Abstract

In this paper, a 4-stage optical time domain multiplexing (OTDM) multiplexer based on plated graded index (GRIN) lens is proposed and experimentally demonstrated. A 10Gbit/s return-to-zero (RZ) signal is upgraded to 160Gbit/s. The time-domain accuracy of the multiplexer is evaluated by analyzing the multiplexed 160Gbit/s signal. Experimental results validate the stability of the optical multiplexing behavior and the polarization insensitivity of the proposed multiplexer. The results also show the advantages of our OTDM multiplexer such as the flexible output signal speeds at the output of different stages, the low insertion loss and the temperature and wavelength stability.

© 2008 Optical Society of America

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References

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  1. A. Chen, C.-T. Lea, and A. K.-S. Wong, “A new optical TDM Ring Architecture,” IEEE Trans. Commun. 55, 2134–2141 (2007).
    [Crossref]
  2. K. M. Feng, C. Y. Wu, D. H. Hsueh, C. S. Ku, C. P. Chang, H. Y. Lin, J. Cheng, and J. Chen, “Demonstration of an optical FIFO multiplexer,” in Optical Fiber Communication Conference and Exposition and National Fiber Optic Engineers Conference, Technical Digest (CD) (Optical Society of America, 2008), paper OMN5. http://www.opticsinfobase.org/abstract.cfm?URI=OFC-2008-OMN5
  3. B. Zhang, L. Zhang, L. S. Yan, I. Fazal, J. Y. Yang, and A. E. Willner, “Continuously-tunable, bit-rate variable OTDM using broadband SBS slow-light delay line,” Opt. Express 15, 8317–8322 (2007).
    [Crossref] [PubMed]
  4. N. Deng, C. K. Chan, and L. K. Chen, “A Hybrid OTDM Scheme with enhanced demultiplexing performance,” IEEE Photon. Technol. Lett. 19, 1454–1456 (2007).
    [Crossref]
  5. H. Murai, M. Kagawa, H. Tsuji, and K. Fujii, “EA-modulator-based optical time division multiplexing/demultiplexing techniques for 160-Gb/s optical signal transmission,” IEEE J. Quantum Electron. 13, 70–78 (2007).
    [Crossref]
  6. T. Ohara, H. Takara, I. Shake, T. Yamada, M. Ishii, I. Ogawa, M. Okamoto, and S. Kawanishi, “Highly stable 160-Gb/s OTDM technologies based on integrated MUX/DEMUX and drift-free PLL-type clock recovery,” IEEE J. Quantum Electron. 13, 40–48 (2007).
    [Crossref]
  7. K. Morishita and K. Takashina, “Polarization properties of fused fiber couplers and polarizing beamsplitters,” J. Lightwave Technol. 9, 1503–1507 (1991).
    [Crossref]
  8. Z. Jing, L. Cai, and T. Li, “Experimental demonstration on 4×10Gb/s optical time domain multiplexing signal,” J. Photon Technol. 8, 18–21 (2005).
  9. Z. Jing, L. Cai, and T. Li, “Experimental investigation in polarization sensitivity of a novel 40-Gbit/s OTDM system,” Proc. SPIE 6019, 1–7 (2005).
  10. L. Cai, Z. Jing, and T. Li, “Function of polarization scrambler in 4×10Gb/s optical time division multiplexing signal generating system,” J. Opt. Commun. Technol. 29, 32–33 (2005).
  11. S. G. Farwell, M. N. Zervas, and R. I. Laming, “2×2 fused fiber null couplers with asymmetric waist cross sections for polarization independent (<0.01dB) switching,” J. Lightwave Technol. 16, 1671–1680 (2006).
    [Crossref]
  12. C. R. Doerr, L. L. Buhl, and W. Lin, “Simple method for mitigation of polarization crosstalk in silica planar lightwave circuit directional couplers,” IEEE Photon. Technol. Lett. 18, 1816–1818 (2006).
    [Crossref]
  13. Andreas Umbach, “4-stage OTDM multiplexer,” http://www.u2t.de/fileadmin/redakteure/Products/Datasheets/DS_OMUX-4_5v0.pdf.
  14. C. Kehua, “Propagation analysis of light rays in combinative optical system of SaFoc lens,” Acta Photonica Sin. 21, 323–329 (1992).
  15. M. Born and E. Wolf, Principles of Optics (Pergamon, 1986).

