Abstract

In the paper, the principle and structure of a pupil-matching optical differential receiver consisting of double 4f confocal lens groups is introduced to overcome atmosphere turbulences in space-to-ground laser communication. Using the scalar diffraction theory, a systematic analysis of 4f lens groups is formulated mathematically. Based on Seidel aberration, lens aberrations produced by the inherent unideal lens and mutual alignment errors of double 4f lens groups primarily caused by relative axial displacement of the foci and vertical position change of the optical axes are studied mathematically and detailed. Under the effects of varying aberrations on the double 4f lens groups, we evaluate the performance of this receiving system by the model of power penalty for a given 109 bit error ratio. Simulated results of the relationship between power penalty and the different root-mean-square errors are concluded in order to put forward the requirement of machining precision of individual components. That will be helpful in optimizing the design of these groups in the optical receiver.

© 2014 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. R. H. Czichy, Z. Sodnik, and B. Furch, “Design of an optical ground station for in-orbit checkout of free-space laser communication payloads,” in Photonics West’95 (International Society for Optics and Photonics, 1995), pp. 26–37.
  2. S. Betti, V. Carrozzo, and E. Duca, “Optical intersatellite system based on DPSK modulation,” in 2nd International Symposium on Wireless Communication Systems (IEEE, 2005), pp. 817–821.
  3. R. G. Marshalek, G. S. Mecherle, and P. Jordan, “System-level comparison of optical and RF technologies for space-to-space and space-to-ground communication links circa 2000,” in Photonics West’96 (International Society for Optics and Photonics, 1996), pp. 134–145.
  4. A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, 1978), Vol. 2.
  5. L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media (SPIE, 2005), Vol. 152.
  6. J. C. Ricklin, S. M. Hammel, F. D. Eaton, and S. L. Lachinova, “Atmospheric channel effects on free-space laser communication,” J. Opt. Fiber Commun. Rep. 3, 111–158 (2006).
    [CrossRef]
  7. L. Zhang, J. Hu, J. Wang, and Y. Feng, “Stimulated-Brillouin-scattering-suppressed high-power single-frequency polarization-maintaining Raman fiber amplifier with longitudinally varied strain for laser guide star,” Opt. Lett. 37, 4796–4798 (2012).
    [CrossRef]
  8. R. Tyson, Principles of Adaptive Optics (CRC Press, 2010).
  9. M. Gregory, F. Heine, H. Kämpfner, R. Lange, K. Saucke, U. Sterr, and R. Meyer, “Inter-satellite and satellite-ground laser communication links based on homodyne BPSK,” in LASE (International Society for Optics and Photonics, 2010), paper 75870E.
  10. A. H. Gnauck and P. J. Winzer, “Optical phase-shift-keyed transmission,” J. Lightwave Technol. 23, 115–130 (2005).
    [CrossRef]
  11. H. Yu, H. Chen, M. Chen, and S. Xie, “Power distribution analysis for multiple modulation formats in an all-optical sampling wavelength division multiplexing system,” Chin. Opt. Lett. 11, 100604 (2013).
    [CrossRef]
  12. Y. a. Zhi, J. Sun, E. Dai, Y. Zhou, L. Wang, W. Lu, P. Hou, and L. Liu, “High-data rate differential phase shift keying receiver for satellite-to-ground optical communication link,” Proc. SPIE 8517, 85170F (2012).
    [CrossRef]
  13. J. Sun, W. Lu, L. Wang, E. Dai, and L. Liu, “High-data rate laser communication field experiment in the turbulence channel,” Proc. SPIE 8517, 851713 (2012).
    [CrossRef]
  14. Z. Sodnik, J. P. Armengol, R. Czichy, and R. Meyer, “Adaptive optics and ESA’s optical ground station,” Proc. SPIE 7464, 746406 (2009).
    [CrossRef]
  15. Z. Luan, Y. Zhou, Y. Zhi, E. Dai, J. Sun, and L. Liu, “An aperture-matched phase-compensated differential phase shift keying receiver with a 90° hybrid,” Proc. SPIE 8162, 81620O (2011).
    [CrossRef]
  16. R. B. Garreis, “90 degree optical hybrid for coherent receivers,” in Munich’91 (Lasers’ 91) (International Society for Optics and Photonics, 1991), pp. 210–219.
  17. J. W. Goodman, Introduction to Fourier Optics (Roberts & Company, 2005).
  18. D. Fink, “Coherent detection signal-to-noise,” Appl. Opt. 14, 689–690 (1975).
    [CrossRef]
  19. R. K. Tyson, “Bit-error rate for free-space adaptive optics laser communications,” J. Opt. Soc. Am. A 19, 753–758 (2002).
    [CrossRef]
  20. J. C. Wyant and K. Creath, “Basic wavefront aberration theory for optical metrology,” Appl. Opt. Opt. Eng. 11, 15–28 (1992).
  21. L. Liu, “Coherent and incoherent synthetic-aperture imaging ladars and laboratory-space experimental demonstrations [Invited],” Appl. Opt. 52, 579–599 (2013).
    [CrossRef]

