Abstract

In this paper we develop a three-dimensional (3D) ray tracing tool based on the ABCD ray transfer matrices. With symmetric optical components and under paraxial approximation, two sets of 2×2 ABCD matrices, each for a two-dimensional subspace, can be used to describe the 3D ray propagation completely. Compared to commercial ray-tracing software packages, our tool requires no tedious drawing, and the results for various conditions, such as different device dimensions and incident angles, can be easily obtained by simply changing the parameter values used for the calculation. We have employed this matrix-based 3D ray tracing tool to model cat’s eye retroreflectors. The cat’s eye performance, including the retroreflection efficiency, acceptance angle (i.e., field of view), and beam divergence and deviation, is fully studied. The application of this 3D ray tracing technique can be further extended to other optical components.

© 2012 Optical Society of America

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    [CrossRef]
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    [CrossRef]
  4. P. G. Goetz, W. S. Rabinovich, R. Mahon, J. L. Murphy, M. S. Ferraro, W. R. Smith, B. B. Xu, H. R. Burris, C. I. Moore, and W. W. Schultz, “Modulating retro-reflector lasercom systems at the Naval Research Laboratory,” in 2010 Military Communications Conference (MILCOM, 2010), pp. 1601–1606.
  5. J. C. Juarez, A. Dwivedi, A. Mammons, S. D. Jones, V. Weerackody, and R. A. Nichols, “Free-space optical communications for next-generation military networks,” IEEE Commun. Mag. 44, 46–51 (2006).
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  8. K. H. Chao, C. D. Liao, B. J. Yang, and J. C. Tsai, “Fabrication and characterization of a micro tunable cat’s eye retro-reflector,” Opt. Commun. 284, 5221–5224 (2011).
    [CrossRef]
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  22. W. F. Harris, “Paraxial ray tracing through noncoaxial astigmatic optical systems, and a 5×5 augmented system matrix,” Opt. Vis. Sci. 71, 282–285 (1994).
  23. P. D. Lin and C. K. Sung, “Matrix-based paraxial skew ray-tracing in 3D systems with non-coplanar optical axis,” Optik 117, 329–340 (2006).
    [CrossRef]
  24. P.-D. Lin and C.-C. Hsueh, “6×6 matrix formalism of optical elements for modeling and analyzing 3D optical systems,” Appl. Phys. B 97, 135–143 (2009).
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  25. H. H. Arsenault and B. Macukow, “Factorization of the transfer matrix for symmetrical optical systems,” J. Opt. Soc. Am. 73, 1350–1359 (1983).
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  26. D. S. Goodman, “1.8 refraction and reflection at interfaces between homogeneous media,” in Handbook of Optics, Volume 1: Geometrical and Physical Optics, Polarized Light, Components and Instruments, 3rd ed. (McGraw-Hill, 2010), pp. 1.23–1.26.
  27. http://www.photonengr.com/software/ .

2011

K. H. Chao, C. D. Liao, B. J. Yang, and J. C. Tsai, “Fabrication and characterization of a micro tunable cat’s eye retro-reflector,” Opt. Commun. 284, 5221–5224 (2011).
[CrossRef]

2009

K. Chen, H. Yang, L. Sun, and G. Jin, “Astigmatism analysis by matrix methods in white cells,” Proc. SPIE 7156, 71560G (2009).

P.-D. Lin and C.-C. Hsueh, “6×6 matrix formalism of optical elements for modeling and analyzing 3D optical systems,” Appl. Phys. B 97, 135–143 (2009).
[CrossRef]

2006

I. Moreno, C. Ferreira, and M. M. Sanchez-Lopez, “Ray matrix analysis of anamorphic fractional Fourier systems,” J. Opt. A Pure Appl. Opt. 8, 427–435 (2006).
[CrossRef]

P. D. Lin and C. K. Sung, “Matrix-based paraxial skew ray-tracing in 3D systems with non-coplanar optical axis,” Optik 117, 329–340 (2006).
[CrossRef]

J. C. Juarez, A. Dwivedi, A. Mammons, S. D. Jones, V. Weerackody, and R. A. Nichols, “Free-space optical communications for next-generation military networks,” IEEE Commun. Mag. 44, 46–51 (2006).
[CrossRef]

2005

W. S. Rabinovich, R. Mahon, H. R. Burris, G. C. Gilbreath, P. G. Goetz, C. I. Moore, M. F. Stell, M. J. Vilcheck, J. L. Witkowsky, L. Swingen, M. R. Suite, E. Oh, and J. Koplow, “Free space optical communications link at 1550 nm using multiple-quantum-well modulating retroreflectors in a marine environment,” Opt. Eng. 44, 56001–56012 (2005).
[CrossRef]

2004

V. A. Handerek and L. C. Laycock, “Feasibility of retroreflective free-space optical communication using retroreflectors with very wide field of view,” Proc. SPIE 5641, 1–9(2004).
[CrossRef]

2002

X. Zhu, V. S. Hsu, and J. M. Kahn, “Optical modeling of MEMS corner cube retroreflectors with misalignment and nonflatness,” IEEE J. Sel. Top. Quantum Electron. 8, 26–32 (2002).
[CrossRef]

2001

G. C. Gilbreath, W. S. Rabinovich, T. J. Meehan, M. J. Vilcheck, R. Mahon, R. Burris, M. Ferraro, I. Solkolsky, J. A. Vasquez, C. S. Bovais, K. Cochrell, K. C. Goins, R. Barbehenn, D. S. Katzer, K. Ikossi-Anastasiou, and M. J. Montes, “Large-aperture multiple quantum well modulating retroreflector for free-space optical data transfer on unmanned aerial vehicles,” Opt. Eng. 40, 1348–1356 (2001).
[CrossRef]

1994

W. F. Harris, “Paraxial ray tracing through noncoaxial astigmatic optical systems, and a 5×5 augmented system matrix,” Opt. Vis. Sci. 71, 282–285 (1994).

