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Optics Express

Optics Express

  • Editor: J. H. Eberly
  • Vol. 9, Iss. 11 — Nov. 19, 2001
  • pp: 592–602
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Mutual alignment errors due to the variation of wave-front aberrations in a free-space laser communication link

Morio Toyoshima, Nobuhiro Takahashi, Takashi Jono, Toshihiko Yamawaki, Keizo Nakagawa, and Akio Yamamoto  »View Author Affiliations


Optics Express, Vol. 9, Issue 11, pp. 592-602 (2001)
http://dx.doi.org/10.1364/OE.9.000592


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Abstract

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.

© Optical Society of America

1. Introduction

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

1. K. Nakagawa and A. Yamamoto, “Preliminary design of Laser Utilizing Communications Experiment (LUCE) installed on Optical Inter-Orbit Communications Engineering Test Satellite (OICETS),” in Free-Space Laser communication Technologies VII, G. S. Mecherle, ed., Proc. SPIE2381, 14–25 (1995).

5

5. T. Jono, M. Toyoda, K. Nakagawa, A. Yamamoto, K. Shiratama, T. Kurii, and Y. Koyama, “Acquisition, tracking and pointing system of OICETS for free space laser communications,” in Acquisition, Tracking, and Pointing XIII, M. K. Masten and L. A. Stockum, eds., Proc. SPIE3692, 41–50 (1999).

]

2. Definition of optical axes

2.A. Received optical axis

Any deformed optical phase Φ(ρ,θ) can be represented using the sum of Zernike polynomials, which are widely used in the analysis of optical phase properties:[6

6. M. Born and E. Wolf, Principles of Optics, 7th Edition, (Cambridge, Cambridge University Press, 1999).

10

10. F. Roddier, Ed., Adaptive Optics in Astronomy, (Cambridge, Cambridge University Press, 1999). [CrossRef]

]

Φ(ρ,θ)=i=1aiZi(2ρD,θ),
(1)
Zi(r,θ)={Zeveni=n+1Rnm(r)2cosmθ,m0Zoddi=n+1Rnm(r)2sinmθ,m0,Zi=n+1Rn0(r),m=0
(2)
Rnm(r)=s=0(nm)2(1)s(ns)!s![(n+m)2s]![(nm)2s]!rn2s,
(3)

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

Φrms2=1πd2rW(Dr2,θ)Φ2(Dr2,θ),
=1πd2rW(Dr2,θ)[i=1aiZi(r,θ)]2,
=1πi=1{d2rW(Dr2,θ)[aiZi(r,θ)]2},
=Φrms,12+Φrms,22+Φrms,32+,
(4)

where W(ρ,θ) is a function of the telescope aperture and given by

{W(ρ,θ)=4πD2forρD2W(ρ,θ)=0forD2<ρ.
(5)

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[11

11. J. W. Goodman, Introduction to Fourier Optics, Second Edition, (New York, McGraw-Hill, 1996).

]

Uf(x,y)=Ajλfejk2f(x2+y2)Ul(u,v)exp[j2πλf(xu+yv)]dudv,
(6)
Ul(u,v)=W(ρ,θ)exp[jΦ(ρ,θ)],
(7)

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,

If(x,y)=A2λ2f2Ul(u,v)exp[j2πλf(xu+yv)]dudv2.
(8)

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[12

12. M. Katzman, Ed., Laser Satellite Communications, (Englewood Cliffs, N.J., Prentice-Hall, 1987).

]

{0If(xX,y)dxdy0If(xX,y)dxdyIf(x,y)dxdy=00If(x,yY)dxdy0If(x,yY)dxdyIf(x,y)dxdy=0.
(9)

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).

Fig. 1. Definition of the coordinate systems.
Fig. 2. Definition of the received optical axis.
Fig. 3. Definition of the transmitted optical axis.