2013

2012

Y. a. Zhi, J. Sun, E. Dai, Y. Zhou, L. Wang, W. Lu, P. Hou, and L. Liu, “High-data rate differential phase shift keying receiver for satellite-to-ground optical communication link,” Proc. SPIE 8517, 85170F (2012).
[CrossRef]

J. Sun, W. Lu, L. Wang, E. Dai, and L. Liu, “High-data rate laser communication field experiment in the turbulence channel,” Proc. SPIE 8517, 851713 (2012).
[CrossRef]

L. Zhang, J. Hu, J. Wang, and Y. Feng, “Stimulated-Brillouin-scattering-suppressed high-power single-frequency polarization-maintaining Raman fiber amplifier with longitudinally varied strain for laser guide star,” Opt. Lett. 37, 4796–4798 (2012).
[CrossRef]

2011

Z. Luan, Y. Zhou, Y. Zhi, E. Dai, J. Sun, and L. Liu, “An aperture-matched phase-compensated differential phase shift keying receiver with a 90° hybrid,” Proc. SPIE 8162, 81620O (2011).
[CrossRef]

2009

Z. Sodnik, J. P. Armengol, R. Czichy, and R. Meyer, “Adaptive optics and ESA’s optical ground station,” Proc. SPIE 7464, 746406 (2009).
[CrossRef]

2006

J. C. Ricklin, S. M. Hammel, F. D. Eaton, and S. L. Lachinova, “Atmospheric channel effects on free-space laser communication,” J. Opt. Fiber Commun. Rep. 3, 111–158 (2006).
[CrossRef]

2005

2002

1992

J. C. Wyant and K. Creath, “Basic wavefront aberration theory for optical metrology,” Appl. Opt. Opt. Eng. 11, 15–28 (1992).

1975

Andrews, L. C.

L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media (SPIE, 2005), Vol. 152.

Armengol, J. P.

Z. Sodnik, J. P. Armengol, R. Czichy, and R. Meyer, “Adaptive optics and ESA’s optical ground station,” Proc. SPIE 7464, 746406 (2009).
[CrossRef]

Betti, S.

S. Betti, V. Carrozzo, and E. Duca, “Optical intersatellite system based on DPSK modulation,” in 2nd International Symposium on Wireless Communication Systems (IEEE, 2005), pp. 817–821.

Carrozzo, V.

S. Betti, V. Carrozzo, and E. Duca, “Optical intersatellite system based on DPSK modulation,” in 2nd International Symposium on Wireless Communication Systems (IEEE, 2005), pp. 817–821.

Chen, H.

Chen, M.

Creath, K.

J. C. Wyant and K. Creath, “Basic wavefront aberration theory for optical metrology,” Appl. Opt. Opt. Eng. 11, 15–28 (1992).

Czichy, R.

