Abstract

We investigate a scheme for a 60GHz radio-over-fiber transmission system by using a dual-electrical Mach–Zehnder modulator (De-MZM) and an optical interleaver. Such a system includes the generation of 60GHz millimeter-wave (mm-wave) signals and the delivery of a remote local oscillator (LO). In the scheme, two lasers with frequency deviation are used as sources, and a De-MZM is biased at the minimum transmission point to realize optical carrier suppression modulation. After fiber transmission, an optical interleaver is used to separate subcarriers. If one laser is modulated with baseband data and the other is not, 60GHz mm-wave signals with and without modulating data can be generated by using photodiodes, one of which can be used in mm-wave communication and the other can be used as a remote LO. In this work, we theoretically analyze the bit-error-rate performance and power degradation due to the lasers’ phase noise. Our scheme uses only a commercially used De-MZM and optical interleaver; therefore, no sensitive equipment and complicated structure is needed, which guarantees the system has steady performance and a cost-effective architecture.

© 2009 Optical Society of America

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References

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  1. ERO, Detailed Spectrum Investigation—First Phase: 3400 MHz to 105 GHz, Copenhagen, Denmark, pp. 27–58, March 1993.
  2. T. Ihara, K. Fujimura, “Research and development trends of millimeter-wave short-range application systems,” IEICE Trans. Commun., vol. E79-B, pp. 1741–1753, Dec. 1996.
  3. Y. Takimoto, “Recent activities on millimeter wave indoor LAN system development in Japan,” in Proc. 1995 IEEE MTT-S Int. Microwave Symp. Dig., May 1995, pp. 405–408.
  4. T. Kuri, K. Kitayama, “Optical heterodyne detection technique for densely multiplexed millimeter-waveband radio-on-fiber system,” J. Lightwave Technol., vol. 21, pp. 3167–3179, 2003.
    [CrossRef]
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    [CrossRef]
  6. G. H. Smith, D. Novak, Z. Ahmed, “Overcoming chromatic-dispersion effects in fiber-wireless systems incorporating external modulators,” IEEE Trans. Microwave Theory Tech., vol. 45, pp. 1410–1415, 1997.
    [CrossRef]
  7. H. Toda, “Demultiplexing using an arrayed-waveguide grating for frequency-interleaved DWDM millimeter-wave radio-on-fiber systems,” J. Lightwave Technol., vol. 21, pp. 1735–1741, Aug. 2003.
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  10. H. Yang, J. Zeng, Y. Zheng, H. D. Jung, B. Huiszoon, J. H. C. van Zantvoort, E. Tangdiongga, A. M. J. Koonen, “Evaluation of effects of MZM nonlinearity on QAM and OFDM signals in RoF transmitter,” in Int. Topical Meeting on Microwave Photonics, 2008 , pp. 90–93.
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    [CrossRef]
  12. K. Y. Lau, J. Park, Ultra-high Frequency Linear Fiber Optic Systems, Springer, 2008, pp. 155–166.
  13. C. Yu, Y. Wang, Z. Pan, T. Luo, S. Kumar, B. Zhang, A. E. Willner, “Carrier-suppressed 160 GHz pulse-train generation using a 40 GHz phase modulator with polarization-maintaining fiber,” Opt. Lett., vol. 34, pp. 1657–1659, 2009.
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    [CrossRef]
  16. H.-C. Ji, H. Kim, Y. C. Chung, “Full-duplex radio-over-fiber system using phase-modulated downlink and intensity-modulated uplink,” IEEE Photon. Technol. Lett., vol. 21, pp. 9–11, 2009.
    [CrossRef]
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2009 (3)

2006 (1)

Y. Jianjun, J. Zhensheng, L. Yi, Y. Su, C. Gee-Kung, W. Ting, “Optical millimeter-wave generation or up-conversion using external modulators,” IEEE Photon. Technol. Lett., vol. 18, pp. 265–267, 2006.
[CrossRef]

2003 (2)

1999 (2)

A. Stohr, “Full-duplex 60 GHz fiber-optic transmission,” Electron. Lett., vol. 35, 1653–1655, 1999.
[CrossRef]

L. N. Langley, M. D. Elkin, C. Edge, M. J. Wale, X. Gliese, X. Huang, A. J. Seeds, “Packaged semiconductor laser optical phase-locked loop (OPLL) for photonic generation, processing and transmission of microwave signals,” IEEE Trans. Microwave Theory Tech., vol. 47, pp. 1257–1264, 1999.
[CrossRef]