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

13. C. C. Chen and C. S. Gardner, “Impact of random pointing and tracking errors on the design of coherent and incoherent optical intersatellite communication links,” IEEE T COMMUN 37, 252–260 (1989). [CrossRef]

,14

14. M. Toyoshima, T. Jono, K. Nakagawa, and A. Yamamoto, “Optimum divergence angle of a Gaussian beam wave in the presence of random jitter in free-space laser communication systems,” J. Opt. Soc. Am. A, (to be published).

] 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:[15

15. L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media, (Bellingham, Washington, SPIE Press, 1998).

]

Iffp(η,ξ)=A2λ2z2W(ρ,θ)exp[ρ2W02jkρ22F0
+jΦ(ρ,θ)]exp[j2πλz(ηu+ξv)]dudv2,
(10)

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

16. B. J. Klein and J. J. Degnan, “Optical Antenna Gain,” Appl. Opt. 13, 2134–2142 (1974). [CrossRef] [PubMed]

,17

17. S. G. Lambert and W. L. Casey, Laser communications in space, (Boston, London, Artech House, 1995).

] 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

16. B. J. Klein and J. J. Degnan, “Optical Antenna Gain,” Appl. Opt. 13, 2134–2142 (1974). [CrossRef] [PubMed]

,17

17. S. G. Lambert and W. L. Casey, Laser communications in space, (Boston, London, Artech House, 1995).

] 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.

Fig. 4. X-axis mutual alignment error due to wave-front aberration (F 0=α, γ=0.0 and α=1.579).
Fig. 5. Y-axis mutual alignment error due to wave-front aberration (F 0=α, γ=0.0 and α=1.579).
Fig. 6. Phase displacement of the Coma aberration (Z7).
Fig. 7. Movie of the received intensity distribution due to the Coma aberration (Z7) generated from the plane wave on the optical sensor as the Zernike coefficient a7 varies with time (γ=0.0 and α=1.579). (906 KB)
Fig. 8. Movie of the transmitted intensity distribution due to the Coma aberration (Z7) generated from the Gaussian beam wave at the far-field as the Zernike coefficient a7 varies with time (F 0=α, γ=0.0 and α=1.579). (847 KB)
Fig. 9. X-axis mutual alignment error due to the Coma aberration (Z7) as a function of the truncation ratio α (F 0=α and γ=0.0).

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.[18

18. T. Jono, M. Toyoshima, K. Nakagawa, and A. Yamamoto, “Design methodology for free-space laser communication terminal onboard a satellite,” Technical report of IEICE, SANE 2000–27, 35–40 (2000).

] 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.

Fig. 10. X-axis mutual alignment error due to wave-front aberrations for the OICETS optical antenna (F 0=α, γ=0.2889 and α=1.579).
Fig. 11. Y-axis mutual alignment error due to wave-front aberrations for the OICETS optical antenna (F 0=α, γ=0.2889 and α=1.579).

4.B. Measurement of wave-front error

Fig. 12. Movie of the wave-front variation of LD1 onboard the OICETS laser terminal measured during the thermal vacuum test. (1.70 MB)
Fig. 13. Movie of the wave-front variation of LD2 for the OICETS laser terminal measured during the thermal vacuum test. (1.53 MB)
Fig. 14. Trend of wave-front errors of LD1 onboard the OICETS laser terminal measured during the thermal vacuum test.
Fig. 15. Trend of wave-front errors of LD2 onboard the OICETS laser terminal measured during the thermal vacuum test.
Fig. 16. Degradation of the transmitted optical power due to wave-front aberrations at the counter terminal for the OICETS laser terminal (F 0=α, γ=0.2889 and α=1.579).

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.

Fig. 17. Trends in the relative peak intensities of the far-field pattern transmitted from LD1 and LD2 onboard the OICETS laser terminal measured during the thermal vacuum test.

5. Conclusion

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.

Acknowledgments

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.

References and links

1.