Z. Sodnik, J. P. Armengol, R. Czichy, and R. Meyer, “Adaptive optics and ESA’s optical ground station,” Proc. SPIE 7464, 746406 (2009).
[CrossRef]

Czichy, R. H.

R. H. Czichy, Z. Sodnik, and B. Furch, “Design of an optical ground station for in-orbit checkout of free-space laser communication payloads,” in Photonics West’95 (International Society for Optics and Photonics, 1995), pp. 26–37.

Dai, E.

Y. a. Zhi, J. Sun, E. Dai, Y. Zhou, L. Wang, W. Lu, P. Hou, and L. Liu, “High-data rate differential phase shift keying receiver for satellite-to-ground optical communication link,” Proc. SPIE 8517, 85170F (2012).
[CrossRef]

J. Sun, W. Lu, L. Wang, E. Dai, and L. Liu, “High-data rate laser communication field experiment in the turbulence channel,” Proc. SPIE 8517, 851713 (2012).
[CrossRef]

Z. Luan, Y. Zhou, Y. Zhi, E. Dai, J. Sun, and L. Liu, “An aperture-matched phase-compensated differential phase shift keying receiver with a 90° hybrid,” Proc. SPIE 8162, 81620O (2011).
[CrossRef]

Duca, E.

S. Betti, V. Carrozzo, and E. Duca, “Optical intersatellite system based on DPSK modulation,” in 2nd International Symposium on Wireless Communication Systems (IEEE, 2005), pp. 817–821.

Eaton, F. D.

J. C. Ricklin, S. M. Hammel, F. D. Eaton, and S. L. Lachinova, “Atmospheric channel effects on free-space laser communication,” J. Opt. Fiber Commun. Rep. 3, 111–158 (2006).
[CrossRef]

Feng, Y.

Fink, D.

Furch, B.

R. H. Czichy, Z. Sodnik, and B. Furch, “Design of an optical ground station for in-orbit checkout of free-space laser communication payloads,” in Photonics West’95 (International Society for Optics and Photonics, 1995), pp. 26–37.

Garreis, R. B.

R. B. Garreis, “90 degree optical hybrid for coherent receivers,” in Munich’91 (Lasers’ 91) (International Society for Optics and Photonics, 1991), pp. 210–219.

Gnauck, A. H.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (Roberts & Company, 2005).

Gregory, M.

M. Gregory, F. Heine, H. Kämpfner, R. Lange, K. Saucke, U. Sterr, and R. Meyer, “Inter-satellite and satellite-ground laser communication links based on homodyne BPSK,” in LASE (International Society for Optics and Photonics, 2010), paper 75870E.

Hammel, S. M.

J. C. Ricklin, S. M. Hammel, F. D. Eaton, and S. L. Lachinova, “Atmospheric channel effects on free-space laser communication,” J. Opt. Fiber Commun. Rep. 3, 111–158 (2006).
[CrossRef]

Heine, F.

M. Gregory, F. Heine, H. Kämpfner, R. Lange, K. Saucke, U. Sterr, and R. Meyer, “Inter-satellite and satellite-ground laser communication links based on homodyne BPSK,” in LASE (International Society for Optics and Photonics, 2010), paper 75870E.

Hou, P.

Y. a. Zhi, J. Sun, E. Dai, Y. Zhou, L. Wang, W. Lu, P. Hou, and L. Liu, “High-data rate differential phase shift keying receiver for satellite-to-ground optical communication link,” Proc. SPIE 8517, 85170F (2012).
[CrossRef]

Hu, J.

Ishimaru, A.

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, 1978), Vol. 2.

Jordan, P.

R. G. Marshalek, G. S. Mecherle, and P. Jordan, “System-level comparison of optical and RF technologies for space-to-space and space-to-ground communication links circa 2000,” in Photonics West’96 (International Society for Optics and Photonics, 1996), pp. 134–145.

Kämpfner, H.

