Impact of an axial non-confocal antenna on the on-orbit lasercom receiver analyzed by the simplified combination method of ray tracing and diffraction theory

Xiaolong Xie; Yanping Zhou; Siyuan Yu; Sheng Zhao; Jing Ma

doi:10.1364/OE.392396

1. Introduction

In recent years, rapidly increasing data transmitting demand of satellite communication is hastening the shift from radio-frequency(RF) carrier to laser carrier [1]. Compared with satellite RF communication, satellite laser communication has unique advantages such as higher and tariff-free bandwidth, lower power (∼1/2 of RF systems), less mass (∼1/2 of RF systems), smaller size (∼1/10 the diameter of RF antenna) and improved channel security [2]. The on-orbit terminals in satellite laser communication network work in the harsh space environment. For receiver’ s optical antenna, space thermal environment and microgravity environment will lead to the optical surface errors and the optical elements’ displacements [3]. The axial non-confocal antenna is a deformed dual-reflector antenna caused by despace of the primary or secondary mirror. This deformation will affect the received beacon light spot’s intensity distribution(BID) which determines the received optical axis, ultimately affecting the pointing, acquisition and tracking(PAT) performance. This deformation will also affect the received signal spot’s intensity distribution which determines the received signal power(RSP), ultimately affecting the bit error rate (BER) performance.

For PAT impact analysis, with a given wavefront aberration, researchers [4–8] have calculated BID by diffraction theory. To express the wavefront aberrations, Toyoshima et al. have used Zernike polynomials in circle region [4], Jianfeng Sun et al. have used modified Zernike polynomials in annular region [5], Liying Tan et al. have used the ellipse Gaussian model for localized wavefront deformations [6], Chao Wang et al. have used the Gaussian random-phase screen for low-frequency error in refractive-type antenna surface [7] and Wanqing Xie et al. have used the wavelet model for localized asymmetrical deformations [8]. For a system with a known wavefront aberration type, one can decide the undetermined coefficients with sampling points by least square method. We call it the wavefront fitting method(WFM) here and it is widely used for wavefront reconstruction [9,10]. However, the wavefront aberration type is always unknown for an on-orbit terminal with a deformed receiver antenna, so that a good fitting model is hard to choose or even hard to find. Besides, the impact of the receiver’s axial non-confocal antenna on PAT is often neglected.

While for the communication impact analysis, some researchers [11–13] have calculated the received signal light intensity distribution at receiver’s entrance pupil and taken encircled power there as the RSP. Wanqing Xie et al. have calculated a uniform intensity distribution by wavelet model [11], Jin Zhao and Minyin Yu et al. have calculated the intensity distribution by ABCD matrices for Gaussian beam propagation [12,13]. However, when the receiver antenna deforms, only part of the received power at the entrance pupil is collected by the photodetector, due to the limited photosensitive area. Ping Jiang et al. have considered the received power in optical detector by ray tracing, limiting spot diagrams to a certain fiber core diameter [14]. However, the method is not suitable for receivers without a fiber coupling system, whose signal light is received directly by photodetectors. As spot diagrams fail to present the intensity distribution in the image plane in focal regions for systems with tiny aberrations [15], the received power in a finite photosensitive area would be unavailable by this raytracing method.

To calculate the intensity distribution in focal regions for both beacon and signal lights, the combination method of ray tracing and diffraction theory(CMRD) [16] is an effective method. In addition, we would like to extend it to calculate the encircled power in focal regions (i.e. the RSP). The CMRD calculates the wavefront (amplitude and phase) in a reference plane in image space with ray tracing in step one and numerically solves the diffraction integration to get the intensity distribution in focal plane in step two [16]. The methods to solve the diffraction integration have been developed by Hopkins, Ludwig and Stamnes et al., discarding different order terms in Taylor series of phase and amplitude in integrand separately [15,17–19]. Hopkins et al. employed a linear approximation to the phase and a constant approximation to the amplitude, Ludwig etc. al. employed both linear approximations to the phase and the amplitude, while Stamnes et al. employed both parabola approximations to the phase and amplitude. In this paper, we employed both constant approximations to the phase and amplitude (i.e. the first-order and higher-order terms in Taylor series are all discarded), We called the simplified method: simplified CMRD (SCMRD).

The purpose of this paper is to determine the influences of the receiver’s axial non-confocal antenna on the BID and the RSP, which can help determine its influences on the PAT and the BER performance, providing a reference for optical design, structural design, thermal design, assembly and adjustment accuracy design and detector selection. Besides, this work is the first step to introduce the SCMRD to analyze the deformed optical antenna impact on the on-orbit lasercom receiver.

In section 2, for an on-orbit lasercom receiver with a dual-reflector antenna, the deformation analysis model is established. In section 3 and 4, for an axial non-confocal antenna, based on the SCMRD, the calculation model of the normalized intensity distribution (NID, i.e. normalized BID) and the normalized received power (NRP, i.e. normalized RSP) are established. In section 5, a comparison calculation model based on the WFM is presented. In section 6, the numerical results of NID and NRP (calculated by the SCMRD and the WFM) are presented to analyze the impact of the non-confocal antenna. The errors induced by undersampling for the SCMRD are estimated as well. Besides, the NID results calculated by the SCMRD are compared with those calculated by the CMRD. At last, the pros and cons of applying the SCMRD, the CMRD and the WFM to analyze the deformed optical antenna impact on the on-orbit lasercom receiver are also discussed.

2. Optical antenna deformation model

The Mersenne telescope, the confocal parabolic dual-reflector telescope, is aplanatic, anastigmatic, achromatic and orthoscopic with an appropriate stop location setting [20]. Combined with a focusing system, it could be a promising optical antenna for the satellite laser communication, where stray light control is more important than the large field of view [21,22].