2007 (5)

N. Deng, C. K. Chan, and L. K. Chen, “A Hybrid OTDM Scheme with enhanced demultiplexing performance,” IEEE Photon. Technol. Lett. 19, 1454–1456 (2007).
[Crossref]

H. Murai, M. Kagawa, H. Tsuji, and K. Fujii, “EA-modulator-based optical time division multiplexing/demultiplexing techniques for 160-Gb/s optical signal transmission,” IEEE J. Quantum Electron. 13, 70–78 (2007).
[Crossref]

T. Ohara, H. Takara, I. Shake, T. Yamada, M. Ishii, I. Ogawa, M. Okamoto, and S. Kawanishi, “Highly stable 160-Gb/s OTDM technologies based on integrated MUX/DEMUX and drift-free PLL-type clock recovery,” IEEE J. Quantum Electron. 13, 40–48 (2007).
[Crossref]

A. Chen, C.-T. Lea, and A. K.-S. Wong, “A new optical TDM Ring Architecture,” IEEE Trans. Commun. 55, 2134–2141 (2007).
[Crossref]

B. Zhang, L. Zhang, L. S. Yan, I. Fazal, J. Y. Yang, and A. E. Willner, “Continuously-tunable, bit-rate variable OTDM using broadband SBS slow-light delay line,” Opt. Express 15, 8317–8322 (2007).
[Crossref] [PubMed]

2006 (2)

S. G. Farwell, M. N. Zervas, and R. I. Laming, “2×2 fused fiber null couplers with asymmetric waist cross sections for polarization independent (<0.01dB) switching,” J. Lightwave Technol. 16, 1671–1680 (2006).
[Crossref]

C. R. Doerr, L. L. Buhl, and W. Lin, “Simple method for mitigation of polarization crosstalk in silica planar lightwave circuit directional couplers,” IEEE Photon. Technol. Lett. 18, 1816–1818 (2006).
[Crossref]

2005 (3)

Z. Jing, L. Cai, and T. Li, “Experimental demonstration on 4×10Gb/s optical time domain multiplexing signal,” J. Photon Technol. 8, 18–21 (2005).

Z. Jing, L. Cai, and T. Li, “Experimental investigation in polarization sensitivity of a novel 40-Gbit/s OTDM system,” Proc. SPIE 6019, 1–7 (2005).

L. Cai, Z. Jing, and T. Li, “Function of polarization scrambler in 4×10Gb/s optical time division multiplexing signal generating system,” J. Opt. Commun. Technol. 29, 32–33 (2005).

1992 (1)

C. Kehua, “Propagation analysis of light rays in combinative optical system of SaFoc lens,” Acta Photonica Sin. 21, 323–329 (1992).

1991 (1)

K. Morishita and K. Takashina, “Polarization properties of fused fiber couplers and polarizing beamsplitters,” J. Lightwave Technol. 9, 1503–1507 (1991).
[Crossref]

Born, M.

M. Born and E. Wolf, Principles of Optics (Pergamon, 1986).

Buhl, L. L.

C. R. Doerr, L. L. Buhl, and W. Lin, “Simple method for mitigation of polarization crosstalk in silica planar lightwave circuit directional couplers,” IEEE Photon. Technol. Lett. 18, 1816–1818 (2006).
[Crossref]

Cai, L.

Z. Jing, L. Cai, and T. Li, “Experimental demonstration on 4×10Gb/s optical time domain multiplexing signal,” J. Photon Technol. 8, 18–21 (2005).

Z. Jing, L. Cai, and T. Li, “Experimental investigation in polarization sensitivity of a novel 40-Gbit/s OTDM system,” Proc. SPIE 6019, 1–7 (2005).

L. Cai, Z. Jing, and T. Li, “Function of polarization scrambler in 4×10Gb/s optical time division multiplexing signal generating system,” J. Opt. Commun. Technol. 29, 32–33 (2005).

Chan, C. K.

N. Deng, C. K. Chan, and L. K. Chen, “A Hybrid OTDM Scheme with enhanced demultiplexing performance,” IEEE Photon. Technol. Lett. 19, 1454–1456 (2007).
[Crossref]

Chang, C. P.

K. M. Feng, C. Y. Wu, D. H. Hsueh, C. S. Ku, C. P. Chang, H. Y. Lin, J. Cheng, and J. Chen, “Demonstration of an optical FIFO multiplexer,” in Optical Fiber Communication Conference and Exposition and National Fiber Optic Engineers Conference, Technical Digest (CD) (Optical Society of America, 2008), paper OMN5. http://www.opticsinfobase.org/abstract.cfm?URI=OFC-2008-OMN5

Chen, A.