M. Gregory, F. Heine, H. Kämpfner, R. Lange, K. Saucke, U. Sterr, and R. Meyer, “Inter-satellite and satellite-ground laser communication links based on homodyne BPSK,” in LASE (International Society for Optics and Photonics, 2010), paper 75870E.

Lachinova, S. L.

J. C. Ricklin, S. M. Hammel, F. D. Eaton, and S. L. Lachinova, “Atmospheric channel effects on free-space laser communication,” J. Opt. Fiber Commun. Rep. 3, 111–158 (2006).
[CrossRef]

Lange, R.

M. Gregory, F. Heine, H. Kämpfner, R. Lange, K. Saucke, U. Sterr, and R. Meyer, “Inter-satellite and satellite-ground laser communication links based on homodyne BPSK,” in LASE (International Society for Optics and Photonics, 2010), paper 75870E.

Liu, L.

L. Liu, “Coherent and incoherent synthetic-aperture imaging ladars and laboratory-space experimental demonstrations [Invited],” Appl. Opt. 52, 579–599 (2013).
[CrossRef]

Y. a. Zhi, J. Sun, E. Dai, Y. Zhou, L. Wang, W. Lu, P. Hou, and L. Liu, “High-data rate differential phase shift keying receiver for satellite-to-ground optical communication link,” Proc. SPIE 8517, 85170F (2012).
[CrossRef]

J. Sun, W. Lu, L. Wang, E. Dai, and L. Liu, “High-data rate laser communication field experiment in the turbulence channel,” Proc. SPIE 8517, 851713 (2012).
[CrossRef]

Z. Luan, Y. Zhou, Y. Zhi, E. Dai, J. Sun, and L. Liu, “An aperture-matched phase-compensated differential phase shift keying receiver with a 90° hybrid,” Proc. SPIE 8162, 81620O (2011).
[CrossRef]

Lu, W.

J. Sun, W. Lu, L. Wang, E. Dai, and L. Liu, “High-data rate laser communication field experiment in the turbulence channel,” Proc. SPIE 8517, 851713 (2012).
[CrossRef]

Y. a. Zhi, J. Sun, E. Dai, Y. Zhou, L. Wang, W. Lu, P. Hou, and L. Liu, “High-data rate differential phase shift keying receiver for satellite-to-ground optical communication link,” Proc. SPIE 8517, 85170F (2012).
[CrossRef]

Luan, Z.

Z. Luan, Y. Zhou, Y. Zhi, E. Dai, J. Sun, and L. Liu, “An aperture-matched phase-compensated differential phase shift keying receiver with a 90° hybrid,” Proc. SPIE 8162, 81620O (2011).
[CrossRef]

Marshalek, R. G.

R. G. Marshalek, G. S. Mecherle, and P. Jordan, “System-level comparison of optical and RF technologies for space-to-space and space-to-ground communication links circa 2000,” in Photonics West’96 (International Society for Optics and Photonics, 1996), pp. 134–145.

Mecherle, G. S.

R. G. Marshalek, G. S. Mecherle, and P. Jordan, “System-level comparison of optical and RF technologies for space-to-space and space-to-ground communication links circa 2000,” in Photonics West’96 (International Society for Optics and Photonics, 1996), pp. 134–145.

Meyer, R.

Z. Sodnik, J. P. Armengol, R. Czichy, and R. Meyer, “Adaptive optics and ESA’s optical ground station,” Proc. SPIE 7464, 746406 (2009).
[CrossRef]

M. Gregory, F. Heine, H. Kämpfner, R. Lange, K. Saucke, U. Sterr, and R. Meyer, “Inter-satellite and satellite-ground laser communication links based on homodyne BPSK,” in LASE (International Society for Optics and Photonics, 2010), paper 75870E.

Phillips, R. L.

L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media (SPIE, 2005), Vol. 152.

Ricklin, J. C.

J. C. Ricklin, S. M. Hammel, F. D. Eaton, and S. L. Lachinova, “Atmospheric channel effects on free-space laser communication,” J. Opt. Fiber Commun. Rep. 3, 111–158 (2006).
[CrossRef]

Saucke, K.