1992

B. Lu, S. Xu, Y. Hu, and B. Cai, “Matrix representation of three-dimensional astigmatic resonators,” Opt. Quantum Electron. 24, 619–630 (1992).
[CrossRef]

A. Nussbaum, “Modernizing the teaching of advanced geometric optics,” Proc. SPIE 1603, 389–400(1992).
[CrossRef]

1983

1980

H. H. Arsenault, “A matrix representation for non-symmetrical optical systems,” J. Opt. 11, 87–91 (1980).
[CrossRef]

1975

Arsenault, H. H.

Barbehenn, R.

G. C. Gilbreath, W. S. Rabinovich, T. J. Meehan, M. J. Vilcheck, R. Mahon, R. Burris, M. Ferraro, I. Solkolsky, J. A. Vasquez, C. S. Bovais, K. Cochrell, K. C. Goins, R. Barbehenn, D. S. Katzer, K. Ikossi-Anastasiou, and M. J. Montes, “Large-aperture multiple quantum well modulating retroreflector for free-space optical data transfer on unmanned aerial vehicles,” Opt. Eng. 40, 1348–1356 (2001).
[CrossRef]

Bovais, C. S.

G. C. Gilbreath, W. S. Rabinovich, T. J. Meehan, M. J. Vilcheck, R. Mahon, R. Burris, M. Ferraro, I. Solkolsky, J. A. Vasquez, C. S. Bovais, K. Cochrell, K. C. Goins, R. Barbehenn, D. S. Katzer, K. Ikossi-Anastasiou, and M. J. Montes, “Large-aperture multiple quantum well modulating retroreflector for free-space optical data transfer on unmanned aerial vehicles,” Opt. Eng. 40, 1348–1356 (2001).
[CrossRef]

Burch, J. M.

A. Gerrard and J. M. Burch, “Matrix methods in paraxial optics,” in Introduction to Matrix Methods in Optics (Dover, 1994), pp. 24–75.

Burris, H. R.

W. S. Rabinovich, R. Mahon, H. R. Burris, G. C. Gilbreath, P. G. Goetz, C. I. Moore, M. F. Stell, M. J. Vilcheck, J. L. Witkowsky, L. Swingen, M. R. Suite, E. Oh, and J. Koplow, “Free space optical communications link at 1550 nm using multiple-quantum-well modulating retroreflectors in a marine environment,” Opt. Eng. 44, 56001–56012 (2005).
[CrossRef]

P. G. Goetz, W. S. Rabinovich, R. Mahon, J. L. Murphy, M. S. Ferraro, W. R. Smith, B. B. Xu, H. R. Burris, C. I. Moore, and W. W. Schultz, “Modulating retro-reflector lasercom systems at the Naval Research Laboratory,” in 2010 Military Communications Conference (MILCOM, 2010), pp. 1601–1606.

P. G. Goetz, W. S. Rabinovich, R. Mahon, L. Swingen, G. C. Gilbreath, J. L. Murphy, H. R. Burris, and M. F. Stell, “Practical considerations of retroreflector choice in modulating retroreflector systems,” in 2005 Digest of the LEOS Summer Topical Meetings (2005), pp. 49–50.

Burris, R.

G. C. Gilbreath, W. S. Rabinovich, T. J. Meehan, M. J. Vilcheck, R. Mahon, R. Burris, M. Ferraro, I. Solkolsky, J. A. Vasquez, C. S. Bovais, K. Cochrell, K. C. Goins, R. Barbehenn, D. S. Katzer, K. Ikossi-Anastasiou, and M. J. Montes, “Large-aperture multiple quantum well modulating retroreflector for free-space optical data transfer on unmanned aerial vehicles,” Opt. Eng. 40, 1348–1356 (2001).
[CrossRef]

Cai, B.

B. Lu, S. Xu, Y. Hu, and B. Cai, “Matrix representation of three-dimensional astigmatic resonators,” Opt. Quantum Electron. 24, 619–630 (1992).
[CrossRef]

Chao, K. H.

K. H. Chao, C. D. Liao, B. J. Yang, and J. C. Tsai, “Fabrication and characterization of a micro tunable cat’s eye retro-reflector,” Opt. Commun. 284, 5221–5224 (2011).
[CrossRef]

K. H. Chao, C. D. Liao, and J. C. Tsai, “Array of cat’s eye retro-reflectors with modulability for an optical identification system,” in 2010 IEEE International Conference on Optical MEMS & Nanophotonics (2010), pp. 7–8.

Chen, K.

K. Chen, H. Yang, L. Sun, and G. Jin, “Astigmatism analysis by matrix methods in white cells,” Proc. SPIE 7156, 71560G (2009).

Cochrell, K.

G. C. Gilbreath, W. S. Rabinovich, T. J. Meehan, M. J. Vilcheck, R. Mahon, R. Burris, M. Ferraro, I. Solkolsky, J. A. Vasquez, C. S. Bovais, K. Cochrell, K. C. Goins, R. Barbehenn, D. S. Katzer, K. Ikossi-Anastasiou, and M. J. Montes, “Large-aperture multiple quantum well modulating retroreflector for free-space optical data transfer on unmanned aerial vehicles,” Opt. Eng. 40, 1348–1356 (2001).
[CrossRef]

Dwivedi, A.

J. C. Juarez, A. Dwivedi, A. Mammons, S. D. Jones, V. Weerackody, and R. A. Nichols, “Free-space optical communications for next-generation military networks,” IEEE Commun. Mag. 44, 46–51 (2006).
[CrossRef]

Ferraro, M.

G. C. Gilbreath, W. S. Rabinovich, T. J. Meehan, M. J. Vilcheck, R. Mahon, R. Burris, M. Ferraro, I. Solkolsky, J. A. Vasquez, C. S. Bovais, K. Cochrell, K. C. Goins, R. Barbehenn, D. S. Katzer, K. Ikossi-Anastasiou, and M. J. Montes, “Large-aperture multiple quantum well modulating retroreflector for free-space optical data transfer on unmanned aerial vehicles,” Opt. Eng. 40, 1348–1356 (2001).
[CrossRef]

Ferraro, M. S.