The configuration of a lasercom receiver’s optical antenna (a Mersenne telescope) with a focusing system (equivalent to an ideal thin lens) is shown in Fig. 1. Taking the primary mirror (M₁) as the reference, the antenna’s axial non-confocal distance, mainly caused by deformation of truss between the primary mirror and the secondary mirror(M₂), can be treated as the M₂’s despace amount. With a reasonable temperature control, the deformation of the focusing system is relatively small thus its influence is neglected.

Fig. 1. Configuration of a lasercom receiver’s optical antenna with a focusing system

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A Mersenne antenna could be identified by three parameters: the aperture of M₁ (D₁), the focal lengths of M₁ (f₁) and M₂ (f₂). The distance between M₁ and M₂ (d₂) is defined as:

(1)$${d_\textrm{2}} = {f_1} - {f_2}\textrm{ + }\varepsilon \textrm{,}$$

where ε is the axial non-confocal distance. Then the compression ratio M = f₁/ f₂. Noted that for the beacon light and the signal light, antennas may have different focal lengths of the ideal thin lens (f₃) and different distances between the M₁ and the ideal thin lens (d₃).

As the light beam is transmitted over a long distance and received by a relatively small aperture, incident light waves in the receiver’s entrance pupil can be approximated as uniform planar light waves. On the image plane, the beacon detector (a camera) and the communication detector (an APD or PIN photodetector) are placed in the beacon light receiving path and the signal light receiving path, respectively.

Owing to the symmetry of the axial non-confocal antenna, light beams (beacon and signal) will be normal incidence to the receiving system when the tracking lock is realized. Thereafter, the BID can be calculated to analyze the impact on the PAT performance and RSP based on the signal intensity distribution(SID) can be calculated to analyze the impact on the BER performance.

3. Normalized intensity distribution calculated by the SCMRD

The CMRD calculates the wavefront (amplitude and phase) in a reference plane in image space with ray tracing in step one and numerically solves the diffraction integration to get the intensity distribution in focal plane in step two [16]. Based on the CMRD, the calculation model of NID is established. As the first-order and higher-order terms in Taylor series of phase and amplitude in integrand are all discarded, we call this method SCMRD.

3.1 Diffraction theory

The plane close to the ideal thin lens in image space is chosen as the reference plane and the first Rayleigh-Sommerfeld scalar diffraction integral is applied to calculate the spot intensity distribution. The light field of a point A₄(x₄, y₄, d₄) shown in Fig. 1 is given by [23]:

(2)$${U_4}({\textrm{A}_\textrm{4}}) = \frac{{{f_3}}}{{\textrm{j}\lambda }}\int\!\!\!\int\limits_S {{U_\textrm{3}}({\textrm{A}_3})\frac{{\exp(\textrm{j}k{d_{34}})}}{{{d_{34}}^2}}\rm{d}s} ,$$

where S is the integral domain at the reference plane in image space, d₃₄ is the norm of direction vector from point A₃(x₃, y₃, d₃) at the reference plane to point A₄, λ is the wavelength, k is the wavenumber and f₃ is the focal length of the thin lens. U₃ is the light field at domain S in image space, which can be written as:

(3)$${U_\textrm{3}}({\textrm{A}_3})\textrm{ = }{u_3}({x_3},{y_3})\exp[\textrm{j}{\varphi _3}({x_3},{y_3})],$$

where u₃ is the amplitude and φ₃ is the phase. So the light intensity at point A₄ (i.e. BID or SID) can be written as:

(4)$$I({\textrm{A}_\textrm{4}}) = {U_\textrm{4}}\mathop {{U_\textrm{4}}}\limits^\ast \textrm{ = }{\left|{\frac{{{f_3}}}{\lambda }\int\!\!\!\int\limits_S {\frac{{{u_3}}}{{{d_{34}}^2}}\exp[{j(k{d_{34}} + {\varphi_3})} ]\rm{d}s} } \right|^2}.$$

To compare the light spots’ intensity distribution with and without the M₂’s axial displacement, the normalized intensity distribution(NID) is defined as follows:

(5)$${I_\textrm{n}}\textrm{(}{\textrm{A}_\textrm{4}}\textrm{) = }{{I\textrm{(}{\textrm{A}_\textrm{4}}\textrm{)}} \mathord{\left/ {\vphantom {{I\textrm{(}{\textrm{A}_\textrm{4}}\textrm{)}} {{I_\textrm{0}}}}} \right.} {{I_\textrm{0}}}},$$

where I₀ is the intensity of Gaussian image point when M₁ and M₂ are confocal.

3.2 Simplified combination of ray tracing and diffraction theory

To compute the integral numerically, the integral domain S is firstly divided into t subdomains s₁, s₂, …, s_t. In each subdomain, first-order and higher-order terms in Taylor series of phase and amplitude are discarded, i.e. the phase in each subdomain is approximated by midpoint’s phase and the amplitude in each subdomain is approximated by midpoint’s amplitude. So Eq. (4) is simplified as:

(6)$$I({\textrm{A}_\textrm{4}}) = \frac{{{f_3}^2}}{{{\lambda ^2}}}\left\{ {{{[\sum\limits_{i = 1}^t {\frac{{{u_{3i}}{\sigma_i}}}{{{d_{34\textrm{,}i}}^2}}\cos (k{d_{34,i}} + {\varphi_3}_i)} ]^2}} + {{[\sum\limits_{i = 1}^t {\frac{{{u_{3i}}{\sigma_i}}}{{{d_{34\textrm{,}i}}^2}}\sin (k{d_{34,i}} + {\varphi_3}_i)} ]^2}}} \right\},$$

where u_3i and φ_3i are the amplitude and phase of midpoint in ith subdomain, respectively. σ_i is the area of the ith subdomain. d_34,i is the distance from midpoint in the ith subdomain to A₄. This simplification helps reduce the computing complexity for each subdomain and precision can be guaranteed by increasing the amount of subdomains.

