Optical devices in free-space laser communication systems are affected by their environment, particularly in relation to the effects of temperature while in orbit. The mutual alignment error between the transmitted and received optical axes is caused by deformation of the optics due to temperature variation in spite of the common optics used for transmission and reception of the optical beams. When a Gaussian beam wave for transmission is aligned at the center of a received plane wave, 3rd-order Coma aberrations have the most influence on the mutual alignment error, which is an inevitable open pointing error under only the Tip/Tilt tracking control. As an example, a mutual alignment error of less than 0.2 µrad is predicted for a laser communication terminal in orbit using the results from space chamber thermal vacuum tests. The relative power penalty due to aberration is estimated to be about 0.4 dB. The results will mitigate surface quality in an optical antenna and contribute to the design of free-space laser communication systems.
© 2001 Optical Society of America
Free-space laser communication systems are expected to be used for ultra-high-data-rate communications in future intersatellite communication networks. Optical devices in orbit are affected by their environment, which includes space radiation, contamination, and temperature variation. Temperature variation strongly affects optical properties, and causes 1) variation of the reflective index, 2) variation of the curvature of the lens surface, 3) variation of the thickness of the lens, and 4) variation in the gap between lenses. If a lens does not have a uniform temperature distribution, the optical wave-front changes locally due to 1), 2) and 3). The optical intensity distributions focused on the imaging array, the acquisition and tracking sensors, degrade due to deformation of the wave-front caused by temperature variation. In free-space laser communications, the optical center of gravity of the received beam changes at the optical tracking sensor, and the transmitted far-field pattern also changes at the counter terminal. The alignment error in free-space laser communications is more important than that in image observations.
In this study, we investigate the effect of the mutual alignment error due to wave-front distortion that originates from temperature variations in the optical tracking system. The received optical axis is obtained by the center of gravity of the received optical power on an optical tracking sensor. The transmitted optical axis is determined by the direction with a peak intensity at a far-field. The difference between the transmitted and received optical axes is caused by the slightly different wave-front distortions between the transmitted Gaussian beam and the received plane waves. This will be referred to in the present paper as the mutual alignment error. First, the optical axes of the transmitted and received optical beams in free-space laser communications are defined in Section 2. The mutual alignment error is evaluated using Zernike polynomials in Section 3. It is found that the 3rd-order Coma aberration has the greatest influence on the mutual alignment error when the transmitting Gaussian beam is aligned at the center of the received plane wave. As an example, wave-front variations for the Optical Inter-orbit Communications Engineering Test Satellite (OICETS) developed by the National Space Development Agency of Japan (NASDA) are analyzed with respect to the mutual alignment error in Section 4.[1–5]
2. Definition of optical axes
2.A. Received optical axis
where Z i(r,θ) are Zernike polynomials, ai are the coefficients of the Zernike polynomials, u=ρcosθ, v=ρsinθ, r=2ρ/D, and D is the diameter of the optical antenna. n is called the radial degree and m is the azimuthal frequency. The rms wave-front error for each Zernike mode is given by
where W(ρ,θ) is a function of the telescope aperture and given by
A complex amplitude of the wave Ul(u,v) passing through a lens with a focal length z=f is focused on the focal plane, and the optical field is given by
where k is the wave number, λ is the wavelength, and A is a constant that represents the strength or amplitude of the wave. Figure 1 gives a definition of the coordinate systems. As can be seen from Eq. (6), the optical distribution on the optical detector is given by a two-dimensional Fourier transform (fu=x/λf, fv=y/λf) of the optical field Ul(u,v) at the input plane as a Fraunhofer diffraction pattern. The intensity distribution If(x,y) on the optical detector is given by the power spectrum,
Since a quadrant detector (QD) is used as the optical tracking sensor in free-space laser communications, the center of gravity (X,Y) of the received optical power on a QD aligned at (x,y)=(0,0) shown in Fig. 2 is given by
The direction of incidence of the received optical beam is obtained as the vector (X,Y,f) with focal length f and the detected position (X,Y) obtained from Eq. (9).
2.B. Transmitted optical axis
A laser beam transmitted from a semiconductor laser tip or an optical fiber generally becomes a Gaussian beam wave.[13,14] Therefore, a transmitted optical beam can be calculated at a far-field in which a Gaussian beam wave with a deformed phase error due to temperature variation at the input plane propagates:
where F 0 is the radius of curvature at the transmitter, W 0 is the half beam-width at an intensity of 1/e2, and the truncation ratio is α=D/(2W 0).[16,17] The transmitted optical axis is determined by the direction with a peak intensity of Iffp(η,ξ)|max=Iffp(η max,ξ max) as shown in Fig. 3. The direction of the transmitted optical beam is obtained as the vector (η max,ξ max, z) using the two-dimensional Fourier transform (fη=u/λz,fζ=v/λz).
The mutual alignment error is defined as the angle between the vectors of the received (X,Y,f) and transmitted (-η max,-ξ max,-z) optical directions. In addition, the relative power degradation due to the mutual alignment error is also obtained as the ratio of the intensities Iffp(-Xz/f,-Yz/f)/Iffp(η max,ξ max) in the far-field pattern.