1998 (1)

U. Gliese, T. N. Nielsen, S. Nrskov, K. E. Stubkjaer, “Multifunction fiber-optic microwave links based on remote heterodyne detection,” IEEE Trans. Microwave Theory Tech., vol. 46, pp. 458–468, 1998.
[CrossRef]

1997 (2)

D. Wake, D. Johansson, D. Moodie, “Passive pico-cell: a new concept in wireless network infrastructure,” Electron. Lett., vol. 33, pp. 404–406, 1997.
[CrossRef]

G. H. Smith, D. Novak, Z. Ahmed, “Overcoming chromatic-dispersion effects in fiber-wireless systems incorporating external modulators,” IEEE Trans. Microwave Theory Tech., vol. 45, pp. 1410–1415, 1997.
[CrossRef]

1996 (1)

T. Ihara, K. Fujimura, “Research and development trends of millimeter-wave short-range application systems,” IEICE Trans. Commun., vol. E79-B, pp. 1741–1753, Dec. 1996.

1992 (1)

H. Ogawa, D. Polifko, S. Banda, “Millimeter-wave fiber optics systems for personal radio communications,” IEEE Trans. Microwave Theory Tech., vol. 40, pp. 2285–2293, Dec. 1992.
[CrossRef]

Ahmed, Z.

G. H. Smith, D. Novak, Z. Ahmed, “Overcoming chromatic-dispersion effects in fiber-wireless systems incorporating external modulators,” IEEE Trans. Microwave Theory Tech., vol. 45, pp. 1410–1415, 1997.
[CrossRef]

Banda, S.

H. Ogawa, D. Polifko, S. Banda, “Millimeter-wave fiber optics systems for personal radio communications,” IEEE Trans. Microwave Theory Tech., vol. 40, pp. 2285–2293, Dec. 1992.
[CrossRef]

Chen, J.

Chi, S.

Chung, Y. C.

H.-C. Ji, H. Kim, Y. C. Chung, “Full-duplex radio-over-fiber system using phase-modulated downlink and intensity-modulated uplink,” IEEE Photon. Technol. Lett., vol. 21, pp. 9–11, 2009.
[CrossRef]

Dai, S.-P.

Edge, C.

L. N. Langley, M. D. Elkin, C. Edge, M. J. Wale, X. Gliese, X. Huang, A. J. Seeds, “Packaged semiconductor laser optical phase-locked loop (OPLL) for photonic generation, processing and transmission of microwave signals,” IEEE Trans. Microwave Theory Tech., vol. 47, pp. 1257–1264, 1999.
[CrossRef]

Elkin, M. D.

L. N. Langley, M. D. Elkin, C. Edge, M. J. Wale, X. Gliese, X. Huang, A. J. Seeds, “Packaged semiconductor laser optical phase-locked loop (OPLL) for photonic generation, processing and transmission of microwave signals,” IEEE Trans. Microwave Theory Tech., vol. 47, pp. 1257–1264, 1999.
[CrossRef]

Fujimura, K.

T. Ihara, K. Fujimura, “Research and development trends of millimeter-wave short-range application systems,” IEICE Trans. Commun., vol. E79-B, pp. 1741–1753, Dec. 1996.

Gee-Kung, C.

Y. Jianjun, J. Zhensheng, L. Yi, Y. Su, C. Gee-Kung, W. Ting, “Optical millimeter-wave generation or up-conversion using external modulators,” IEEE Photon. Technol. Lett., vol. 18, pp. 265–267, 2006.
[CrossRef]

Gliese, U.

U. Gliese, T. N. Nielsen, S. Nrskov, K. E. Stubkjaer, “Multifunction fiber-optic microwave links based on remote heterodyne detection,” IEEE Trans. Microwave Theory Tech., vol. 46, pp. 458–468, 1998.
[CrossRef]

Gliese, X.

L. N. Langley, M. D. Elkin, C. Edge, M. J. Wale, X. Gliese, X. Huang, A. J. Seeds, “Packaged semiconductor laser optical phase-locked loop (OPLL) for photonic generation, processing and transmission of microwave signals,” IEEE Trans. Microwave Theory Tech., vol. 47, pp. 1257–1264, 1999.
[CrossRef]

Huang, X.