The light field in each subdomain at the reference plane comes from a corresponding subdomain at the entrance pupil, from the energy conservation theory [15] the following equation can be obtained:

(7)$${u_0}^2{\xi _i} = {u_{3i}}^2{\sigma _i},$$

where ξ_i is the area of the ith subdomain at the entrance pupil and u₀ is the amplitude of incident light at the entrance pupil. Suppose the axial displacement is so tiny that ξ_i is proportional to σ_i as follows:

(8)$${\xi _i} = {\eta ^2}{\sigma _i},$$

where η is the ratio of radial length. η equals to M when M₁ and M₂ are confocal, and changes with M₂ shifting away from M₁. With ray tracing, η can be determined by the ratio of a marginal ray’s radial coordinate at the entrance pupil to that at the reference plane.

From equations (6) and(8), the intensity at point A₄ (i.e. BID or SID) is given as follows:

(9)$$I({\textrm{A}_\textrm{4}}) = \frac{{{f_3}^2{u_0}^2}}{{{\lambda ^2}{\eta ^2}}}\left\{ {{{[\sum\limits_{i = 1}^t {\frac{{{\xi_i}}}{{{d_{34\textrm{,}i}}^2}}\cos (k{d_{34,i}} + {\varphi_3}_i)} ]^2}} + {{[\sum\limits_{i = 1}^t {\frac{{{\xi_i}}}{{{d_{34\textrm{,}i}}^2}}\sin (k{d_{34,i}} + {\varphi_3}_i)} ]^2}}} \right\}.$$

The partitioning scheme at the entrance pupil determines the raytracing sampling. One of the typical partitioning scheme divides entrance pupil into t_ρ equally spaced annuli and t_θ equally spaced sectors as shown in Fig. 2. So the sum of the subdomains is t = t_ρt_θ. t_ρ and t_θ are called the radial sampling number and azimuthal sampling number, respectively. The subdomain (a, b) marked as the ith subdomain and the polar coordinates of its midpoint A_1i (ρ_1i, θ_1i) are given as:

(10)$$\begin{array}{l} {\rho _{1i}}\textrm{ = }[{D_2}\textrm{ + (2}a\textrm{ - 1}){\mu _{\mathrm{\rho}}}{D_1}]/2,a = 1,2,\ldots ,{t_{\mathrm{\rho}}},\\ {\theta _{1i}}\textrm{ = (2}b\textrm{ - }1){\mu _{\mathrm{\theta}}}/2,b = 1,2,\ldots ,{t_{\mathrm{\theta}}}, \end{array}$$

where index i = (a-1)t_θ+b, index a and b represent the ath annulus and bth sector, respectively. The normalized radial span μ_ρ and azimuth span μ_θ are given as follows:

(11)$${\mu _{\mathrm{\rho}}}\textrm{ = }{{({D_1}\textrm{ - }{D_2})} \mathord{\left/ {\vphantom {{({D_1}\textrm{ - }{D_2})} {(2{D_1}{t_{\mathrm{\rho}}})}}} \right.} {(2{D_1}{t_{\mathrm{\rho}}})}},{\mu _{\mathrm{\theta}}}\textrm{ = }2\pi /{t_{\mathrm{\theta}}}.$$

The area of the subdomain (a, b) is given as:

(12)$${\xi _i}\textrm{ = }{\xi _{ab}}\textrm{ = }{\mu _{\mathrm{\theta}}}{\mu _{\mathrm{\rho}}}{D_1}\textrm{[2}{D_2}\textrm{ + (2}a\textrm{ - }1){\mu _{\mathrm{\rho}}}{D_1}]/8.$$

Fig. 2. Sampling points at the entrance pupil

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4. Normalized received power

For a satellite laser communication system with intensity modulation and direct detection (IM/DD), a simple receiving scheme is that the signal light is focused directly to an APD photodetector. The effective receiving radius at image plane (APD’s photosensitive surface) directly determines the encircled power (i.e. the receiving signal power), affecting the signal-to-noise ratio (SNR) and bit error rate (BER) performance.

The encircled power is defined as the receiving light power within the circle with effective receiving radius based on SID. Since the circular symmetry of SID, the encircled power can be written as:

(13)$$P({l_\textrm{e}}) = 2\pi \int_0^{{l_\textrm{e}}} {I(l)l\textrm{d}l} ,$$

where l_e is the effective receiving radius at image plane, I(l) is the intensity of the image point with a distance l to the spot center. According to the composite trapezoidal integral formula, Eq. (13) can be written as:

(14)$$P({l_\textrm{e}}) \approx {\mu _\textrm{l}}\pi \sum\limits_{{i_\textrm{m}}\textrm{ = 1}}^{{m_\textrm{l}}} {[I({l_{{i_\textrm{m}}\textrm{ - 1}}}){l_{{i_\textrm{m}}\textrm{ - 1}}}\textrm{ + }I({l_{{i_\textrm{m}}}}){l_{{i_\textrm{m}}}}]} ,$$

where l_i=i_mμ_l, the sampling interval μ_l= l_e/m_l. Based on the SCMRD, I(l) is calculated by (9) with l being the radial coordinate of A₄.

To show the power coupling efficiency of the optical antenna, NRP is defined as the rate of the encircled power to the incident light power through the entrance pupil [24]:

(15)$${P_\textrm{n}}({l_\textrm{e}}) = {{P({l_\textrm{e}})} \mathord{\left/ {\vphantom {{P({l_\textrm{e}})} {{P_{\textrm{in}}}}}} \right.} {{P_{\textrm{in}}}}},$$

where P_in is the incident light power through the entrance pupil, considering the obscuration ratio.