P. G. Goetz, W. S. Rabinovich, R. Mahon, J. L. Murphy, M. S. Ferraro, W. R. Smith, B. B. Xu, H. R. Burris, C. I. Moore, and W. W. Schultz, “Modulating retro-reflector lasercom systems at the Naval Research Laboratory,” in 2010 Military Communications Conference (MILCOM, 2010), pp. 1601–1606.

Ferreira, C.

I. Moreno, C. Ferreira, and M. M. Sanchez-Lopez, “Ray matrix analysis of anamorphic fractional Fourier systems,” J. Opt. A Pure Appl. Opt. 8, 427–435 (2006).
[CrossRef]

Gerrard, A.

A. Gerrard and J. M. Burch, “Matrix methods in paraxial optics,” in Introduction to Matrix Methods in Optics (Dover, 1994), pp. 24–75.

Gilbreath, G. C.

W. S. Rabinovich, R. Mahon, H. R. Burris, G. C. Gilbreath, P. G. Goetz, C. I. Moore, M. F. Stell, M. J. Vilcheck, J. L. Witkowsky, L. Swingen, M. R. Suite, E. Oh, and J. Koplow, “Free space optical communications link at 1550 nm using multiple-quantum-well modulating retroreflectors in a marine environment,” Opt. Eng. 44, 56001–56012 (2005).
[CrossRef]

G. C. Gilbreath, W. S. Rabinovich, T. J. Meehan, M. J. Vilcheck, R. Mahon, R. Burris, M. Ferraro, I. Solkolsky, J. A. Vasquez, C. S. Bovais, K. Cochrell, K. C. Goins, R. Barbehenn, D. S. Katzer, K. Ikossi-Anastasiou, and M. J. Montes, “Large-aperture multiple quantum well modulating retroreflector for free-space optical data transfer on unmanned aerial vehicles,” Opt. Eng. 40, 1348–1356 (2001).
[CrossRef]

P. G. Goetz, W. S. Rabinovich, R. Mahon, L. Swingen, G. C. Gilbreath, J. L. Murphy, H. R. Burris, and M. F. Stell, “Practical considerations of retroreflector choice in modulating retroreflector systems,” in 2005 Digest of the LEOS Summer Topical Meetings (2005), pp. 49–50.

Goetz, P. G.

W. S. Rabinovich, R. Mahon, H. R. Burris, G. C. Gilbreath, P. G. Goetz, C. I. Moore, M. F. Stell, M. J. Vilcheck, J. L. Witkowsky, L. Swingen, M. R. Suite, E. Oh, and J. Koplow, “Free space optical communications link at 1550 nm using multiple-quantum-well modulating retroreflectors in a marine environment,” Opt. Eng. 44, 56001–56012 (2005).
[CrossRef]

P. G. Goetz, W. S. Rabinovich, R. Mahon, J. L. Murphy, M. S. Ferraro, W. R. Smith, B. B. Xu, H. R. Burris, C. I. Moore, and W. W. Schultz, “Modulating retro-reflector lasercom systems at the Naval Research Laboratory,” in 2010 Military Communications Conference (MILCOM, 2010), pp. 1601–1606.

P. G. Goetz, W. S. Rabinovich, R. Mahon, L. Swingen, G. C. Gilbreath, J. L. Murphy, H. R. Burris, and M. F. Stell, “Practical considerations of retroreflector choice in modulating retroreflector systems,” in 2005 Digest of the LEOS Summer Topical Meetings (2005), pp. 49–50.

Goins, K. C.

G. C. Gilbreath, W. S. Rabinovich, T. J. Meehan, M. J. Vilcheck, R. Mahon, R. Burris, M. Ferraro, I. Solkolsky, J. A. Vasquez, C. S. Bovais, K. Cochrell, K. C. Goins, R. Barbehenn, D. S. Katzer, K. Ikossi-Anastasiou, and M. J. Montes, “Large-aperture multiple quantum well modulating retroreflector for free-space optical data transfer on unmanned aerial vehicles,” Opt. Eng. 40, 1348–1356 (2001).
[CrossRef]

Goodman, D. S.

D. S. Goodman, “1.8 refraction and reflection at interfaces between homogeneous media,” in Handbook of Optics, Volume 1: Geometrical and Physical Optics, Polarized Light, Components and Instruments, 3rd ed. (McGraw-Hill, 2010), pp. 1.23–1.26.

Handerek, V. A.

V. A. Handerek and L. C. Laycock, “Feasibility of retroreflective free-space optical communication using retroreflectors with very wide field of view,” Proc. SPIE 5641, 1–9(2004).
[CrossRef]

Harris, W. F.

W. F. Harris, “Paraxial ray tracing through noncoaxial astigmatic optical systems, and a 5×5 augmented system matrix,” Opt. Vis. Sci. 71, 282–285 (1994).

Haus, H. A.

H. A. Haus, “5.5 The ABCD matrix in ray optics,” in Waves and Fields in Optoelectronics (Prentice-Hall, 1984), pp. 132–136.

Hayes, D. J.

D. J. Hayes, “Cat’s eye retro-reflector array coding device and method of fabrication,” U.S. patent 7,152,984 B1 (26December2006).

Hecht, E.

E. Hecht, “6.2 Analytical ray tracing,” in Optics, 4th ed.(Addison-Wesley, 2002), pp. 246–253.

Hsu, V. S.

X. Zhu, V. S. Hsu, and J. M. Kahn, “Optical modeling of MEMS corner cube retroreflectors with misalignment and nonflatness,” IEEE J. Sel. Top. Quantum Electron. 8, 26–32 (2002).
[CrossRef]

Hsueh, C.-C.

P.-D. Lin and C.-C. Hsueh, “6×6 matrix formalism of optical elements for modeling and analyzing 3D optical systems,” Appl. Phys. B 97, 135–143 (2009).
[CrossRef]

Hu, Y.

B. Lu, S. Xu, Y. Hu, and B. Cai, “Matrix representation of three-dimensional astigmatic resonators,” Opt. Quantum Electron. 24, 619–630 (1992).
[CrossRef]

Ikossi-Anastasiou, K.