A. Chen, C.-T. Lea, and A. K.-S. Wong, “A new optical TDM Ring Architecture,” IEEE Trans. Commun. 55, 2134–2141 (2007).
[Crossref]

Chen, J.

K. M. Feng, C. Y. Wu, D. H. Hsueh, C. S. Ku, C. P. Chang, H. Y. Lin, J. Cheng, and J. Chen, “Demonstration of an optical FIFO multiplexer,” in Optical Fiber Communication Conference and Exposition and National Fiber Optic Engineers Conference, Technical Digest (CD) (Optical Society of America, 2008), paper OMN5. http://www.opticsinfobase.org/abstract.cfm?URI=OFC-2008-OMN5

Chen, L. K.

N. Deng, C. K. Chan, and L. K. Chen, “A Hybrid OTDM Scheme with enhanced demultiplexing performance,” IEEE Photon. Technol. Lett. 19, 1454–1456 (2007).
[Crossref]

Cheng, J.

K. M. Feng, C. Y. Wu, D. H. Hsueh, C. S. Ku, C. P. Chang, H. Y. Lin, J. Cheng, and J. Chen, “Demonstration of an optical FIFO multiplexer,” in Optical Fiber Communication Conference and Exposition and National Fiber Optic Engineers Conference, Technical Digest (CD) (Optical Society of America, 2008), paper OMN5. http://www.opticsinfobase.org/abstract.cfm?URI=OFC-2008-OMN5

Deng, N.

N. Deng, C. K. Chan, and L. K. Chen, “A Hybrid OTDM Scheme with enhanced demultiplexing performance,” IEEE Photon. Technol. Lett. 19, 1454–1456 (2007).
[Crossref]

Doerr, C. R.

C. R. Doerr, L. L. Buhl, and W. Lin, “Simple method for mitigation of polarization crosstalk in silica planar lightwave circuit directional couplers,” IEEE Photon. Technol. Lett. 18, 1816–1818 (2006).
[Crossref]

Farwell, S. G.

Fazal, I.

Feng, K. M.

K. M. Feng, C. Y. Wu, D. H. Hsueh, C. S. Ku, C. P. Chang, H. Y. Lin, J. Cheng, and J. Chen, “Demonstration of an optical FIFO multiplexer,” in Optical Fiber Communication Conference and Exposition and National Fiber Optic Engineers Conference, Technical Digest (CD) (Optical Society of America, 2008), paper OMN5. http://www.opticsinfobase.org/abstract.cfm?URI=OFC-2008-OMN5

Fujii, K.

H. Murai, M. Kagawa, H. Tsuji, and K. Fujii, “EA-modulator-based optical time division multiplexing/demultiplexing techniques for 160-Gb/s optical signal transmission,” IEEE J. Quantum Electron. 13, 70–78 (2007).
[Crossref]

Hsueh, D. H.

K. M. Feng, C. Y. Wu, D. H. Hsueh, C. S. Ku, C. P. Chang, H. Y. Lin, J. Cheng, and J. Chen, “Demonstration of an optical FIFO multiplexer,” in Optical Fiber Communication Conference and Exposition and National Fiber Optic Engineers Conference, Technical Digest (CD) (Optical Society of America, 2008), paper OMN5. http://www.opticsinfobase.org/abstract.cfm?URI=OFC-2008-OMN5

Ishii, M.

T. Ohara, H. Takara, I. Shake, T. Yamada, M. Ishii, I. Ogawa, M. Okamoto, and S. Kawanishi, “Highly stable 160-Gb/s OTDM technologies based on integrated MUX/DEMUX and drift-free PLL-type clock recovery,” IEEE J. Quantum Electron. 13, 40–48 (2007).
[Crossref]

Jing, Z.

Z. Jing, L. Cai, and T. Li, “Experimental demonstration on 4×10Gb/s optical time domain multiplexing signal,” J. Photon Technol. 8, 18–21 (2005).

Z. Jing, L. Cai, and T. Li, “Experimental investigation in polarization sensitivity of a novel 40-Gbit/s OTDM system,” Proc. SPIE 6019, 1–7 (2005).