5. NID and NRP calculated by the wavefront fitting method

For comparisons, results of the NID and the NRP calculated by the wavefront fitting method (WFM) are also presented.

With aberrations induced by the receiving antenna, the complex amplitude of the optical field before the focusing system is given as:

(16)$${U_\textrm{L}}({\rho _3},{\theta _3}) = {u_\textrm{3}}\exp[{j\phi ({\rho_3},{\theta_3})} ],$$

where u₃ is the amplitude, ϕ is the phase of the distorted wavefront, ρ₃ and θ₃ are the radial coordinate and the azimuthal coordinate, respectively. The wave U_L (ρ, θ) passing through the focusing system with a focal length f₃ is focused on the focal plane, and the intensity distribution is given by [23,24]:

(17)$$I({\textrm{A}_\textrm{4}}) = \frac{1}{{{\lambda ^2}{f_3}^2}}{\left|{\int_0^{2{\pi }} {\int_{{D_{3\textrm{i}}}/2}^{{D_3}/2} {{U_\textrm{L}}({\rho_3},{\theta_3})\exp \left[ { - \textrm{j}\frac{{2{\pi }{\rho_3}l}}{{\lambda {f_3}}}\cos({\theta_3} - \beta )} \right]{\rho_3}\textrm{d}{\rho_3}\textrm{d}{\theta_3}} } } \right|^2},$$

where D₃ and D_3i are the outer diameter and inner diameter of light beams before the focusing system, respectively. l and β are the radial coordinate and azimuthal coordinate of A₄, respectively. As η²u₀²= u₃², Eq. (17) can be written as:

(18)$$I({\textrm{A}_\textrm{4}}) = \frac{{{\eta ^2}{u_0}^2}}{{{\lambda ^2}{f_3}^2}}{\left|{\int_0^{2{\pi }} {\int_{{D_{3\textrm{i}}}/2}^{{D_3}/2} {\exp \left[ {\textrm{j}\phi ({\rho_3},{\theta_3}) - \textrm{j}\frac{{2{\pi }{\rho_3}l}}{{\lambda {f_3}}}\cos({\theta_3} - \beta )} \right]{\rho_3}\textrm{d}{\rho_3}\textrm{d}{\theta_3}} } } \right|^2}.$$

The encircled power is given by

(19)$$P({l_e}) = \frac{{{\eta ^2}{u_0}^2}}{{{\lambda ^2}{f_3}^2}}{\int_0^{2{\pi }} {\int_0^{{l_e}} {\left|{\int_0^{2{\pi }} {\int_{{D_{3\textrm{i}}}\textrm{/2}}^{{D_3}\textrm{/2}} {\exp \left[ {\textrm{j}\phi ({\rho_3},{\theta_3}) - \textrm{j}\frac{{2{\pi }{\rho_3}l}}{{\lambda {f_3}}}\cos({\theta_3} - \beta )} \right]{\rho_3}\textrm{d}{\rho_3}\textrm{d}{\theta_3}} } } \right|} } ^2}l\textrm{d}l\textrm{d}\beta .$$

Then the NID and NRP are calculated by Eq. (5) and (15), respectively.

The sampling points’ phases ϕ before focusing system are calculated by the raytracing method. Then the wavefront aberration function before the focusing system is fitted by the least square method. The wavefront aberration function expressed by Zernike polynomials as follows [24,25]:

(20)$$\phi ({\rho _3},{\theta _3}) = \frac{{2{\pi }}}{\lambda }W({\rho _{30}},\theta )\textrm{ = }\frac{{2{\pi }}}{\lambda }\sum\limits_{\tau = 1}^N {{a_\tau }{Z_\tau }({\rho _{30}},\theta )} ,$$

(21)$${Z_\tau }({\rho _{30}},\theta ) = \left\{ {\begin{array}{{ll}} {{Z_{\textrm{even }\tau }} = \sqrt {n + 1} R_n^m({\rho_{30}})\sqrt 2 \cos (m{\theta_3}),}&{m \ne 0}\\ {{Z_{\textrm{odd }\tau }} = \sqrt {n + 1} R_n^m({\rho_{30}})\sqrt 2 \sin (m{\theta_3}),}&{m \ne 0}\\ {{Z_\tau } = \sqrt {n + 1} R_n^0({\rho_{30}}),}&{m = 0} \end{array}} \right.,$$

(22)$$R_n^m({\rho _{30}}) = \sum\limits_{s = 0}^{(n - m)/2} {\frac{{{{( - 1)}^s}(n - s)!}}{{s![(n + m)/2 - s]![(n - m)/2 - s]!}}} {\rho _{30}}^{n - 2s},$$

where Z_τ are Zernike polynomials in a unit circle, a_τ are the coefficients of the Zernike polynomials, ρ₃₀=2ρ₃/D₃. n is called the radial degree and m is the azimuthal frequency (n, m ∈ Z⁺, m ≤ n, and n - m is even). The index τ is a Zernike mode ordering number and N is the number of Zernike modes. In the Zemax standard ordering scheme [25], the modes, Z_τ, are ordered such that even τ corresponds to the symmetric modes defined by cos(mθ₃), while odd τ corresponds to the antisymmetric modes given by sin(mθ₃). For a given n, modes with a lower value of m are ordered first.

