## Abstract

Divergence angle of antenna is an important parameter in laser optical communication. It determines the power of the receiver terminal. In this paper, the influence of temperature on the divergence angle is discussed. Theoretical analysis and experiment results demonstrate that the relationship between the variance of temperature and of divergence angle is linear.

© 2012 OSA

## 1. Introduction

Compared to traditional approach using microwave communication system, the use of optical wavelength for communication between ground and satellite offers several advantages, such as smaller size and mass of terminal, less power consumption, more immune to interference, lower probability of interception, larger data rate, broader modulation bandwidths, higher antenna gain with relatively smaller aperture size, denser satellite orbit population, and the like. For this, laser communication has been under active development over the past three decades [1–7].

As we all know, for the long distance between ground and satellite, the divergence angle of laser beam transmitted by laser communication optical system must be very small to maintain links, typically on the order of tens of microradians, else the signal and beam energy at the receiving site would be significantly degraded resulting in large number of error codes or even no signal. Thus, optical system plays an important role in laser communication system, especially antenna [8].

However, the divergence angle of laser beam transmitted by laser communication optical system is often in unstable state which is not our expectation because of the change of environment temperature. Although thermal control strategies are often implemented on laser communication system, it is inevitable that the complexity of the whole system be greatly increased resulting in the increment of the total mass, uncertainty of the performance and the launch cost of satellite. And often, using these thermal control strategies, the operating temperature of antenna can be limited to a certain temperature range instead of a fixed temperature. Hence, it is necessary and imperative to make clear that what’s the relationship between temperature and beam divergence angle of antenna on laser communication system and how much does divergence angle can be affected by temperature.

Generally, beam divergence angle of antenna is mainly influenced by three factors: a) the distance between primary mirror and secondary mirror; b) curvature change of primary mirror; c) curvature change of secondary mirror. Unfortunately, all these factors are impacted by temperature [9]. Therefore, not only the coefficient of thermal expansion (CTE) of material selected for the primary and secondary mirrors should be very low, but also the material of the link between the primary and secondary mirrors. T. P. O’Brien *et al*. have studied the influence of temperature on the link length between mirrors and the resulting defocus of camera [10].

In this paper, we’ll calculate the theoretical influence of temperature on the antenna divergence angle of laser communication optical system in the next section. Then we’ll go on to the emulation of the impact using optical design software Code V. In section 4, we’ll show our experiment result. At last, the conclusion will be given.

## 2. Theoretical analysis

Sketch map of the antenna of laser communication optical system is shown in Fig. 1 .

M_{1} and M_{2} denotes secondary and primary mirrors, respectively. M_{1} is spherical and its radius is *r*_{1}, while M_{2} is a high-order aspheric surface with vertex radius *r*_{2}, which can be expressed using Eq. (1):

*K*is conic constant, and

*A*,

*B*are high-order coefficients of aspheric surface with $h=\sqrt{{x}^{2}+{y}^{2}}$ as depicted in Fig. 2 . P

^{′}is the projection P(

*z*

_{2},

*h*

_{2}) on

*xoy*plane and its coordinate is (

*x*

_{2},

*y*

_{2}).

The distance between M_{1} and M_{2} is *d*. The height of marginal ray on M_{1} and M_{2} are *h*_{1} and *h*_{2} respectively. The incident angle of marginal ray on M_{1} and M_{2} are *i*_{1} and *i*_{2} respectively. Suppose *α*_{1} and *α*_{2} are thermal coefficient of the substrate of M_{1} and M_{2}, and *α*_{3} is thermal coefficient of supporting frame material between M_{1} and M_{2}. Thus, when temperature changes Δ*T*, *r*_{1}, *r*_{2} and *d* will change to *r*_{1}′, *r*_{2}′ and *d*′ expressed by Eqs. (2), (3) and (4) respectively (high-order terms of aspherical polynomial of M_{2} are omitted, because they are very small compared to the change of radius):

The incident angle of marginal ray on M_{1} changes to *i*_{1}′ as Eq. (5) depicts:

Suppose the marginal ray height on M_{2} is *h*_{2}′. Then, the relationship between *h*_{2}′ and *d*′ is:

According to Eqs. (1) and (6), we can get P(*z*_{2}′, *h*_{2}′), the point of intersection between marginal ray and M_{2}.