G. C. Gilbreath, W. S. Rabinovich, T. J. Meehan, M. J. Vilcheck, R. Mahon, R. Burris, M. Ferraro, I. Solkolsky, J. A. Vasquez, C. S. Bovais, K. Cochrell, K. C. Goins, R. Barbehenn, D. S. Katzer, K. Ikossi-Anastasiou, and M. J. Montes, “Large-aperture multiple quantum well modulating retroreflector for free-space optical data transfer on unmanned aerial vehicles,” Opt. Eng. 40, 1348–1356 (2001).
[CrossRef]

Jin, G.

K. Chen, H. Yang, L. Sun, and G. Jin, “Astigmatism analysis by matrix methods in white cells,” Proc. SPIE 7156, 71560G (2009).

Jones, S. D.

J. C. Juarez, A. Dwivedi, A. Mammons, S. D. Jones, V. Weerackody, and R. A. Nichols, “Free-space optical communications for next-generation military networks,” IEEE Commun. Mag. 44, 46–51 (2006).
[CrossRef]

Juarez, J. C.

J. C. Juarez, A. Dwivedi, A. Mammons, S. D. Jones, V. Weerackody, and R. A. Nichols, “Free-space optical communications for next-generation military networks,” IEEE Commun. Mag. 44, 46–51 (2006).
[CrossRef]

Kahn, J. M.

X. Zhu, V. S. Hsu, and J. M. Kahn, “Optical modeling of MEMS corner cube retroreflectors with misalignment and nonflatness,” IEEE J. Sel. Top. Quantum Electron. 8, 26–32 (2002).
[CrossRef]

Katzer, D. S.

G. C. Gilbreath, W. S. Rabinovich, T. J. Meehan, M. J. Vilcheck, R. Mahon, R. Burris, M. Ferraro, I. Solkolsky, J. A. Vasquez, C. S. Bovais, K. Cochrell, K. C. Goins, R. Barbehenn, D. S. Katzer, K. Ikossi-Anastasiou, and M. J. Montes, “Large-aperture multiple quantum well modulating retroreflector for free-space optical data transfer on unmanned aerial vehicles,” Opt. Eng. 40, 1348–1356 (2001).
[CrossRef]

Koplow, J.

W. S. Rabinovich, R. Mahon, H. R. Burris, G. C. Gilbreath, P. G. Goetz, C. I. Moore, M. F. Stell, M. J. Vilcheck, J. L. Witkowsky, L. Swingen, M. R. Suite, E. Oh, and J. Koplow, “Free space optical communications link at 1550 nm using multiple-quantum-well modulating retroreflectors in a marine environment,” Opt. Eng. 44, 56001–56012 (2005).
[CrossRef]

Laycock, L. C.

V. A. Handerek and L. C. Laycock, “Feasibility of retroreflective free-space optical communication using retroreflectors with very wide field of view,” Proc. SPIE 5641, 1–9(2004).
[CrossRef]

Liao, C. D.

K. H. Chao, C. D. Liao, B. J. Yang, and J. C. Tsai, “Fabrication and characterization of a micro tunable cat’s eye retro-reflector,” Opt. Commun. 284, 5221–5224 (2011).
[CrossRef]

K. H. Chao, C. D. Liao, and J. C. Tsai, “Array of cat’s eye retro-reflectors with modulability for an optical identification system,” in 2010 IEEE International Conference on Optical MEMS & Nanophotonics (2010), pp. 7–8.

Lin, P. D.

P. D. Lin and C. K. Sung, “Matrix-based paraxial skew ray-tracing in 3D systems with non-coplanar optical axis,” Optik 117, 329–340 (2006).
[CrossRef]

Lin, P.-D.

P.-D. Lin and C.-C. Hsueh, “6×6 matrix formalism of optical elements for modeling and analyzing 3D optical systems,” Appl. Phys. B 97, 135–143 (2009).
[CrossRef]

Lu, B.

B. Lu, S. Xu, Y. Hu, and B. Cai, “Matrix representation of three-dimensional astigmatic resonators,” Opt. Quantum Electron. 24, 619–630 (1992).
[CrossRef]

Macukow, B.

Mahon, R.

W. S. Rabinovich, R. Mahon, H. R. Burris, G. C. Gilbreath, P. G. Goetz, C. I. Moore, M. F. Stell, M. J. Vilcheck, J. L. Witkowsky, L. Swingen, M. R. Suite, E. Oh, and J. Koplow, “Free space optical communications link at 1550 nm using multiple-quantum-well modulating retroreflectors in a marine environment,” Opt. Eng. 44, 56001–56012 (2005).
[CrossRef]

G. C. Gilbreath, W. S. Rabinovich, T. J. Meehan, M. J. Vilcheck, R. Mahon, R. Burris, M. Ferraro, I. Solkolsky, J. A. Vasquez, C. S. Bovais, K. Cochrell, K. C. Goins, R. Barbehenn, D. S. Katzer, K. Ikossi-Anastasiou, and M. J. Montes, “Large-aperture multiple quantum well modulating retroreflector for free-space optical data transfer on unmanned aerial vehicles,” Opt. Eng. 40, 1348–1356 (2001).
[CrossRef]

P. G. Goetz, W. S. Rabinovich, R. Mahon, J. L. Murphy, M. S. Ferraro, W. R. Smith, B. B. Xu, H. R. Burris, C. I. Moore, and W. W. Schultz, “Modulating retro-reflector lasercom systems at the Naval Research Laboratory,” in 2010 Military Communications Conference (MILCOM, 2010), pp. 1601–1606.

P. G. Goetz, W. S. Rabinovich, R. Mahon, L. Swingen, G. C. Gilbreath, J. L. Murphy, H. R. Burris, and M. F. Stell, “Practical considerations of retroreflector choice in modulating retroreflector systems,” in 2005 Digest of the LEOS Summer Topical Meetings (2005), pp. 49–50.

Mammons, A.