6. Numerical results and discussion

A typical Mersenne antenna for the on-orbit lasercom receiver is taken as an example to analyze the impact of non-confocal antenna on NID and NRP. Parameters of the receiving system are shown in Table 1. The wavelengths of incident beacon and signal light are 775nm and 1550nm, respectively. The antenna is assumed to have the same f₃ and d₃ for both beacon and signal light receiving paths. NID is calculated by the SCMRD with sampling parameters t_ρ=81, t_θ=81. NRP of signal light is calculated by the SCMRD with sampling parameters t_ρ=81, t_θ=81 and μ_l=1μm. Besides, numerical results of NID and NRP calculated by WFM are also presented for comparison in section 6.1 and 6.2. Numerical results of NID calculated by the CMRD and the SCMRD are compared in section 6.4.

Table 1. Parameters of Receiving System with Mersenne Antenna

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6.1 Normalized intensity distribution

The NID of light spot plotted as a function of the distance from point A₄ to the spot center is shown in Fig. 3. The maximum difference of NID between the lines (calculated by the SCMRD) and circles (calculated by WFM) is 1.9‰ for 775nm and 1.8‰ for 1550nm. As the antenna’s axial non-confocal distance increases, the light spot flattens and the intensity of the light spot center decreases. The decrease speed gradually increases first and then gradually decreases which can be seen more obviously in Fig. 4. When the axial displacement reaches a certain value, the spot center’s intensity is no longer the peak of the light intensity distribution, i.e. the ring light spot appears. If the peak of the intensity is far away from the center, we should consider if it is still within the detector’s receiving surface. But as the maximum normalized intensity of the ring light spot is less than 0.1, we could decide the tolerance of the antenna’s axial non-confocal distance according the intensity of the beacon light spot center in most cases.

Fig. 3. NID of light spot (The solid line and dash lines are calculated by the SCMRD, values of mark ‘o’ are calculated by the WFM)

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Fig. 4. Normalized intensity of light spot’s center and peak point

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Owing to the symmetry of the axial non-confocal antenna, NID of the beacon light is symmetrical. As long as the sensor can detect the beacon light, the receiving optical antenna can point right to the transmitter. The normalized intensities of the beacon light spot center and peak points vary with antenna’s axial non-confocal distance are shown in Fig. 4. According to it, the tolerance of antenna’s axial non-confocal distance should be determined for a given beacon light detection threshold. For example, when the beacon light detection threshold is set as 10% of I₀, the tolerance of the antenna’s axial non-confocal distance will be about 119 μm for 775 nm and 238 μm for 1550 nm. The tolerance of the antenna’s axial non-confocal distance is proportional to the beacon light’s wavelength approximatively.

6.2 Normalized received power

When the receiving optical antenna points right to the transmitter and tracking lock is realized, the signal light spot center is located to the center of the photodetector’s photosensitive surface. Figure 5(a)) displays the calculated NRP of the signal light as a function of both antenna’s axial non-confocal distance and effective receiving radius (ε and l_e). It is found that the NRP is strongly dependent on both antenna’s axial non-confocal distance and effective receiving radius.

Fig. 5. NRP of the signal light a) surface plot, b) NRP curves versus ε, c) NRP curves versus l_e. The solid line and dash lines are calculated by the SCMRD, values of mark ‘o’ and ‘×’ are calculated by the WFM.

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NRP curves versus antenna’s non-confocal distance from Fig. 5(a)) are shown in Fig. 5(b)). The maximum difference of NRP between the lines (calculated by the SCMRD) and crosses (calculated by WFM) is 9.4‰. As the antenna’s axial non-confocal distance increases, each NRP curve decreases. It can be inferred that each curve is a section of long curve that undergoes a process of slowly decreasing, rapidly linear decreasing and finally stabilizing. The larger effective receiving radius at image plane, the slower received power declines, offering a higher tolerance of antenna’s axial non-confocal distance.

NRP curves versus effective receiving radius at image plane from Fig. 5(a)) are shown in Fig. 5(c)). The maximum difference of NRP between the lines (calculated by the SCMRD) and circles (calculated by WFM) is 9.4‰. As the effective receiving radius at image plane increases, each NRP curve increases from 0 and approaches to a certain value between 0.9 and 1. Each curve roughly undergoes a process of slowly increasing, rapidly increasing, and finally stabilizing. Provided that effective receiving radius is big enough, the EP can be approximately to the incident light power through the whole entrance pupil.

During the design stage, according to the received signal power requirement, Fig. 5 will help to determine the radius of photosensitive surface of detector and tolerance of antenna’s axial non-confocal distance. For example, if the received signal power requirement is set to be 0.5 of the incident power, the designers should either control the antenna’s non-confocal distance less than 130um to allow a smaller detector with an effective receiving radius no less than 16um (from Fig. 5(b)), or choose a detector with an effective receiving radius larger than 44um to allow the antenna’s non-confocal distance to reach 350um (from Fig. 5(c)). The trade-off will depend on the difficulties and cost of getting a larger detector and realizing high-precision thermal control and adjustment.

6.3 Errors estimation and sampling parameters selection

In this section, for numerical results of NID and NRP which are calculated by the SCMRD, the errors induced by undersampling are estimated.

6.3.1 Errors Estimation of NID

The errors of NID induced by undersampling at entrance are defined as:

(23)$${\delta _{\mathrm{\rho}}}_{\mathrm{\theta}}(l,{t_{\mathrm{\rho}}},{t_{\mathrm{\theta}}}) = |{{I_n}(l,{t_{\mathrm{\rho}}},{t_{\mathrm{\theta}}}) - {I_n}(l,3{t_{\mathrm{\rho}}},3{t_{\mathrm{\theta}}})} |,$$

where I_n(l, t_ρ, t_θ) is I_n(l) calculated with a combination of two sampling parameters t_ρ and t_θ at the entrance pupil. δ_ρθ is reduced by increasing t_ρ and t_θ step by step. To ensure the new sampling points in each step cover the former sampling points, the new adopted t_ρ and t_θ should be three times the former ones.