L. Cai, Z. Jing, and T. Li, “Function of polarization scrambler in 4×10Gb/s optical time division multiplexing signal generating system,” J. Opt. Commun. Technol. 29, 32–33 (2005).

Kagawa, M.

H. Murai, M. Kagawa, H. Tsuji, and K. Fujii, “EA-modulator-based optical time division multiplexing/demultiplexing techniques for 160-Gb/s optical signal transmission,” IEEE J. Quantum Electron. 13, 70–78 (2007).
[Crossref]

Kawanishi, S.

T. Ohara, H. Takara, I. Shake, T. Yamada, M. Ishii, I. Ogawa, M. Okamoto, and S. Kawanishi, “Highly stable 160-Gb/s OTDM technologies based on integrated MUX/DEMUX and drift-free PLL-type clock recovery,” IEEE J. Quantum Electron. 13, 40–48 (2007).
[Crossref]

Kehua, C.

C. Kehua, “Propagation analysis of light rays in combinative optical system of SaFoc lens,” Acta Photonica Sin. 21, 323–329 (1992).

Ku, C. S.

K. M. Feng, C. Y. Wu, D. H. Hsueh, C. S. Ku, C. P. Chang, H. Y. Lin, J. Cheng, and J. Chen, “Demonstration of an optical FIFO multiplexer,” in Optical Fiber Communication Conference and Exposition and National Fiber Optic Engineers Conference, Technical Digest (CD) (Optical Society of America, 2008), paper OMN5. http://www.opticsinfobase.org/abstract.cfm?URI=OFC-2008-OMN5

Laming, R. I.

Lea, C.-T.

A. Chen, C.-T. Lea, and A. K.-S. Wong, “A new optical TDM Ring Architecture,” IEEE Trans. Commun. 55, 2134–2141 (2007).
[Crossref]

Li, T.

L. Cai, Z. Jing, and T. Li, “Function of polarization scrambler in 4×10Gb/s optical time division multiplexing signal generating system,” J. Opt. Commun. Technol. 29, 32–33 (2005).

Z. Jing, L. Cai, and T. Li, “Experimental investigation in polarization sensitivity of a novel 40-Gbit/s OTDM system,” Proc. SPIE 6019, 1–7 (2005).

Z. Jing, L. Cai, and T. Li, “Experimental demonstration on 4×10Gb/s optical time domain multiplexing signal,” J. Photon Technol. 8, 18–21 (2005).

Lin, H. Y.

K. M. Feng, C. Y. Wu, D. H. Hsueh, C. S. Ku, C. P. Chang, H. Y. Lin, J. Cheng, and J. Chen, “Demonstration of an optical FIFO multiplexer,” in Optical Fiber Communication Conference and Exposition and National Fiber Optic Engineers Conference, Technical Digest (CD) (Optical Society of America, 2008), paper OMN5. http://www.opticsinfobase.org/abstract.cfm?URI=OFC-2008-OMN5

Lin, W.

C. R. Doerr, L. L. Buhl, and W. Lin, “Simple method for mitigation of polarization crosstalk in silica planar lightwave circuit directional couplers,” IEEE Photon. Technol. Lett. 18, 1816–1818 (2006).
[Crossref]

Morishita, K.

K. Morishita and K. Takashina, “Polarization properties of fused fiber couplers and polarizing beamsplitters,” J. Lightwave Technol. 9, 1503–1507 (1991).
[Crossref]

Murai, H.

H. Murai, M. Kagawa, H. Tsuji, and K. Fujii, “EA-modulator-based optical time division multiplexing/demultiplexing techniques for 160-Gb/s optical signal transmission,” IEEE J. Quantum Electron. 13, 70–78 (2007).
[Crossref]

Ogawa, I.

T. Ohara, H. Takara, I. Shake, T. Yamada, M. Ishii, I. Ogawa, M. Okamoto, and S. Kawanishi, “Highly stable 160-Gb/s OTDM technologies based on integrated MUX/DEMUX and drift-free PLL-type clock recovery,” IEEE J. Quantum Electron. 13, 40–48 (2007).
[Crossref]

Ohara, T.

T. Ohara, H. Takara, I. Shake, T. Yamada, M. Ishii, I. Ogawa, M. Okamoto, and S. Kawanishi, “Highly stable 160-Gb/s OTDM technologies based on integrated MUX/DEMUX and drift-free PLL-type clock recovery,” IEEE J. Quantum Electron. 13, 40–48 (2007).
[Crossref]

Okamoto, M.