M. Gregory, F. Heine, H. Kämpfner, R. Lange, K. Saucke, U. Sterr, and R. Meyer, “Inter-satellite and satellite-ground laser communication links based on homodyne BPSK,” in LASE (International Society for Optics and Photonics, 2010), paper 75870E.

Sodnik, Z.

Z. Sodnik, J. P. Armengol, R. Czichy, and R. Meyer, “Adaptive optics and ESA’s optical ground station,” Proc. SPIE 7464, 746406 (2009).
[CrossRef]

R. H. Czichy, Z. Sodnik, and B. Furch, “Design of an optical ground station for in-orbit checkout of free-space laser communication payloads,” in Photonics West’95 (International Society for Optics and Photonics, 1995), pp. 26–37.

Sterr, U.

M. Gregory, F. Heine, H. Kämpfner, R. Lange, K. Saucke, U. Sterr, and R. Meyer, “Inter-satellite and satellite-ground laser communication links based on homodyne BPSK,” in LASE (International Society for Optics and Photonics, 2010), paper 75870E.

Sun, J.

Y. a. Zhi, J. Sun, E. Dai, Y. Zhou, L. Wang, W. Lu, P. Hou, and L. Liu, “High-data rate differential phase shift keying receiver for satellite-to-ground optical communication link,” Proc. SPIE 8517, 85170F (2012).
[CrossRef]

J. Sun, W. Lu, L. Wang, E. Dai, and L. Liu, “High-data rate laser communication field experiment in the turbulence channel,” Proc. SPIE 8517, 851713 (2012).
[CrossRef]

Z. Luan, Y. Zhou, Y. Zhi, E. Dai, J. Sun, and L. Liu, “An aperture-matched phase-compensated differential phase shift keying receiver with a 90° hybrid,” Proc. SPIE 8162, 81620O (2011).
[CrossRef]

Tyson, R.

R. Tyson, Principles of Adaptive Optics (CRC Press, 2010).

Tyson, R. K.

Wang, J.

Wang, L.

Y. a. Zhi, J. Sun, E. Dai, Y. Zhou, L. Wang, W. Lu, P. Hou, and L. Liu, “High-data rate differential phase shift keying receiver for satellite-to-ground optical communication link,” Proc. SPIE 8517, 85170F (2012).
[CrossRef]

J. Sun, W. Lu, L. Wang, E. Dai, and L. Liu, “High-data rate laser communication field experiment in the turbulence channel,” Proc. SPIE 8517, 851713 (2012).
[CrossRef]

Winzer, P. J.

Wyant, J. C.

J. C. Wyant and K. Creath, “Basic wavefront aberration theory for optical metrology,” Appl. Opt. Opt. Eng. 11, 15–28 (1992).

Xie, S.

Yu, H.

Zhang, L.

Zhi, Y.

Z. Luan, Y. Zhou, Y. Zhi, E. Dai, J. Sun, and L. Liu, “An aperture-matched phase-compensated differential phase shift keying receiver with a 90° hybrid,” Proc. SPIE 8162, 81620O (2011).
[CrossRef]

Zhi, Y. a.

Y. a. Zhi, J. Sun, E. Dai, Y. Zhou, L. Wang, W. Lu, P. Hou, and L. Liu, “High-data rate differential phase shift keying receiver for satellite-to-ground optical communication link,” Proc. SPIE 8517, 85170F (2012).
[CrossRef]

Zhou, Y.

Y. a. Zhi, J. Sun, E. Dai, Y. Zhou, L. Wang, W. Lu, P. Hou, and L. Liu, “High-data rate differential phase shift keying receiver for satellite-to-ground optical communication link,” Proc. SPIE 8517, 85170F (2012).
[CrossRef]

Z. Luan, Y. Zhou, Y. Zhi, E. Dai, J. Sun, and L. Liu, “An aperture-matched phase-compensated differential phase shift keying receiver with a 90° hybrid,” Proc. SPIE 8162, 81620O (2011).
[CrossRef]

Appl. Opt.