3. Mutual alignment error due to wave-front deformation
The mutual alignment error for each wave-front aberration is shown in Figs. 4 and 5. In the figures, γ means the obscuration ratio of the diameters of the secondary to the primary mirrors with a Cassegrain-type telescope.[16,17] For the lower-order Zernike modes, mutual alignment errors are due to aberrations with m=1 or m=3, without Tip/tilt aberrations. Since Tip/tilt aberrations merely involve sloping in a plane, the transmitted and received optical axes completely coincide with each other. The 3rd-order Coma aberration has the greatest influence on the mutual alignment error. On the other hand, the higher-order Zernike modes have less of an influence on the mutual alignment error because the wave-front error at higher-order Zernike modes is less sensitive to the center of gravity of the intensity distribution.
For example, the Coma aberration (Z7) is shown in Fig. 6 and the variation of the intensity received at the optical detector, generated from the plane wave, is shown in Fig. 7 as the Zernike coefficient a7 varies with time. As seen from Fig. 7, the intensity distribution of the Coma aberration is asymmetric because the Coma aberration shown in Fig. 6 is asymmetric. On the other hand, the far-field pattern of the transmitted optical beam, which is generated from the Gaussian beam wave, is shown in Fig. 8. Note that the variation in Fig. 8 is different from that in Fig. 7. The mutual alignment error is caused by this difference between the direction of the center of gravity in Fig. 7 and the direction of the peak intensity in Fig. 8. The influence of each Zernike mode will be changed according to how to align the transmitting Gaussian beam against the received optical axis. The aberration near the center of the received axis mostly affects on the mutual alignment error, that is, the Coma aberration has the most influence in this case.
Figure 9 shows the change in the mutual alignment error due to the Coma aberration (Z7) against the truncation ratio α, which is the ratio of the diameter of the telescope to the beam-width of the Gaussian beam wave. The mutual alignment error increases at a larger truncation ratio because the truncated Gaussian beam distribution for transmission deviates more from the received optical intensity distribution.
4. Mutual alignment error for the OICETS laser terminal
4.A. Analysis of mutual alignment error
As an example, the mutual alignment errors for the laser terminal onboard the OICETS were calculated using the following values: diameter of the optical antenna D=0.26 m, truncation ratio α=1.579, obscuration ratio γ=0.2889, and wavelength λ=0.847µm. The results are shown in Figs. 10 and 11. As expected, the Coma aberrations (Z7,Z8) have the greatest influence on the mutual alignment error for the OICETS laser terminal.
4.B. Measurement of wave-front error
The optical wave-front variations for the orbit model laser terminal were measured in the 6-mϕ space chamber at the Tsukuba Space Center during the thermal vacuum test (TVT). There are two laser diodes (LD1 and LD2) onboard the OICETS laser terminal for redundancy. The wave-front variations for LD1 and LD2 during the TVT are shown in Figs. 12 and 13, where the optical intensities were modulated by a 50-Mbps pseudo noise signal. The trends of the wave-front errors for LD1 and LD2 are shown in Figs. 14 and 15, respectively. The abscissa shows the simulated thermal conditions in orbit. The total wave-front errors of both the transmitted laser beams were less than λ/10. The wave-front error for the Coma aberration of LD1 varied from 0.0407λ to 0.0518λ. Therefore, a mutual alignment error of less than 0.2 µrad can be expected in orbit, based on Figs. 10 and 11, and this can be considered a pointing error.
4.C. Relative power degradation due to wave-front error
Figure 16 shows the optical power degradation due to the mutual alignment error at the counter terminal for the OICETS laser terminal. The Coma aberrations have the most influence on the power penalty. The others except for Tip/Tilt aberrations also have the larger power penalty at larger wave-front errors. The optical power variation due only to the Coma aberration is predicted to be about 0.4 dB with the wave-front variation measured in the TVT. It is difficult to extract only this power penalty component in the real measurement, however, the variations of the peak optical intensities of the far-field pattern transmitted from LD1 and LD2 measured in the TVT are shown in Fig. 17 for reference. Based on these results, the degradation of the peak optical intensity due to wave-front deformation in the OICETS laser terminal may be adequately suppressed, since the optical antenna for OICETS is made of a glass material with a small coefficient of expansion.
The effect of the mutual alignment error between the transmitted and received optical axes due to wave-front distortion that originated in temperature variation in a free-space laser communication terminal was evaluated using Zernike polynomials. It was found that the 3rd-order Coma aberration had the greatest influence on the mutual alignment error when the transmitting Gaussian beam was aligned at the center of the received plane wave. The wave-front variations for the OICETS laser terminal measured in the TVT were presented and compared with the results of the theoretical analysis. Optical system modelers will have to pay the most attention to compensate the variation of the Coma aberrations since this type of open pointing errors cannot be eliminated without such an adaptive optics system. The results will mitigate surface quality of an optical antenna and contribute to the design of free-space laser communication systems.
The authors gratefully acknowledge the helpful discussions with Prof. T. Takano of the Institute of Space Astronautical Science of Tokyo University. We would like to thank Mr. Y. Koyama and Mr. K. Shiratama of NEC Corporation for their assistance with the thermal vacuum test and for acquiring the optical properties of the laser communication terminal.
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