T. Ohara, H. Takara, I. Shake, T. Yamada, M. Ishii, I. Ogawa, M. Okamoto, and S. Kawanishi, “Highly stable 160-Gb/s OTDM technologies based on integrated MUX/DEMUX and drift-free PLL-type clock recovery,” IEEE J. Quantum Electron. 13, 40–48 (2007).
[Crossref]

Shake, I.

T. Ohara, H. Takara, I. Shake, T. Yamada, M. Ishii, I. Ogawa, M. Okamoto, and S. Kawanishi, “Highly stable 160-Gb/s OTDM technologies based on integrated MUX/DEMUX and drift-free PLL-type clock recovery,” IEEE J. Quantum Electron. 13, 40–48 (2007).
[Crossref]

Takara, H.

T. Ohara, H. Takara, I. Shake, T. Yamada, M. Ishii, I. Ogawa, M. Okamoto, and S. Kawanishi, “Highly stable 160-Gb/s OTDM technologies based on integrated MUX/DEMUX and drift-free PLL-type clock recovery,” IEEE J. Quantum Electron. 13, 40–48 (2007).
[Crossref]

Takashina, K.

K. Morishita and K. Takashina, “Polarization properties of fused fiber couplers and polarizing beamsplitters,” J. Lightwave Technol. 9, 1503–1507 (1991).
[Crossref]

Tsuji, H.

H. Murai, M. Kagawa, H. Tsuji, and K. Fujii, “EA-modulator-based optical time division multiplexing/demultiplexing techniques for 160-Gb/s optical signal transmission,” IEEE J. Quantum Electron. 13, 70–78 (2007).
[Crossref]

Umbach, Andreas

Andreas Umbach, “4-stage OTDM multiplexer,” http://www.u2t.de/fileadmin/redakteure/Products/Datasheets/DS_OMUX-4_5v0.pdf.

Willner, A. E.

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Pergamon, 1986).

Wong, A. K.-S.

A. Chen, C.-T. Lea, and A. K.-S. Wong, “A new optical TDM Ring Architecture,” IEEE Trans. Commun. 55, 2134–2141 (2007).
[Crossref]

Wu, C. Y.

K. M. Feng, C. Y. Wu, D. H. Hsueh, C. S. Ku, C. P. Chang, H. Y. Lin, J. Cheng, and J. Chen, “Demonstration of an optical FIFO multiplexer,” in Optical Fiber Communication Conference and Exposition and National Fiber Optic Engineers Conference, Technical Digest (CD) (Optical Society of America, 2008), paper OMN5. http://www.opticsinfobase.org/abstract.cfm?URI=OFC-2008-OMN5

Yamada, T.

T. Ohara, H. Takara, I. Shake, T. Yamada, M. Ishii, I. Ogawa, M. Okamoto, and S. Kawanishi, “Highly stable 160-Gb/s OTDM technologies based on integrated MUX/DEMUX and drift-free PLL-type clock recovery,” IEEE J. Quantum Electron. 13, 40–48 (2007).
[Crossref]

Yan, L. S.

Yang, J. Y.

Zervas, M. N.

Zhang, B.

Zhang, L.

Acta Photonica Sin. (1)

C. Kehua, “Propagation analysis of light rays in combinative optical system of SaFoc lens,” Acta Photonica Sin. 21, 323–329 (1992).

IEEE J. Quantum Electron. (2)

H. Murai, M. Kagawa, H. Tsuji, and K. Fujii, “EA-modulator-based optical time division multiplexing/demultiplexing techniques for 160-Gb/s optical signal transmission,” IEEE J. Quantum Electron. 13, 70–78 (2007).
[Crossref]

T. Ohara, H. Takara, I. Shake, T. Yamada, M. Ishii, I. Ogawa, M. Okamoto, and S. Kawanishi, “Highly stable 160-Gb/s OTDM technologies based on integrated MUX/DEMUX and drift-free PLL-type clock recovery,” IEEE J. Quantum Electron. 13, 40–48 (2007).
[Crossref]

IEEE Photon. Technol. Lett. (2)

N. Deng, C. K. Chan, and L. K. Chen, “A Hybrid OTDM Scheme with enhanced demultiplexing performance,” IEEE Photon. Technol. Lett. 19, 1454–1456 (2007).
[Crossref]