L. N. Langley, M. D. Elkin, C. Edge, M. J. Wale, X. Gliese, X. Huang, A. J. Seeds, “Packaged semiconductor laser optical phase-locked loop (OPLL) for photonic generation, processing and transmission of microwave signals,” IEEE Trans. Microwave Theory Tech., vol. 47, pp. 1257–1264, 1999.
[CrossRef]

Huiszoon, B.

H. Yang, J. Zeng, Y. Zheng, H. D. Jung, B. Huiszoon, J. H. C. van Zantvoort, E. Tangdiongga, A. M. J. Koonen, “Evaluation of effects of MZM nonlinearity on QAM and OFDM signals in RoF transmitter,” in Int. Topical Meeting on Microwave Photonics, 2008 , pp. 90–93.

Ihara, T.

T. Ihara, K. Fujimura, “Research and development trends of millimeter-wave short-range application systems,” IEICE Trans. Commun., vol. E79-B, pp. 1741–1753, Dec. 1996.

Ji, H.-C.

H.-C. Ji, H. Kim, Y. C. Chung, “Full-duplex radio-over-fiber system using phase-modulated downlink and intensity-modulated uplink,” IEEE Photon. Technol. Lett., vol. 21, pp. 9–11, 2009.
[CrossRef]

Jiang, W.

Jianjun, Y.

Y. Jianjun, J. Zhensheng, L. Yi, Y. Su, C. Gee-Kung, W. Ting, “Optical millimeter-wave generation or up-conversion using external modulators,” IEEE Photon. Technol. Lett., vol. 18, pp. 265–267, 2006.
[CrossRef]

Johansson, D.

D. Wake, D. Johansson, D. Moodie, “Passive pico-cell: a new concept in wireless network infrastructure,” Electron. Lett., vol. 33, pp. 404–406, 1997.
[CrossRef]

Jung, H. D.

H. Yang, J. Zeng, Y. Zheng, H. D. Jung, B. Huiszoon, J. H. C. van Zantvoort, E. Tangdiongga, A. M. J. Koonen, “Evaluation of effects of MZM nonlinearity on QAM and OFDM signals in RoF transmitter,” in Int. Topical Meeting on Microwave Photonics, 2008 , pp. 90–93.

Kim, H.

H.-C. Ji, H. Kim, Y. C. Chung, “Full-duplex radio-over-fiber system using phase-modulated downlink and intensity-modulated uplink,” IEEE Photon. Technol. Lett., vol. 21, pp. 9–11, 2009.
[CrossRef]

Kitayama, K.

Koonen, A. M. J.

H. Yang, J. Zeng, Y. Zheng, H. D. Jung, B. Huiszoon, J. H. C. van Zantvoort, E. Tangdiongga, A. M. J. Koonen, “Evaluation of effects of MZM nonlinearity on QAM and OFDM signals in RoF transmitter,” in Int. Topical Meeting on Microwave Photonics, 2008 , pp. 90–93.

Kumar, S.

Kuri, T.

Langley, L. N.

L. N. Langley, M. D. Elkin, C. Edge, M. J. Wale, X. Gliese, X. Huang, A. J. Seeds, “Packaged semiconductor laser optical phase-locked loop (OPLL) for photonic generation, processing and transmission of microwave signals,” IEEE Trans. Microwave Theory Tech., vol. 47, pp. 1257–1264, 1999.
[CrossRef]

Lau, K. Y.

K. Y. Lau, J. Park, Ultra-high Frequency Linear Fiber Optic Systems, Springer, 2008, pp. 155–166.

Lin, C.-T.

Lin, Y.-M.

Luo, T.

Moodie, D.

D. Wake, D. Johansson, D. Moodie, “Passive pico-cell: a new concept in wireless network infrastructure,” Electron. Lett., vol. 33, pp. 404–406, 1997.
[CrossRef]

Nielsen, T. N.

U. Gliese, T. N. Nielsen, S. Nrskov, K. E. Stubkjaer, “Multifunction fiber-optic microwave links based on remote heterodyne detection,” IEEE Trans. Microwave Theory Tech., vol. 46, pp. 458–468, 1998.
[CrossRef]

Novak, D.

G. H. Smith, D. Novak, Z. Ahmed, “Overcoming chromatic-dispersion effects in fiber-wireless systems incorporating external modulators,” IEEE Trans. Microwave Theory Tech., vol. 45, pp. 1410–1415, 1997.
[CrossRef]

Nrskov, S.

