing and tracking errors, they would increase as the distortion becomes wide.

## 4. Numerical results and analysis

To show the advantages of ellipse Gaussian model, the comparison between ellipse Gaussian function and Zernike polynomials to represent the localized deformation is addressed in Figs. 4 and 5. For the localized deformation which is expressed accurately by ellipse Gaussian function, we represent it using Zernike polynomials with different terms. The term numbers are *N*=20, 40 and 60, respectively. The results are in the Fig. 4. When *N*=20 the result of Zernike polynomials is very poor, and when *N*=40 the result becomes better. When *N*=60 the result is close to that of Gaussian function. The results show that it does need many terms for Zernike polynomials to express the localized deformation with less error, which will complicate the calculation. Fig. 5 gives the results of Zernike polynomials with *N*=40 for different *a*/*D*. As can be seen that the result is better for large value of *a*/*D* than for small value of *a*/*D*. In a word, by comparison with Zernike polynomials, ellipse Gaussian model can really simplify the calculation due to its simple expression, especially for small value of *a*/*D*.

Based on ellipse Gaussian model, the numerical results of the effects of localized wave-front deformation on pointing and tracking errors are given in Figs. 6 and 7. In the calculation process, the parameters are *D*=2*R*
_{1}=2*r*
_{1}=250 mm, *R*
_{2}=*r*
_{2}=40 mm, *λ*=800 nm, *ω*
_{0}=125 mm, and *f*=1000 mm. The distance of the two satellites is taken to be *z _{f}*=50,000 km. Fig. 6 shows how pointing and tracking errors vary with the center deepness

*h*, the radiuses

*a*and

*b*, and the distance

*d*. In calculation we only consider the condition that the Gaussian deformation is totally in the aperture of the antenna, and the center of Gaussian distortion is in

*x*axis. As can be seen from Fig. 6, pointing and tracking errors do not monotonically rise with

*h*increasing as generally expected, but fluctuates like damped oscillation. On the other hand, pointing and tracking errors monotonically increases as

*a*rises. In other words, the wider localized distortion, the stronger influence on pointing and tracking errors. With the distance

*d*increasing, tracking error increases monotonically, while pointing error increases monotonically at first and then decreases secondly. The difference is considered that the beam contributing to tracking error is plane beam, while that contributing to pointing error is Gaussian beam whose intensity decreases with

*d*increasing. Fig. 7 shows clearly the fluctuation of pointing and tracking errors with

*h*and rms rising. The peak appears around

*h*=0.2

*λ*(rms=

*π*),

*h*=0.75

*λ*(rms=3

*π*),

*h*=1.25

*λ*(rms=5

*π*),

*et. al.*. The fluctuation period for rms value is 2

*π*. The results show that to reduce the impact of localized deformation on pointing and tracking errors, the center deepness

*h*should be more less than 0.2

*λ*, namely the machining accuracy of the optical devices should be more greater than 0.2

*λ*. Moreover, the influence of localized deformation is up to 0.7

*µ*rad for pointing error, and 0.5

*µ*rad for tracking error.

The comparison of pointing and tracking errors for localized deformation expressed by ellipse Gaussian function and Zernike polynomials are shown in Fig. 8. As can be seen that pointing and tracking errors due to wave-front aberrations described by Zernike polynomials, are gradually close to that expressed by Gaussian function with *N* increasing. Figs. 10(a) and 10(d) show that Zernike results are better for small value of *h* than for large value *h*. The reason is that, for small value of *h*, the localized deformation plays an important role, and the effect of Zernike error is comparatively weak. With *h* rising, the influence of the localized deformation reduces, then the impact of Zernike error gradually increases. Furthermore, as shown in Figs. 8(b) and 8(e), Zernike results are obviously worse for small value *a* than for large value *a*. The reason is that Zernike error is large for small value *a*/*D* than large value *a*/*D*, which is shown in Fig. 5.

From above numerical analysis, we can conclude that ellipse Gaussian model is an effective method for the localized distortion, especially for that with small values of *a*/*D*. To weaken the effect of localized deformation on pointing and tracking errors, processing precision of optical devices should be more than 0.2*λ*. If we have to use the optical devices with localized deformation, we may select them according to the following principles: (1) The deepness *h* is more less than 0.2*λ*; (2) The radiuses *a* and *b* are small; (3) The center position (*x*
_{0},*y*
_{0}) is near by the center of the optical device. In addition, if we know the localized deformation before laser beam transmitting/receiving, we can adjust pointing direction to compensate the pointing and tracking errors caused by localized aberrations.

## 5. Conclusion

To research localized deformation on pointing and tracking errors in inter-satellite lasercom, ellipse Gaussian model is proposed, which can simplify the calculation especially for small value of *a*/*D* by comparison with Zernike polynomials. It is found that pointing and tracking errors due to localized deformation are mainly determined by the center deepness *h*, the radiuses *a* and *b*, and the distance *d*. With the increasing of the deepness *h*, both of pointing and tracking errors fluctuate like damped oscillation with peak values around *h*=0.2*λ* (rms=*π*), *h*=0.75*λ* (rms=3*π*), *h*=1.25*λ* (rms=5*π*), *et al.*. The wider the localized deformation is, the more for the influence on pointing and tracking errors being. With the distance *d* rising, tracking error increases monotonically, while pointing error increases monotonically at first and then decreases monotonically. The effects of localized deformation is up to 0.7urad for pointing error, and 0.5urad for tracking error. To reduce the impact of localized deformation on pointing and tracking errors, the processing accuracy of optical devices should be more greater than 0.2*λ*. The principle of choosing the optical devices with localized distortion is presented, and the method that adjusts the pointing direction to compensate pointing and tracking errors is given. We hope the conclusion can be used in the design of inter-satellite lasercom systems.

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