J. C. Juarez, A. Dwivedi, A. Mammons, S. D. Jones, V. Weerackody, and R. A. Nichols, “Free-space optical communications for next-generation military networks,” IEEE Commun. Mag. 44, 46–51 (2006).
[CrossRef]

Meehan, T. J.

G. C. Gilbreath, W. S. Rabinovich, T. J. Meehan, M. J. Vilcheck, R. Mahon, R. Burris, M. Ferraro, I. Solkolsky, J. A. Vasquez, C. S. Bovais, K. Cochrell, K. C. Goins, R. Barbehenn, D. S. Katzer, K. Ikossi-Anastasiou, and M. J. Montes, “Large-aperture multiple quantum well modulating retroreflector for free-space optical data transfer on unmanned aerial vehicles,” Opt. Eng. 40, 1348–1356 (2001).
[CrossRef]

Montes, M. J.

G. C. Gilbreath, W. S. Rabinovich, T. J. Meehan, M. J. Vilcheck, R. Mahon, R. Burris, M. Ferraro, I. Solkolsky, J. A. Vasquez, C. S. Bovais, K. Cochrell, K. C. Goins, R. Barbehenn, D. S. Katzer, K. Ikossi-Anastasiou, and M. J. Montes, “Large-aperture multiple quantum well modulating retroreflector for free-space optical data transfer on unmanned aerial vehicles,” Opt. Eng. 40, 1348–1356 (2001).
[CrossRef]

Moore, C. I.

W. S. Rabinovich, R. Mahon, H. R. Burris, G. C. Gilbreath, P. G. Goetz, C. I. Moore, M. F. Stell, M. J. Vilcheck, J. L. Witkowsky, L. Swingen, M. R. Suite, E. Oh, and J. Koplow, “Free space optical communications link at 1550 nm using multiple-quantum-well modulating retroreflectors in a marine environment,” Opt. Eng. 44, 56001–56012 (2005).
[CrossRef]

P. G. Goetz, W. S. Rabinovich, R. Mahon, J. L. Murphy, M. S. Ferraro, W. R. Smith, B. B. Xu, H. R. Burris, C. I. Moore, and W. W. Schultz, “Modulating retro-reflector lasercom systems at the Naval Research Laboratory,” in 2010 Military Communications Conference (MILCOM, 2010), pp. 1601–1606.

Moreno, I.

I. Moreno, C. Ferreira, and M. M. Sanchez-Lopez, “Ray matrix analysis of anamorphic fractional Fourier systems,” J. Opt. A Pure Appl. Opt. 8, 427–435 (2006).
[CrossRef]

Murphy, J. L.

P. G. Goetz, W. S. Rabinovich, R. Mahon, L. Swingen, G. C. Gilbreath, J. L. Murphy, H. R. Burris, and M. F. Stell, “Practical considerations of retroreflector choice in modulating retroreflector systems,” in 2005 Digest of the LEOS Summer Topical Meetings (2005), pp. 49–50.

P. G. Goetz, W. S. Rabinovich, R. Mahon, J. L. Murphy, M. S. Ferraro, W. R. Smith, B. B. Xu, H. R. Burris, C. I. Moore, and W. W. Schultz, “Modulating retro-reflector lasercom systems at the Naval Research Laboratory,” in 2010 Military Communications Conference (MILCOM, 2010), pp. 1601–1606.

Nichols, R. A.

J. C. Juarez, A. Dwivedi, A. Mammons, S. D. Jones, V. Weerackody, and R. A. Nichols, “Free-space optical communications for next-generation military networks,” IEEE Commun. Mag. 44, 46–51 (2006).
[CrossRef]

Nussbaum, A.

A. Nussbaum, “Modernizing the teaching of advanced geometric optics,” Proc. SPIE 1603, 389–400(1992).
[CrossRef]

Oh, E.

W. S. Rabinovich, R. Mahon, H. R. Burris, G. C. Gilbreath, P. G. Goetz, C. I. Moore, M. F. Stell, M. J. Vilcheck, J. L. Witkowsky, L. Swingen, M. R. Suite, E. Oh, and J. Koplow, “Free space optical communications link at 1550 nm using multiple-quantum-well modulating retroreflectors in a marine environment,” Opt. Eng. 44, 56001–56012 (2005).
[CrossRef]

Rabinovich, W. S.

W. S. Rabinovich, R. Mahon, H. R. Burris, G. C. Gilbreath, P. G. Goetz, C. I. Moore, M. F. Stell, M. J. Vilcheck, J. L. Witkowsky, L. Swingen, M. R. Suite, E. Oh, and J. Koplow, “Free space optical communications link at 1550 nm using multiple-quantum-well modulating retroreflectors in a marine environment,” Opt. Eng. 44, 56001–56012 (2005).
[CrossRef]

G. C. Gilbreath, W. S. Rabinovich, T. J. Meehan, M. J. Vilcheck, R. Mahon, R. Burris, M. Ferraro, I. Solkolsky, J. A. Vasquez, C. S. Bovais, K. Cochrell, K. C. Goins, R. Barbehenn, D. S. Katzer, K. Ikossi-Anastasiou, and M. J. Montes, “Large-aperture multiple quantum well modulating retroreflector for free-space optical data transfer on unmanned aerial vehicles,” Opt. Eng. 40, 1348–1356 (2001).
[CrossRef]

P. G. Goetz, W. S. Rabinovich, R. Mahon, J. L. Murphy, M. S. Ferraro, W. R. Smith, B. B. Xu, H. R. Burris, C. I. Moore, and W. W. Schultz, “Modulating retro-reflector lasercom systems at the Naval Research Laboratory,” in 2010 Military Communications Conference (MILCOM, 2010), pp. 1601–1606.

P. G. Goetz, W. S. Rabinovich, R. Mahon, L. Swingen, G. C. Gilbreath, J. L. Murphy, H. R. Burris, and M. F. Stell, “Practical considerations of retroreflector choice in modulating retroreflector systems,” in 2005 Digest of the LEOS Summer Topical Meetings (2005), pp. 49–50.

Sanchez-Lopez, M. M.