U. Gliese, T. N. Nielsen, S. Nrskov, K. E. Stubkjaer, “Multifunction fiber-optic microwave links based on remote heterodyne detection,” IEEE Trans. Microwave Theory Tech., vol. 46, pp. 458–468, 1998.
[CrossRef]

Ogawa, H.

H. Ogawa, D. Polifko, S. Banda, “Millimeter-wave fiber optics systems for personal radio communications,” IEEE Trans. Microwave Theory Tech., vol. 40, pp. 2285–2293, Dec. 1992.
[CrossRef]

Pan, Z.

Park, J.

K. Y. Lau, J. Park, Ultra-high Frequency Linear Fiber Optic Systems, Springer, 2008, pp. 155–166.

Peng, P.-C.

Polifko, D.

H. Ogawa, D. Polifko, S. Banda, “Millimeter-wave fiber optics systems for personal radio communications,” IEEE Trans. Microwave Theory Tech., vol. 40, pp. 2285–2293, Dec. 1992.
[CrossRef]

Seeds, A. J.

L. N. Langley, M. D. Elkin, C. Edge, M. J. Wale, X. Gliese, X. Huang, A. J. Seeds, “Packaged semiconductor laser optical phase-locked loop (OPLL) for photonic generation, processing and transmission of microwave signals,” IEEE Trans. Microwave Theory Tech., vol. 47, pp. 1257–1264, 1999.
[CrossRef]

Shih, P. T.

Smith, G. H.

G. H. Smith, D. Novak, Z. Ahmed, “Overcoming chromatic-dispersion effects in fiber-wireless systems incorporating external modulators,” IEEE Trans. Microwave Theory Tech., vol. 45, pp. 1410–1415, 1997.
[CrossRef]

Stohr, A.

A. Stohr, “Full-duplex 60 GHz fiber-optic transmission,” Electron. Lett., vol. 35, 1653–1655, 1999.
[CrossRef]

Stubkjaer, K. E.

U. Gliese, T. N. Nielsen, S. Nrskov, K. E. Stubkjaer, “Multifunction fiber-optic microwave links based on remote heterodyne detection,” IEEE Trans. Microwave Theory Tech., vol. 46, pp. 458–468, 1998.
[CrossRef]

Su, Y.

Y. Jianjun, J. Zhensheng, L. Yi, Y. Su, C. Gee-Kung, W. Ting, “Optical millimeter-wave generation or up-conversion using external modulators,” IEEE Photon. Technol. Lett., vol. 18, pp. 265–267, 2006.
[CrossRef]

Takimoto, Y.

Y. Takimoto, “Recent activities on millimeter wave indoor LAN system development in Japan,” in Proc. 1995 IEEE MTT-S Int. Microwave Symp. Dig., May 1995, pp. 405–408.

Tangdiongga, E.

H. Yang, J. Zeng, Y. Zheng, H. D. Jung, B. Huiszoon, J. H. C. van Zantvoort, E. Tangdiongga, A. M. J. Koonen, “Evaluation of effects of MZM nonlinearity on QAM and OFDM signals in RoF transmitter,” in Int. Topical Meeting on Microwave Photonics, 2008 , pp. 90–93.

Ting, W.

Y. Jianjun, J. Zhensheng, L. Yi, Y. Su, C. Gee-Kung, W. Ting, “Optical millimeter-wave generation or up-conversion using external modulators,” IEEE Photon. Technol. Lett., vol. 18, pp. 265–267, 2006.
[CrossRef]

Toda, H.

van Zantvoort, J. H. C.

H. Yang, J. Zeng, Y. Zheng, H. D. Jung, B. Huiszoon, J. H. C. van Zantvoort, E. Tangdiongga, A. M. J. Koonen, “Evaluation of effects of MZM nonlinearity on QAM and OFDM signals in RoF transmitter,” in Int. Topical Meeting on Microwave Photonics, 2008 , pp. 90–93.

Wake, D.

D. Wake, D. Johansson, D. Moodie, “Passive pico-cell: a new concept in wireless network infrastructure,” Electron. Lett., vol. 33, pp. 404–406, 1997.
[CrossRef]

Wale, M. J.