C. R. Doerr, L. L. Buhl, and W. Lin, “Simple method for mitigation of polarization crosstalk in silica planar lightwave circuit directional couplers,” IEEE Photon. Technol. Lett. 18, 1816–1818 (2006).
[Crossref]

IEEE Trans. Commun. (1)

A. Chen, C.-T. Lea, and A. K.-S. Wong, “A new optical TDM Ring Architecture,” IEEE Trans. Commun. 55, 2134–2141 (2007).
[Crossref]

J. Lightwave Technol. (2)

K. Morishita and K. Takashina, “Polarization properties of fused fiber couplers and polarizing beamsplitters,” J. Lightwave Technol. 9, 1503–1507 (1991).
[Crossref]

S. G. Farwell, M. N. Zervas, and R. I. Laming, “2×2 fused fiber null couplers with asymmetric waist cross sections for polarization independent (<0.01dB) switching,” J. Lightwave Technol. 16, 1671–1680 (2006).
[Crossref]

J. Opt. Commun. Technol. (1)

L. Cai, Z. Jing, and T. Li, “Function of polarization scrambler in 4×10Gb/s optical time division multiplexing signal generating system,” J. Opt. Commun. Technol. 29, 32–33 (2005).

J. Photon Technol. (1)

Z. Jing, L. Cai, and T. Li, “Experimental demonstration on 4×10Gb/s optical time domain multiplexing signal,” J. Photon Technol. 8, 18–21 (2005).

Opt. Express (1)

Proc. SPIE (1)

Z. Jing, L. Cai, and T. Li, “Experimental investigation in polarization sensitivity of a novel 40-Gbit/s OTDM system,” Proc. SPIE 6019, 1–7 (2005).

Other (3)

K. M. Feng, C. Y. Wu, D. H. Hsueh, C. S. Ku, C. P. Chang, H. Y. Lin, J. Cheng, and J. Chen, “Demonstration of an optical FIFO multiplexer,” in Optical Fiber Communication Conference and Exposition and National Fiber Optic Engineers Conference, Technical Digest (CD) (Optical Society of America, 2008), paper OMN5. http://www.opticsinfobase.org/abstract.cfm?URI=OFC-2008-OMN5

Andreas Umbach, “4-stage OTDM multiplexer,” http://www.u2t.de/fileadmin/redakteure/Products/Datasheets/DS_OMUX-4_5v0.pdf.

M. Born and E. Wolf, Principles of Optics (Pergamon, 1986).

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Figures (8)

Fig. 1.
Fig. 1.

Structure of the plated GRIN lens and the inner optical paths.

Fig. 2.
Fig. 2.

One stage of OTDM multiplexer.

Fig. 3.
Fig. 3.

Experimental 160Gbit/s OTDM signal generating system.

Fig. 4.
Fig. 4.

Back-to-back results on the oscilloscope: (a) Waveform of 80GHz pulse train (12.5ps/div), (b) Eye diagram of 80Gbit/s RZ signal (10ps/div).

Fig. 5.
Fig. 5.

Back-to-back results on the autocorrelator: (a) 160GHz pulse train (The autocorrelation window is 200ps, so there are 32 continuous pulses in the window), (b) 160Gbit/s RZ signal.

Fig. 6.
Fig. 6.

Time-domain accuracy at the fourth stage of the multiplexer (5ps/div).

Fig. 7.
Fig. 7.

Output of the conventional multiplexer without (a) and with (b) polarization scrambler (10ps/div).

Fig. 8.
Fig. 8.

Back-to-back 80Gbit/s RZ signal without (a) and with (b) polarization scrambler (5ps/div).

Equations (4)

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

n = n 0 ( 1 1 2 A r 2 )
[ x 2 θ 2 ] = [ cos ( A 2 L 2 ) n 01 n 02 A 2 sin ( A 2 L 2 ) n 02 A 2 sin ( A 2 L 2 ) n 01 cos ( A 2 L 2 ) ] [ cos ( A 1 L 1 ) 1 n 01 A 1 sin ( A 1 L 1 ) A 1 sin ( A 1 L 1 ) 1 n 01 cos ( A 1 L 1 ) ] [ x 0 θ 0 ] ,
[ x 2 θ 2 ] = [ 0 n 01 n 02 A 2 n 02 A 2 0 ] [ 0 1 n 01 A 1 A 1 0 ] [ x 0 θ 0 ]
{ x 2 = n 01 n 02 x 0 = x 0 θ 2 = n 02 n 01 θ 0 = θ 0

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