K. Nakagawa and A. Yamamoto, “Preliminary design of Laser Utilizing Communications Experiment (LUCE) installed on Optical Inter-Orbit Communications Engineering Test Satellite (OICETS),” in Free-Space Laser communication Technologies VII, G. S. Mecherle, ed., Proc. SPIE2381, 14–25 (1995).

2.

K. Nakagawa, A. Yamamoto, and Y. Suzuki, “OICETS optical link communications experiment in space,” in Semiconductor Laser II, S. Forouhar and Q. Wang, eds., Proc. SPIE2886, 172–183 (1996).

3.

K. Nakagawa and A. Yamamoto, “Engineering model test of LUCE (Laser Utilizing Communications Equipment),” in Free-Space Laser communication Technologies VIII, G. S. Mecherle, ed., Proc. SPIE2699, 114–120 (1996).

4.

Y. Suzuki, K. Nakagawa, T. Jono, and A. Yamamoto, “Current status of OICETS laser-communication-terminal development: development of laser diodes and sensors for OICETS program,” in Free-Space Laser communication Technologies IX, G. S. Mecherle, ed., Proc. SPIE2990, 31–37 (1997).

5.

T. Jono, M. Toyoda, K. Nakagawa, A. Yamamoto, K. Shiratama, T. Kurii, and Y. Koyama, “Acquisition, tracking and pointing system of OICETS for free space laser communications,” in Acquisition, Tracking, and Pointing XIII, M. K. Masten and L. A. Stockum, eds., Proc. SPIE3692, 41–50 (1999).

6.

M. Born and E. Wolf, Principles of Optics, 7th Edition, (Cambridge, Cambridge University Press, 1999).

7.

R. J. Noll, “Zernike polynomials and atmospheric turbulence,” J. Opt. Soc. Am. 66, 207–211 (1976). [CrossRef]

8.

J. Y. Wang and D. E. Silva, “Wave-front interpretation with Zernike polynomials,” Appl. Opt. 19, 1510–1517 (1980). [CrossRef] [PubMed]

9.

N. Roddier, “Atmospheric wavefront simulation using Zernike polynomials,” Opt. Eng.29, 1174–1180 (1990). [CrossRef]

10.

F. Roddier, Ed., Adaptive Optics in Astronomy, (Cambridge, Cambridge University Press, 1999). [CrossRef]

11.

J. W. Goodman, Introduction to Fourier Optics, Second Edition, (New York, McGraw-Hill, 1996).

12.

M. Katzman, Ed., Laser Satellite Communications, (Englewood Cliffs, N.J., Prentice-Hall, 1987).

13.

C. C. Chen and C. S. Gardner, “Impact of random pointing and tracking errors on the design of coherent and incoherent optical intersatellite communication links,” IEEE T COMMUN 37, 252–260 (1989). [CrossRef]

14.

M. Toyoshima, T. Jono, K. Nakagawa, and A. Yamamoto, “Optimum divergence angle of a Gaussian beam wave in the presence of random jitter in free-space laser communication systems,” J. Opt. Soc. Am. A, (to be published).

15.

L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media, (Bellingham, Washington, SPIE Press, 1998).

16.

B. J. Klein and J. J. Degnan, “Optical Antenna Gain,” Appl. Opt. 13, 2134–2142 (1974). [CrossRef] [PubMed]

17.

S. G. Lambert and W. L. Casey, Laser communications in space, (Boston, London, Artech House, 1995).

18.

T. Jono, M. Toyoshima, K. Nakagawa, and A. Yamamoto, “Design methodology for free-space laser communication terminal onboard a satellite,” Technical report of IEICE, SANE 2000–27, 35–40 (2000).