Appl. Opt. Opt. Eng.

J. C. Wyant and K. Creath, “Basic wavefront aberration theory for optical metrology,” Appl. Opt. Opt. Eng. 11, 15–28 (1992).

Chin. Opt. Lett.

J. Lightwave Technol.

J. Opt. Fiber Commun. Rep.

J. C. Ricklin, S. M. Hammel, F. D. Eaton, and S. L. Lachinova, “Atmospheric channel effects on free-space laser communication,” J. Opt. Fiber Commun. Rep. 3, 111–158 (2006).
[CrossRef]

J. Opt. Soc. Am. A

Opt. Lett.

Proc. SPIE

Y. a. Zhi, J. Sun, E. Dai, Y. Zhou, L. Wang, W. Lu, P. Hou, and L. Liu, “High-data rate differential phase shift keying receiver for satellite-to-ground optical communication link,” Proc. SPIE 8517, 85170F (2012).
[CrossRef]

J. Sun, W. Lu, L. Wang, E. Dai, and L. Liu, “High-data rate laser communication field experiment in the turbulence channel,” Proc. SPIE 8517, 851713 (2012).
[CrossRef]

Z. Sodnik, J. P. Armengol, R. Czichy, and R. Meyer, “Adaptive optics and ESA’s optical ground station,” Proc. SPIE 7464, 746406 (2009).
[CrossRef]

Z. Luan, Y. Zhou, Y. Zhi, E. Dai, J. Sun, and L. Liu, “An aperture-matched phase-compensated differential phase shift keying receiver with a 90° hybrid,” Proc. SPIE 8162, 81620O (2011).
[CrossRef]

Other

R. B. Garreis, “90 degree optical hybrid for coherent receivers,” in Munich’91 (Lasers’ 91) (International Society for Optics and Photonics, 1991), pp. 210–219.

J. W. Goodman, Introduction to Fourier Optics (Roberts & Company, 2005).

R. Tyson, Principles of Adaptive Optics (CRC Press, 2010).

M. Gregory, F. Heine, H. Kämpfner, R. Lange, K. Saucke, U. Sterr, and R. Meyer, “Inter-satellite and satellite-ground laser communication links based on homodyne BPSK,” in LASE (International Society for Optics and Photonics, 2010), paper 75870E.

R. H. Czichy, Z. Sodnik, and B. Furch, “Design of an optical ground station for in-orbit checkout of free-space laser communication payloads,” in Photonics West’95 (International Society for Optics and Photonics, 1995), pp. 26–37.

S. Betti, V. Carrozzo, and E. Duca, “Optical intersatellite system based on DPSK modulation,” in 2nd International Symposium on Wireless Communication Systems (IEEE, 2005), pp. 817–821.

R. G. Marshalek, G. S. Mecherle, and P. Jordan, “System-level comparison of optical and RF technologies for space-to-space and space-to-ground communication links circa 2000,” in Photonics West’96 (International Society for Optics and Photonics, 1996), pp. 134–145.

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, 1978), Vol. 2.

L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media (SPIE, 2005), Vol. 152.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (14)

Fig. 1.
Fig. 1.

Schematic of the pupil-matching optical differential receiver (bold line, electrical signal; fine line, optical signal; blue line, polarization state; λ/2: half-wave plate; λ/4, quarter-wave plate; PBS, polarization beam splitter).

Fig. 2.
Fig. 2.

Pupil-matching DI with double 4f systems.

Fig. 3.
Fig. 3.

Diagram of the long-arm branch of 4f systems.

Fig. 4.
Fig. 4.

Power penalty ΔP as a function of RMS wavefront error for 109 BER in the long-arm branch. The inset shows a magnified view of power penalty.

Fig. 5.
Fig. 5.