The errors of NID induced by undersampling at the entrance pupil are shown in Fig. 6(a)) and Fig. 6(b)). Error curves in each subgraph show a convergence trend as t_ρ and t_θ increase. In section 6.1, the NID of beacon light is calculated with sampling parameters t_ρ=81, t_θ=81. So from Fig. 6(a)) and (b)), it can be seen that the errors induced by undersampling at the entrance pupil are less than 7.209e-5.

Fig. 6. Errors of beacon lighr’s NID induced by undersampling at the entrance pupil. t_ρ and t_θ are increased simultaneously in a) and b); only t_ρ is increased in c), d); only t_θ is increased in e), f). In a), c) and e), for a given ε, the ordinate value is the maximum error when l_e varies from 0 to 0.048 mm. In b), d) and f), for a given l_e, the ordinate value is the maximum error when ε varies from 0 to 0.175 mm.

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If we want to increase the precision of NID, increasing t_ρ would be more effective than increasing t_θ. This can be concluded by the following study.

The errors of NID induced by insufficient radial sampling number t_ρ and azimuthal sampling number t_θ at the entrance pupil are defined as:

(24)$${\delta _{\mathrm{\rho}}}(l,{t_{\mathrm{\rho}}},{t_{\mathrm{\theta}}}) = |{{I_n}(l,{t_{\mathrm{\rho}}},{t_{\mathrm{\theta}}}) - {I_n}(l,3{t_{\mathrm{\rho}}},{t_{\mathrm{\theta}}})} |,$$

(25)$${\delta _{\mathrm{\theta}}}(l,{t_{\mathrm{\rho}}},{t_{\mathrm{\theta}}}) = |{{I_n}(l,{t_{\mathrm{\rho}}},{t_{\mathrm{\theta}}}) - {I_n}(l,{t_{\mathrm{\rho}}},3{t_{\mathrm{\theta}}})} |.$$

Note that this time the t_ρ and t_θ are increased separately.

Errors of NID induced by insufficient t_ρ and t_θ at the entrance pupil are shown in Fig. 6(c)) to (f)). In Fig. 6(c)) and Fig. 6(d)), δ_ρ,max decreases distinctly by increasing t_ρ. While by increasing t_θ in Fig. 6(e)) and Fig. 6(f)), δ_θ,max remain lower than 1e-10 when t_θ over 27. So the errors of NID induced by undersampling at the entrance pupil mainly affected by t_ρ when t_θ greater than a certain value. It can be inferred Fig. 6(f)) that this value is related to the l_e. For example, if we only concern about conditions with l_e less than 0.01mm, t_θ=9 would be sufficient.

6.3.2 Errors Estimation of NRP

The errors of NRP induced by undersampling at the entrance pupil are defined as:

(26)$${\zeta _{{{\mathrm{\rho}} {\mathrm{\theta}}}}}({l_\textrm{e}},{\mu _\textrm{l}},{t_{\mathrm{\rho}}},{t_{\mathrm{\theta}}}) = |{{P_\textrm{n}}({l_\textrm{e}},{\mu_\textrm{l}},{t_{\mathrm{\rho}}},{t_{\mathrm{\theta}}}) - {P_\textrm{n}}({l_\textrm{e}},{\mu_\textrm{l}},3{t_{\mathrm{\rho}}},3{t_{\mathrm{\theta}}})} |,$$

and the errors of NRP induced by undersampling at the image plane are defined as:

(27)$${\zeta _\textrm{l}}({l_\textrm{e}},{\mu _\textrm{l}},{t_{\mathrm{\rho}}},{t_{\mathrm{\theta}}}) = |{{P_\textrm{n}}({l_\textrm{e}},{\mu_\textrm{l}},{t_{\mathrm{\rho}}},{t_{\mathrm{\theta}}}) - {P_\textrm{n}}({l_\textrm{e}},0.5{\mu_\textrm{l}},{t_{\mathrm{\rho}}},{t_{\mathrm{\theta}}})} |.$$

where P_n(l_e, μ_l, t_ρ, t_θ) is P_n(l_e) calculated with a combination of three sampling parameters: t_ρ, t_θ at the entrance pupil and μ_l at the image plane. ζ_ρθ and ζ_l are reduced by increasing t_ρ, t_θ and decreasing μ_l step by step. To ensure the new sampling points in each step cover the former sampling points, the new adopted t_ρ and t_θ should be three times the former ones, and the new adopted sampling interval μ_l at the image plane should be half of the former one.

The errors of NRP induced by undersampling at the entrance pupil are shown in Fig. 7(a)) and (b)). The errors of NRP induced by undersampling at the image plane are shown in Fig. 7(c)) and (d)). Error curves in each subgraph show a convergence trend. In section 6.2, the NRP of signal light is calculated with sampling parameters t_ρ=81, t_θ=81 and μ_l=1μm. So from Fig. 7(a)) and Fig. 7(b)), it can be seen that the errors induced by undersampling at the entrance pupil are less than 1.313e-4. From Fig. 7(c)) and Fig. 7(d)), the errors induced by undersampling at the image plane are less than 1.981e-3. So the errors of NRP induced by undersampling are less than 1.981e-3.

Fig. 7. Errors of NRP induced by undersampling. t_ρ and t_θ are increased simultaneously in a) and b); only μ_l is decreased in c) and d); only t_ρ is increased in e) and f) only t_θ is increased in g) and h). In a), c), e) and g), for a given ε, the ordinate value is the maximum error when l_e varies from 0 to 0.096 mm. In b), d), f) and h), for a given l_e, the ordinate value is the maximum error when ε varies from 0 to 0.35 mm.

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If we want to increase the precision of the NRP, decreasing μ_l would be more effective than increasing t_ρ and t_θ. This can be concluded by the following study.