I. Moreno, C. Ferreira, and M. M. Sanchez-Lopez, “Ray matrix analysis of anamorphic fractional Fourier systems,” J. Opt. A Pure Appl. Opt. 8, 427–435 (2006).
[CrossRef]

Schultz, W. W.

P. G. Goetz, W. S. Rabinovich, R. Mahon, J. L. Murphy, M. S. Ferraro, W. R. Smith, B. B. Xu, H. R. Burris, C. I. Moore, and W. W. Schultz, “Modulating retro-reflector lasercom systems at the Naval Research Laboratory,” in 2010 Military Communications Conference (MILCOM, 2010), pp. 1601–1606.

Smith, W. R.

P. G. Goetz, W. S. Rabinovich, R. Mahon, J. L. Murphy, M. S. Ferraro, W. R. Smith, B. B. Xu, H. R. Burris, C. I. Moore, and W. W. Schultz, “Modulating retro-reflector lasercom systems at the Naval Research Laboratory,” in 2010 Military Communications Conference (MILCOM, 2010), pp. 1601–1606.

Snyder, J. J.

Solkolsky, I.

G. C. Gilbreath, W. S. Rabinovich, T. J. Meehan, M. J. Vilcheck, R. Mahon, R. Burris, M. Ferraro, I. Solkolsky, J. A. Vasquez, C. S. Bovais, K. Cochrell, K. C. Goins, R. Barbehenn, D. S. Katzer, K. Ikossi-Anastasiou, and M. J. Montes, “Large-aperture multiple quantum well modulating retroreflector for free-space optical data transfer on unmanned aerial vehicles,” Opt. Eng. 40, 1348–1356 (2001).
[CrossRef]

Stell, M. F.

W. S. Rabinovich, R. Mahon, H. R. Burris, G. C. Gilbreath, P. G. Goetz, C. I. Moore, M. F. Stell, M. J. Vilcheck, J. L. Witkowsky, L. Swingen, M. R. Suite, E. Oh, and J. Koplow, “Free space optical communications link at 1550 nm using multiple-quantum-well modulating retroreflectors in a marine environment,” Opt. Eng. 44, 56001–56012 (2005).
[CrossRef]

P. G. Goetz, W. S. Rabinovich, R. Mahon, L. Swingen, G. C. Gilbreath, J. L. Murphy, H. R. Burris, and M. F. Stell, “Practical considerations of retroreflector choice in modulating retroreflector systems,” in 2005 Digest of the LEOS Summer Topical Meetings (2005), pp. 49–50.

Suite, M. R.

W. S. Rabinovich, R. Mahon, H. R. Burris, G. C. Gilbreath, P. G. Goetz, C. I. Moore, M. F. Stell, M. J. Vilcheck, J. L. Witkowsky, L. Swingen, M. R. Suite, E. Oh, and J. Koplow, “Free space optical communications link at 1550 nm using multiple-quantum-well modulating retroreflectors in a marine environment,” Opt. Eng. 44, 56001–56012 (2005).
[CrossRef]

Sun, L.

K. Chen, H. Yang, L. Sun, and G. Jin, “Astigmatism analysis by matrix methods in white cells,” Proc. SPIE 7156, 71560G (2009).

Sung, C. K.

P. D. Lin and C. K. Sung, “Matrix-based paraxial skew ray-tracing in 3D systems with non-coplanar optical axis,” Optik 117, 329–340 (2006).
[CrossRef]

Swingen, L.

W. S. Rabinovich, R. Mahon, H. R. Burris, G. C. Gilbreath, P. G. Goetz, C. I. Moore, M. F. Stell, M. J. Vilcheck, J. L. Witkowsky, L. Swingen, M. R. Suite, E. Oh, and J. Koplow, “Free space optical communications link at 1550 nm using multiple-quantum-well modulating retroreflectors in a marine environment,” Opt. Eng. 44, 56001–56012 (2005).
[CrossRef]

P. G. Goetz, W. S. Rabinovich, R. Mahon, L. Swingen, G. C. Gilbreath, J. L. Murphy, H. R. Burris, and M. F. Stell, “Practical considerations of retroreflector choice in modulating retroreflector systems,” in 2005 Digest of the LEOS Summer Topical Meetings (2005), pp. 49–50.

Tsai, J. C.

K. H. Chao, C. D. Liao, B. J. Yang, and J. C. Tsai, “Fabrication and characterization of a micro tunable cat’s eye retro-reflector,” Opt. Commun. 284, 5221–5224 (2011).
[CrossRef]

K. H. Chao, C. D. Liao, and J. C. Tsai, “Array of cat’s eye retro-reflectors with modulability for an optical identification system,” in 2010 IEEE International Conference on Optical MEMS & Nanophotonics (2010), pp. 7–8.

Vasquez, J. A.

G. C. Gilbreath, W. S. Rabinovich, T. J. Meehan, M. J. Vilcheck, R. Mahon, R. Burris, M. Ferraro, I. Solkolsky, J. A. Vasquez, C. S. Bovais, K. Cochrell, K. C. Goins, R. Barbehenn, D. S. Katzer, K. Ikossi-Anastasiou, and M. J. Montes, “Large-aperture multiple quantum well modulating retroreflector for free-space optical data transfer on unmanned aerial vehicles,” Opt. Eng. 40, 1348–1356 (2001).
[CrossRef]

Vilcheck, M. J.

W. S. Rabinovich, R. Mahon, H. R. Burris, G. C. Gilbreath, P. G. Goetz, C. I. Moore, M. F. Stell, M. J. Vilcheck, J. L. Witkowsky, L. Swingen, M. R. Suite, E. Oh, and J. Koplow, “Free space optical communications link at 1550 nm using multiple-quantum-well modulating retroreflectors in a marine environment,” Opt. Eng. 44, 56001–56012 (2005).
[CrossRef]

G. C. Gilbreath, W. S. Rabinovich, T. J. Meehan, M. J. Vilcheck, R. Mahon, R. Burris, M. Ferraro, I. Solkolsky, J. A. Vasquez, C. S. Bovais, K. Cochrell, K. C. Goins, R. Barbehenn, D. S. Katzer, K. Ikossi-Anastasiou, and M. J. Montes, “Large-aperture multiple quantum well modulating retroreflector for free-space optical data transfer on unmanned aerial vehicles,” Opt. Eng. 40, 1348–1356 (2001).
[CrossRef]

Weerackody, V.