Power penalty ΔP affected by field curvature from both branches. The inset shows a magnified view of power penalty.

Fig. 6.
Fig. 6.

Power penalty ΔP affected by spherical aberration from both branches of the lens. The inset shows a magnified view of power penalty.

Fig. 7.
Fig. 7.

Power penalty ΔP affected by astigmatism from both branches. The inset shows a magnified view of power penalty.

Fig. 8.
Fig. 8.

Power penalty ΔP as a function of rotation angle α of lens t3 with astigmatism only in the long-arm branch.

Fig. 9.
Fig. 9.

Diagram of single lens t3 with tilt angle θ.

Fig. 10.
Fig. 10.

Power penalty ΔP as a function of tilt angle θ of lens in both branches.

Fig. 11.
Fig. 11.

Schematic of the center shift of the lens.

Fig. 12.
Fig. 12.

Power penalty ΔP as a function of center shift (Δx,Δy) of lens t3 in one branch. (a) 3D diagram of shift (Δx,Δy) from the optical axis in x- and y-directions. (b) Section view of 3D diagram in x-direction.

Fig. 13.
Fig. 13.

Diagram of defocusing distance ΔL between lenses.

Fig. 14.
Fig. 14.

Effect of defocusing distance ΔL on power penalty.

Tables (1)

Tables Icon

Table 1. Wavefront Aberration Items

Equations (20)

Equations on this page are rendered with MathJax. Learn more.

UT(x,y;t0)=AT(x,y;t0)exp[j(φ(t0)+θ0)],
Δφ(ti)=φ(ti)φ(tiT)={0data bit0πdata bit1,
US(x,y;ti)=AS(x,y;ti)exp[j(φ(ti)+kW(x,y;ti)+θi)],
I(i)=eηA2hvZ0[US(x,y;ti)+US(x,y;tiT)]2={eηAhvZ0AS2(x,y;ti)[1+cos(θiθiT)]data bit0eηAhvZ0AS2(x,y;ti)[1cos(θiθiT)]data bit1
ηH(i)=AUS(x,y;ti)US*(x,y;tiT)dAAUS*(x,y;ti)US(x,y;tiT)dAA|US(x,y;ti)|2dAA|US(x,y;tiT)|2dA,
P0(x0,y0,r0)={0whenx02+y02<r01otherwise.
h(x,y)=exp(jkf1)jλf1exp[jk2f1(x2+y2)],
U1_L(x1,y1)=U0(x0,y0)h(x1x0,y1y0)dx0dy0.
t1(x1,y1)=exp[jk2f1(x12+y12)]×exp(jkΔW1)·Pt(x1,y1,t1).
U3_L(x3,y3)=exp(jk2f1)jλ2f1exp(jk4f1(x32+y32))×U1(x1,y1)t1(x1,y1)exp[jk(x12+y12)4f1]×exp[jk(x1x3+y1y3)2f1]dx1dy1.
U5_L(x5,y5)=exp(jkf1)jλf1exp(jk2f1(x52+y52))×U3(x3,y3)t3(x3,y3)exp[jk(x32+y32)2f1]×exp[jk(x5x3+y3y3)f1]dx3dy3.
U5_L(x5,y5)=FF[U0(x0,y0)]=U0(x0,y0),
U5(x5,y5)=U5_L(x5,y5)+U5_S(x5,y5).
SNR=ηP02hvBηH,
BER=12erfc(SNR22)2πexp(SNR/8)SNR,
ΔP=10log10(PnP0).
u=u0exp(jkW(ρ,θ)),
W(ρ,θ)=l,m,nwl,m,nx0lρmcosnθ,
σ2=1SareaSarea[ΔW(x,y)ΔW(x,y)¯]2dS=1SareaSareaΔW2(x,y)dS(1Sarea)2[SareaΔW(x,y)dS]2,
t3(x,y)=exp(jk2(x2f1cos2θ+y2f1))circle((xrtcosθ)2+(yrt)2).

Metrics