OCIS Codes
(010.3310) Atmospheric and oceanic optics : Laser beam transmission
(060.4510) Fiber optics and optical communications : Optical communications

ToC Category:
Research Papers

History
Original Manuscript: September 27, 2001
Published: November 19, 2001

Citation
Morio Toyoshima, Nobuhiro Takahashi, Takashi Jono, Toshihiko Yamawaki, Keizo Nakagawa, and Akio Yamamoto, "Mutual alignment errors due to the variation of wave-front aberrations in a free-space laser communication link," Opt. Express 9, 592-602 (2001)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-9-11-592


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References

  1. K. Nakagawa and A. Yamamoto, "Preliminary design of Laser Utilizing Communications Experiment (LUCE) installed on Optical Inter-Orbit Communications Engineering Test Satellite (OICETS)," in Free-Space Laser communication Technologies VII, G. S. Mecherle, ed., Proc. SPIE 2381, 14-25 (1995).
  2. K. Nakagawa, A. Yamamoto and Y. Suzuki, "OICETS optical link communications experiment in space," in Semiconductor Laser II, S. Forouhar and Q. Wang, eds., Proc. SPIE 2886, 172-183 (1996).
  3. K. Nakagawa and A. Yamamoto, "Engineering model test of LUCE (Laser Utilizing Communications Equipment)," in Free-Space Laser communication Technologies III, G. S. Mecherle, ed., Proc. SPIE 2699, 114-120 (1996).
  4. Y. Suzuki, K. Nakagawa, T. Jono and A. Yamamoto, "Current status of OICETS laser-communication-terminal development: development of laser diodes and sensors for OICETS program," in Free-Space Laser communication Technologies IX, G. S. Mecherle, ed., Proc. SPIE 2990, 31-37 (1997).
  5. T. Jono, M. Toyoda, K. Nakagawa, A. Yamamoto, K. Shiratama, T. Kurii and Y. Koyama, "Acquisition, tracking and pointing system of OICETS for free space laser communications," in Acquisition, Tracking, and Pointing XIII, M. K. Masten and L. A. Stockum, eds., Proc. SPIE 3692, 41-50 (1999).
  6. M. Born and E. Wolf, Principles of Optics, 7th Edition, (Cambridge, Cambridge University Press, 1999).
  7. R. J. Noll, "Zernike polynomials and atmospheric turbulence," J. Opt. Soc. Am. 66, 207-211 (1976). [CrossRef]
  8. J. Y. Wang and D. E. Silva, "Wave-front interpretation with Zernike polynomials," Appl. Opt. 19, 1510-1517 (1980). [CrossRef] [PubMed]
  9. N. Roddier, "Atmospheric wavefront simulation using Zernike polynomials," Opt. Eng. 29, 1174-1180 (1990). [CrossRef]
  10. F. Roddier, Ed., Adaptive Optics in Astronomy, (Cambridge, Cambridge University Press, 1999). [CrossRef]
  11. J. W. Goodman, Introduction to Fourier Optics, Second Edition, (New York, McGraw-Hill, 1996).
  12. M. Katzman, Ed., Laser Satellite Communications, (Englewood Cliffs, N.J., Prentice-Hall, 1987).
  13. C. C. Chen and C. S. Gardner, "Impact of random pointing and tracking errors on the design of coherent and incoherent optical intersatellite communication links," IEEE T Commun. 37, 252-260 (1989). [CrossRef]
  14. M. Toyoshima, T. Jono, K. Nakagawa, and A. Yamamoto, "Optimum divergence angle of a Gaussian beam wave in the presence of random jitter in free-space laser communication systems," J. Opt. Soc. Am. A, (to be published).
  15. L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media, (Bellingham, Washington, SPIE Press, 1998).
  16. B. J. Klein and J. J. Degnan, "Optical Antenna Gain," Appl. Opt. 13, 2134-2142 (1974). [CrossRef] [PubMed]
  17. S. G. Lambert and W. L. Casey, Laser communications in space, (Boston, London, Artech House, 1995).
  18. T. Jono, M. Toyoshima, K. Nakagawa, and A. Yamamoto, "Design methodology for free-space laser communication terminal onboard a satellite," Technical report of IEICE, SANE 2000-27, 35-40 (2000).

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