J. C. Juarez, A. Dwivedi, A. Mammons, S. D. Jones, V. Weerackody, and R. A. Nichols, “Free-space optical communications for next-generation military networks,” IEEE Commun. Mag. 44, 46–51 (2006).
[CrossRef]

Witkowsky, J. L.

W. S. Rabinovich, R. Mahon, H. R. Burris, G. C. Gilbreath, P. G. Goetz, C. I. Moore, M. F. Stell, M. J. Vilcheck, J. L. Witkowsky, L. Swingen, M. R. Suite, E. Oh, and J. Koplow, “Free space optical communications link at 1550 nm using multiple-quantum-well modulating retroreflectors in a marine environment,” Opt. Eng. 44, 56001–56012 (2005).
[CrossRef]

Xu, B. B.

P. G. Goetz, W. S. Rabinovich, R. Mahon, J. L. Murphy, M. S. Ferraro, W. R. Smith, B. B. Xu, H. R. Burris, C. I. Moore, and W. W. Schultz, “Modulating retro-reflector lasercom systems at the Naval Research Laboratory,” in 2010 Military Communications Conference (MILCOM, 2010), pp. 1601–1606.

Xu, S.

B. Lu, S. Xu, Y. Hu, and B. Cai, “Matrix representation of three-dimensional astigmatic resonators,” Opt. Quantum Electron. 24, 619–630 (1992).
[CrossRef]

Yang, B. J.

K. H. Chao, C. D. Liao, B. J. Yang, and J. C. Tsai, “Fabrication and characterization of a micro tunable cat’s eye retro-reflector,” Opt. Commun. 284, 5221–5224 (2011).
[CrossRef]

Yang, H.

K. Chen, H. Yang, L. Sun, and G. Jin, “Astigmatism analysis by matrix methods in white cells,” Proc. SPIE 7156, 71560G (2009).

Zhu, X.

X. Zhu, V. S. Hsu, and J. M. Kahn, “Optical modeling of MEMS corner cube retroreflectors with misalignment and nonflatness,” IEEE J. Sel. Top. Quantum Electron. 8, 26–32 (2002).
[CrossRef]

Appl. Opt.

Appl. Phys. B

P.-D. Lin and C.-C. Hsueh, “6×6 matrix formalism of optical elements for modeling and analyzing 3D optical systems,” Appl. Phys. B 97, 135–143 (2009).
[CrossRef]

IEEE Commun. Mag.

J. C. Juarez, A. Dwivedi, A. Mammons, S. D. Jones, V. Weerackody, and R. A. Nichols, “Free-space optical communications for next-generation military networks,” IEEE Commun. Mag. 44, 46–51 (2006).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron.

X. Zhu, V. S. Hsu, and J. M. Kahn, “Optical modeling of MEMS corner cube retroreflectors with misalignment and nonflatness,” IEEE J. Sel. Top. Quantum Electron. 8, 26–32 (2002).
[CrossRef]

J. Opt.

H. H. Arsenault, “A matrix representation for non-symmetrical optical systems,” J. Opt. 11, 87–91 (1980).
[CrossRef]

J. Opt. A Pure Appl. Opt.

I. Moreno, C. Ferreira, and M. M. Sanchez-Lopez, “Ray matrix analysis of anamorphic fractional Fourier systems,” J. Opt. A Pure Appl. Opt. 8, 427–435 (2006).
[CrossRef]

J. Opt. Soc. Am.

Opt. Commun.

K. H. Chao, C. D. Liao, B. J. Yang, and J. C. Tsai, “Fabrication and characterization of a micro tunable cat’s eye retro-reflector,” Opt. Commun. 284, 5221–5224 (2011).
[CrossRef]

Opt. Eng.

W. S. Rabinovich, R. Mahon, H. R. Burris, G. C. Gilbreath, P. G. Goetz, C. I. Moore, M. F. Stell, M. J. Vilcheck, J. L. Witkowsky, L. Swingen, M. R. Suite, E. Oh, and J. Koplow, “Free space optical communications link at 1550 nm using multiple-quantum-well modulating retroreflectors in a marine environment,” Opt. Eng. 44, 56001–56012 (2005).
[CrossRef]

G. C. Gilbreath, W. S. Rabinovich, T. J. Meehan, M. J. Vilcheck, R. Mahon, R. Burris, M. Ferraro, I. Solkolsky, J. A. Vasquez, C. S. Bovais, K. Cochrell, K. C. Goins, R. Barbehenn, D. S. Katzer, K. Ikossi-Anastasiou, and M. J. Montes, “Large-aperture multiple quantum well modulating retroreflector for free-space optical data transfer on unmanned aerial vehicles,” Opt. Eng. 40, 1348–1356 (2001).
[CrossRef]

Opt. Quantum Electron.

B. Lu, S. Xu, Y. Hu, and B. Cai, “Matrix representation of three-dimensional astigmatic resonators,” Opt. Quantum Electron. 24, 619–630 (1992).
[CrossRef]

Opt. Vis. Sci.

W. F. Harris, “Paraxial ray tracing through noncoaxial astigmatic optical systems, and a 5×5 augmented system matrix,” Opt. Vis. Sci. 71, 282–285 (1994).

Optik

P. D. Lin and C. K. Sung, “Matrix-based paraxial skew ray-tracing in 3D systems with non-coplanar optical axis,” Optik 117, 329–340 (2006).
[CrossRef]

Proc. SPIE

K. Chen, H. Yang, L. Sun, and G. Jin, “Astigmatism analysis by matrix methods in white cells,” Proc. SPIE 7156, 71560G (2009).