The errors of NRP induced by insufficient radial sampling number t_ρ and azimuthal sampling number t_θ at the entrance pupil are defined as:

(28)$${\zeta _{\mathrm{\rho}}}({l_\textrm{e}},{\mu _\textrm{l}},{t_{\mathrm{\rho}}},{t_{\mathrm{\theta}}}) = |{{P_\textrm{n}}({l_\textrm{e}},{\mu_\textrm{l}},{t_{\mathrm{\rho}}},{t_{\mathrm{\theta}}}) - {P_\textrm{n}}({l_\textrm{e}},{\mu_\textrm{l}},3{t_{\mathrm{\rho}}},{t_{\mathrm{\theta}}})} |,$$

(29)$${\zeta _{\mathrm{\theta}}}({l_\textrm{e}},{\mu _\textrm{l}},{t_{\mathrm{\rho}}},{t_{\mathrm{\theta}}}) = |{{P_\textrm{n}}({l_\textrm{e}},{\mu_\textrm{l}},{t_{\mathrm{\rho}}},{t_{\mathrm{\theta}}}) - {P_\textrm{n}}({l_\textrm{e}},{\mu_\textrm{l}},{t_{\mathrm{\rho}}},3{t_{\mathrm{\theta}}})} |.$$

Note that this time the t_ρ and t_θ are increased separately.

The Errors of NRP induced by insufficient t_ρ and t_θ at the entrance pupil are shown in Fig. 7(e)) to Fig. 7(h)). In Fig. 7(e)) and Fig. 7(f)), ζ_ρ,max decreases distinctly by increasing t_ρ. While by increasing t_θ in Fig. 7(g)) and Fig. 7(h)), ζ_θ,max remain lower than 1e-10 when t_θ over 27. So the errors of NRP induced by undersampling at the entrance pupil mainly affected by t_ρ when t_θ greater than a certain value. It can be inferred that this value is related to the l_e. For example, if we only concern about conditions with l_e less than 0.02 mm, t_θ=9 would be sufficient.

Besides, by changing μ_l (form 1um to 4um), results of ζ_ρ,max and ζ_θ,max above change a little. So that we could decide these two sampling parameters at the entrance pupil first and then consider the sampling parameters at the image plane.

By comparing ζ_ρθ,max, ζ_ρ,max, ζ_θ,max and ζ_l,max, the precision of the NRP is mainly affected by undersampling at the image plane as ζ_l,max is the maximal.

Errors estimation shows that with the SCMRD, errors of beacon light’s NID are less than 7.209e-5 and increasing t_ρ to 243 would further reduce these errors to 8.008e-6. Errors of signal light’s NRP are less than 1.981e-3 and decreasing μ_l to 0.5um would further reduce these errors to 4.953e-4. If the precision of NID required are lower than the current accuracy, we could refer to the Fig. 8(c)) in section 6.4 to select the sampling parameters.

Fig. 8. Maximum errors as a function of the radial sampling number t_ρ and azimuthal sampling number t_θ.

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6.4 Compared with the WFM and CMRD

From Fig. 3, we know that NID calculated by the SCMRD have the maximum difference of 1.9‰ with those calculated by the WFM. From Fig. 5, we know that NRP calculated by the SCMRD have the maximum difference of 9.4‰ with those calculated by the WFM. Therefore, both the SCMRD and WFM are suitable for analyzing the impact of axial non-confocal antenna on on-orbit lasercom receiver when the accuracy requirements are 1% for NID and NRP.

Both methods use ray tracing method and diffraction theory. The main difference between them is that WFM requires fitting the received wavefront by choosing an appropriate function but the SCMRD does not.

The advantage of WFM is that the integration can be solved by well-developed numerical method after wavefront fitting. The convergence of the numerical integration would be guaranteed more easily. However, the precision of the results depends on the accuracy of wavefront fitting. Different distorted wavefront need to be fitted by different function. For example, wavefront aberration induced by localized deformation [6] and low-frequency error [7] could not be represented well by the widely used Zernike polynomials.

Therefore, the advantage of the SCMRD is that there is no need to worry about the poor wavefront fitting. As long as the sampling points at entrance pupil and at image plane are adequate, NID and NRP calculated by the SCMRD can achieve a quite high accuracy. If it is difficult to choose an appropriate function to fit the distorted received wavefront well, especially for a deformed optical antenna, the SCMRD could be a good alternative method.

The results of NID are also calculated by the CMRD, specifically with the Ludwig algorithm [18] and Stamnes-Spjelkavik-Pedersen(SSP) algorithm [19] separately. They are compared with those calculated by the SCMRD with t_ρ=243, t_θ=81(From section 6.3 we know the errors are less than 8.008e-6. So the precision of the results would be better than 1.6e-5). The maximum errors as a function of the radial sampling number t_ρ and azimuthal sampling number t_θ are presented in Fig. 8. For a given precision, the SCMRD requires less sampling subdomains than the CMRD with the Ludwig algorithm and SSP algorithm.

At the first glance that the CMRD with higher-order approximation should requires less sampling subdomains than the SCMRD with zero-order approximation. But the contrary result is still understandable when we look into the approximations used by these algorithms, which are illustrated in Fig. 9 [26].