According to Eq. (1), differential coefficient of *z* can be expressed by Eq. (7):

*z*′ is the normal line of the surface on point P(

*z*

_{2}′,

*h*

_{2}′). So the angle between

*z*axis and the normal line of marginal point P(

*z*

_{2}′,

*h*

_{2}′) on M

_{2}can be expressed using Eq. (8):

Due to the changes of *r*_{1} and *r*_{2}, the incident angle of marginal ray on M_{2} changes to *i*_{2}′:

For the ideal telescopic antenna, we know that, *i*_{1} is equal to *i*_{2}, and the angle of divergence *θ* is zero. Hence, the change of divergence angle Δ*θ* induced by the temperature change is:

According to the Eqs. (9) and (10), we can get Eq. (11):

Equation (11) is the relationship between the change of antenna divergence angle and the change of temperature. The case of Δ*θ*<0 means that the beam emitted from the antenna is convergent, while the beam is divergent in case of Δ*θ*>0. The result is based on condition that the design effective aperture is smaller than actual aperture. Otherwise in the former case, the marginal ray on M_{1} will not hit M_{2}.

For our system, *r*_{1} = 25mm, *r*_{2} = 300mm, *h*_{1} = 3.6mm, *d* = 137.5mm, *K* = −0.9713, and aperture *D* = 87.2346mm. The materials and CTE of the substrate of M_{1}, M_{2} and the link between M_{1} and M_{2}, are listed in Table 1
.

We calculated Δ*θ* as the function of Δ*T* with the step of 0.2°C. The results are shown in Fig. 3
and Table 2
which indicate that the relationship between Δ*θ* and Δ*T* is linear. And Δ*θ*/Δ*T* equals to −0.574μard/°C.

## 3. Simulation result

We input the parameters mentioned in section 2 into Code V to establish the system model and simulated its divergence angle at different temperatures from 22°C to 24°C. The results are as follows in Table 3
. The simulation result perfectly matched the theory result with just a little difference because of the omission of high-order terms of aspherical polynomial of M_{2}.

Suppose all components of the system are made of aluminum(CET = 23.4E-6/°C). Define a, b, and c are contribution factor of changes of *d*, *r*_{2} and *r*_{1} respectively. The contribution of each factor to the overall variance of divergence angle with 1 degree temperature change is shown in Table 4
. The result shows that the contribution of factor a and c is negative and the contribution of b is positive. Factor *d* and *r*_{2} are major influences on afocal telescope divergence angle due to temperature change. The overall variance of divergence angle of all-aluminum system is even smaller than that of our elaborate system.

As we know, materials with small CET are very expensive and hard to manufacture. From the result above we know that it is unnecessary to choose small-CET material in making a fine system. Chosen appropriate materials, it is possible to make a telescope without temperature influence.

## 4. Experiment result

In order to verify the precision of the deduced result above, we measured the result of our system in laboratory under different temperature conditions from 22°C to 24°C. Figure 4
shows the geometrical relationship among *f*, *D* and sag which is expressed by “Power” in MetrPro software. The measurements were done using Zygo interferometer and MetrPro. The experiment results are showed in Fig. 5
.

From Fig. 4 we can get the relationship among *f*, *D* and sag expressed as Eq. (12):

Because the interferometer beam goes through the optics twice, then

Using Eq. (13) different *θ* value at different temperatures from 22°C to 24°C are calculated in Table 5
and their relationship is shown in Fig. 6
. The results also illustrate that the relationship between Δ*θ* and Δ*T* is linear. And Δ*θ*/Δ*T* equals to −0.611μard/°C. The experiment result exactly matches the theory. The experiment result of Δ*θ*/Δ*T* is a little larger than that of the theory because of the M_{2} mount is made of titanium alloy which has a lager CTE and few of it is between M_{1} and M_{2}.

## 5. Conclusion

A method of analyzing temperature influence on divergence angle of afocal telescope used in laser communication has been proposed. Based on it, two different models have been established using Code V. The results have shown that the relationship between the variance of temperature and of divergence angle is linear and perfect telescope without temperature influence can be manufactured choosing appropriate materials. The linear relationship between the variance of temperature and of divergence angle has been perfectly verified by experimental results. The results are useful in optics athermal design.

## Acknowledgments

This work is supported by Beijing Institute of Space Mechanics and Electricity and National Key Laboratory of Tunable Laser Technology, Harbin Institute of Technology. The analysis of the theory is under the discussion of all the authors. The experiment datum was made by the communication author when he was pursuing Ph. D Harbin Institute of Technology.

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