A. Nussbaum, “Modernizing the teaching of advanced geometric optics,” Proc. SPIE 1603, 389–400(1992).
[CrossRef]

V. A. Handerek and L. C. Laycock, “Feasibility of retroreflective free-space optical communication using retroreflectors with very wide field of view,” Proc. SPIE 5641, 1–9(2004).
[CrossRef]

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P. G. Goetz, W. S. Rabinovich, R. Mahon, L. Swingen, G. C. Gilbreath, J. L. Murphy, H. R. Burris, and M. F. Stell, “Practical considerations of retroreflector choice in modulating retroreflector systems,” in 2005 Digest of the LEOS Summer Topical Meetings (2005), pp. 49–50.

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A. Gerrard and J. M. Burch, “Matrix methods in paraxial optics,” in Introduction to Matrix Methods in Optics (Dover, 1994), pp. 24–75.

H. A. Haus, “5.5 The ABCD matrix in ray optics,” in Waves and Fields in Optoelectronics (Prentice-Hall, 1984), pp. 132–136.

P. G. Goetz, W. S. Rabinovich, R. Mahon, J. L. Murphy, M. S. Ferraro, W. R. Smith, B. B. Xu, H. R. Burris, C. I. Moore, and W. W. Schultz, “Modulating retro-reflector lasercom systems at the Naval Research Laboratory,” in 2010 Military Communications Conference (MILCOM, 2010), pp. 1601–1606.

D. J. Hayes, “Cat’s eye retro-reflector array coding device and method of fabrication,” U.S. patent 7,152,984 B1 (26December2006).

K. H. Chao, C. D. Liao, and J. C. Tsai, “Array of cat’s eye retro-reflectors with modulability for an optical identification system,” in 2010 IEEE International Conference on Optical MEMS & Nanophotonics (2010), pp. 7–8.

http://www.reflexite.com/refl/americas/en/traffic-control .

D. S. Goodman, “1.8 refraction and reflection at interfaces between homogeneous media,” in Handbook of Optics, Volume 1: Geometrical and Physical Optics, Polarized Light, Components and Instruments, 3rd ed. (McGraw-Hill, 2010), pp. 1.23–1.26.

http://www.photonengr.com/software/ .

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Figures (12)

Fig. 1.
Fig. 1.

Paraxial ray hitting a spherical surface in a 3D space.

Fig. 2.
Fig. 2.

Paraxial ray reflecting off a spherical surface in a 3D space—cross-sectional view. The relation among s, k, and k is depicted.

Fig. 3.
Fig. 3.

Cat’s eye structure presented in [8].

Fig. 4.
Fig. 4.

Schematic drawing—a square grid of optical rays is incident on the cat’s eye.

Fig. 5.
Fig. 5.

Comparison of the efficiency-incident angle relations among different simulation methods, when there is (a) no lateral offset/misalignment; (b) an x offset/misalignment of 0.02mm, or (c) a y offset/misalignment of 0.02mm.

Fig. 6.
Fig. 6.

Retroreflection efficiency versus incident angle for (a) different x offsets (misalignments) and (b) different y offsets. Please refer to the insets of Fig. 5 for the lateral shift’s direction of a certain misalignment amount.

Fig. 7.
Fig. 7.

Simulation results of efficiency versus incident angle for an ideal cat’s eye calculated by our matrix-based 3D technique and FRED, respectively.

Fig. 8.
Fig. 8.

Efficiency versus incident angle when a lateral (a) x or (b) y offset (misalignment) is introduced, making the cat’s eye not ideal anymore.

Fig. 9.
Fig. 9.

Effciency versus incident angle when (a) R2<R (1.268 mm) and (b) R2>R (1.268 mm).

Fig. 10.
Fig. 10.

Efficiency, divergence angle, and angular deviatoin versus the air gap dfill for three different R2 values, (a) R, (b) 0.8R, and (c) . The incident angle is 5°.

Fig. 11.
Fig. 11.

Efficiency, divergence angle, and angular deviation versus angle of incidence for several cat’s eye structures with different (R2, dfill): (a) (0.8R, 0.8d); (b) (0.8R, d); (c) (0.8R, 1.33d); (d) (R, 0.8d); (e) (R, d); (f) (R, 1.33d); (g) (1.33R, 0.8d); (h) (1.33R, d); (i) (1.33R, 1.33d), and (j) (0.635R, 0.785d). The characteristics of the ideal cat’s eye in Subsection 4.B and the cat’s eye in Subsection 4.A are shown again in (e) and (j), respectively, for comparison.

Fig. 12.
Fig. 12.

Characteristics of nonideal cat’s eyes with specially selected values for (R2, dfill), which are (a) (0.85R, 0.8d) and (b) (1.2R, 1.27d), respectively. The efficiency-incident angle relations under different x misalignments are also shown.

Equations (28)

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

S:x2+y2+z2=R2.
s=(xR,yR,zR).
s=(xR,yR,1),
k=(α,β,1),
k=(α,β,1).
n(s×k)=n(s×k).
n(yRβ,α+xR,βxαyR)=n(yRβ,α+xR,βxαyR)
α=(nnn)xR+nnα,
β=(nnn)yR+nnβ.
(xα)=(10nnR·nnn)·(xα),
(yβ)=(10nnR·nnn)·(yβ)
s=(xR,yR,1)
k=(α,β,1),
k=(α,β,1).
ks=k+s.
α=α2xR,
β=β2yR
x=x,
y=y.
(xα)=(102R1)·(xα),
(yβ)=(102R1)·(yβ).
(1d01)
Lens=(10nairnlens,1nairR11),Mirror=(102nlens,2nairR21),Gap=(1nairdfillnfill01),Substrate1=(1naird1nsubstrate,101),Substrate2=(1naird2nsubstrate,201).
Lens·Substrate1·Gap·Substrate2·Mirror·Substrate2·Gap·Substrate1·Lens.
dfill=R1·nfillnlens,1naird1·nfillnsubstrate,1d2·nfillnsubstrate,2,
R2=R1·nlens,2nlens,1nair.
(1001).
θ=±D(nlens,1nair)2nairR1,

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