Fig. 9. Illustration of the different approximation to the phase (imaginary part of Eq. (4)). Hopkins’ tangent approximation, Ludwig’s chord approximation, SSP’s best linear approximation and SCMRD’s ladder approximation

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For Ludwig algorithm, a higher-order approximation does not guarantee a better integral result in each subdomain. It depends on the curvature of the actual phases. For SSP algorithm, as stated by the proposers [26], for a two-dimensional diffraction integral, there is little to be gained by using second- or higher- order approximations to the phase functions, since the resulting approximate integral cannot then be evaluated in terms of known functions. Therefore, SSP algorithm calculates the first integration for some constant values of the other integration variable, and then uses the integrated values as an input to the standard integration routine called Gauss-Legendre integration. As we discuss cases that the wavefront aberration type is unknown, the Gauss points in the reference plane for Gauss-Legendre integration need to be estimated by interpolation, after raytracing from the sampling points at the entrance pupil to the reference plane. This interpolation sampling could cause errors to the first integration. Besides, to do the first integration, SSP algorithm would apply different simplified calculation for different cases to increase the processing efficiency [19]. These simplified calculations would also cause errors. For example, the best linear approximation instead of parabola approximation to the phase would be adopted in some cases. So it is possible that the SCMRD requires less subdomains than the CMRD. For other deformation cases, it is hard to say which method would need the minimum sampling subdomains.

To achieve the precision of 1‰, the processing time of these algorithms are list in Table 2. The SCMRD takes the least time. The same algorithm can be implemented by different programing codes, so the processing efficiency would be different. The processing time here is the best result we can achieve so far.

Table 2. Processing time of three algorithm to achieve certain precision (1‰)^a

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It is not surprising in principle that the SCMRD has the least mean time consuming per subdomain because it simplifies the calculation. The other two need to determine the coefficients for linear or parabola approximations first and then calculate the double integration. As sampling subdomains required by the SCMRD are also less than those CMRD methods for a certain precision. We could say the SCMRD has the highest processing efficiency for this case study.

As stated by Stamnes [19,26], the Hopkins’ algorithm and Ludwig algorithm can only accurately handle rectangular apertures by using Cartesian coordinates, and circular and elliptical apertures by using polar coordinates. But the SSP algorithm could also work well for a pupil of general shape. We would stress that the SCMRD could also work well for a pupil of general shape. Because the integral interval for each subdomain can be determined independently, which can avoid the mismatch between the boundaries of the integration domain and the exit pupil domain.

As the Hopkins’ algorithm only works for cases with known wavefront type (it need sampling points’ phases and their derivatives with respect to x and y) [17], the results calculated by it are not presented here.

We summarize the application range of different algorithms in Table 3. For different cases, we could choose different algorithms. As the wavefront aberration type is always unknown for an on-orbit terminal with a deformed receiver antenna, we would recommend in order: SCMRD, CMRD with SSP algorithm and CMRD with Ludwig algorithm. The SCMRD has the advantages of both SSP algorithm and Ludwig’s algorithm as shown in Table 3.

Table 3. The application range of different algorithms

View Table | View all tables in this article

7. Conclusion

The impact of an axial non-confocal antenna on an on-orbit lasercom receiver is analyzed in terms of NID and NRP, which are calculated by the calculation model based on the SCMRD. The case study results show that, the axial non-confocal antenna will flatten the received light spot. The peak intensity of the beacon light spot declines with the axial non-confocal distance increases and when it is lower than a threshold, the communication link will be interrupted. The dispersion of the signal spot will decrease the encircled power (i.e. received power), deteriorating the BER performance. To ensure the PAT and BER performance, the effective receiving radius and the tolerance of the axial non-confocal distance should be determined with comprehensive considerations. The quantitative relations of NID and NRP versus the axial non-confocal distance would provide a reference for optical design, structural design, thermal design, assembly and adjustment accuracy design and detector selection.

In the calculation model based on the SCMRD, first-order and higher-order terms in Taylor series of phase and amplitude in integrand are all discarded. NID and NRP calculated by this model have sufficient accuracy compared with those calculated by the WFM, CMRD. It is meaningful to develop this calculation model for analyzing other deformed optical antenna impact on the on-orbit lasercom receiver because it has the intrinsic potential to achieve a higher accuracy than the WFM. Besides, compared with the CMRD, it simplifies the calculation, increases the processing efficiency and has the advantages of both SSP algorithm and Ludwig’s algorithm. This calculation model based on the SCMRD is expected to contribute to the integrated Thermal/Structural/Optical (TSO) analysis for satellite laser communication.

Funding

National Natural Science Foundation of China (91838302).

Disclosures

The authors declare that there are no conflicts of interest related to this article.

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algorithm	mean processing time per subdomain\μs	sampling numbers (for precision of 1‰)	total processing time\ms
Ludwig	121.19	262144	31769.23
SSP ^a	>121.19	≥1024	>124.10
SCMRD	53.53	512	27.41

algorithm	mean processing time per subdomain\μs	sampling numbers (for precision of 1‰)	total processing time\ms
Ludwig	121.19	262144	31769.23
SSP ^a	>121.19	≥1024	>124.10
SCMRD	53.53	512	27.41

Impact of an axial non-confocal antenna on the on-orbit lasercom receiver analyzed by the simplified combination method of ray tracing and diffraction theory

Abstract

1. Introduction

2. Optical antenna deformation model

3. Normalized intensity distribution calculated by the SCMRD

3.1 Diffraction theory

3.2 Simplified combination of ray tracing and diffraction theory

4. Normalized received power

5. NID and NRP calculated by the wavefront fitting method

6. Numerical results and discussion

6.1 Normalized intensity distribution

6.2 Normalized received power

6.3 Errors estimation and sampling parameters selection

6.3.1 Errors Estimation of NID

6.3.2 Errors Estimation of NRP

6.4 Compared with the WFM and CMRD

7. Conclusion

Funding

Disclosures

References

Cited By

Figures (9)

Tables (3)

Equations (29)

Optics Express

algorithm	known wavefront type	unknown wavefront type		general pupil
algorithm	known wavefront type	sample once	sample and then interpolate	general pupil
WFM	√	×	×	×
Hopkins	√	×	×	×
Ludwig	√	√	No need	×
SSP	√	×	√	√
SCMRD	√	√	No need	√