L. N. Langley, M. D. Elkin, C. Edge, M. J. Wale, X. Gliese, X. Huang, A. J. Seeds, “Packaged semiconductor laser optical phase-locked loop (OPLL) for photonic generation, processing and transmission of microwave signals,” IEEE Trans. Microwave Theory Tech., vol. 47, pp. 1257–1264, 1999.
[CrossRef]

Wang, Y.

Willner, A. E.

Yang, H.

H. Yang, J. Zeng, Y. Zheng, H. D. Jung, B. Huiszoon, J. H. C. van Zantvoort, E. Tangdiongga, A. M. J. Koonen, “Evaluation of effects of MZM nonlinearity on QAM and OFDM signals in RoF transmitter,” in Int. Topical Meeting on Microwave Photonics, 2008 , pp. 90–93.

Yi, L.

Y. Jianjun, J. Zhensheng, L. Yi, Y. Su, C. Gee-Kung, W. Ting, “Optical millimeter-wave generation or up-conversion using external modulators,” IEEE Photon. Technol. Lett., vol. 18, pp. 265–267, 2006.
[CrossRef]

Yu, C.

Zeng, J.

H. Yang, J. Zeng, Y. Zheng, H. D. Jung, B. Huiszoon, J. H. C. van Zantvoort, E. Tangdiongga, A. M. J. Koonen, “Evaluation of effects of MZM nonlinearity on QAM and OFDM signals in RoF transmitter,” in Int. Topical Meeting on Microwave Photonics, 2008 , pp. 90–93.

Zhang, B.

Zheng, Y.

H. Yang, J. Zeng, Y. Zheng, H. D. Jung, B. Huiszoon, J. H. C. van Zantvoort, E. Tangdiongga, A. M. J. Koonen, “Evaluation of effects of MZM nonlinearity on QAM and OFDM signals in RoF transmitter,” in Int. Topical Meeting on Microwave Photonics, 2008 , pp. 90–93.

Zhensheng, J.

Y. Jianjun, J. Zhensheng, L. Yi, Y. Su, C. Gee-Kung, W. Ting, “Optical millimeter-wave generation or up-conversion using external modulators,” IEEE Photon. Technol. Lett., vol. 18, pp. 265–267, 2006.
[CrossRef]

Electron. Lett. (2)

D. Wake, D. Johansson, D. Moodie, “Passive pico-cell: a new concept in wireless network infrastructure,” Electron. Lett., vol. 33, pp. 404–406, 1997.
[CrossRef]

A. Stohr, “Full-duplex 60 GHz fiber-optic transmission,” Electron. Lett., vol. 35, 1653–1655, 1999.
[CrossRef]

IEEE Photon. Technol. Lett. (2)

Y. Jianjun, J. Zhensheng, L. Yi, Y. Su, C. Gee-Kung, W. Ting, “Optical millimeter-wave generation or up-conversion using external modulators,” IEEE Photon. Technol. Lett., vol. 18, pp. 265–267, 2006.
[CrossRef]

H.-C. Ji, H. Kim, Y. C. Chung, “Full-duplex radio-over-fiber system using phase-modulated downlink and intensity-modulated uplink,” IEEE Photon. Technol. Lett., vol. 21, pp. 9–11, 2009.
[CrossRef]

IEEE Trans. Microwave Theory Tech. (4)

G. H. Smith, D. Novak, Z. Ahmed, “Overcoming chromatic-dispersion effects in fiber-wireless systems incorporating external modulators,” IEEE Trans. Microwave Theory Tech., vol. 45, pp. 1410–1415, 1997.
[CrossRef]

L. N. Langley, M. D. Elkin, C. Edge, M. J. Wale, X. Gliese, X. Huang, A. J. Seeds, “Packaged semiconductor laser optical phase-locked loop (OPLL) for photonic generation, processing and transmission of microwave signals,” IEEE Trans. Microwave Theory Tech., vol. 47, pp. 1257–1264, 1999.
[CrossRef]

U. Gliese, T. N. Nielsen, S. Nrskov, K. E. Stubkjaer, “Multifunction fiber-optic microwave links based on remote heterodyne detection,” IEEE Trans. Microwave Theory Tech., vol. 46, pp. 458–468, 1998.
[CrossRef]

H. Ogawa, D. Polifko, S. Banda, “Millimeter-wave fiber optics systems for personal radio communications,” IEEE Trans. Microwave Theory Tech., vol. 40, pp. 2285–2293, Dec. 1992.
[CrossRef]

IEICE Trans. Commun. (1)

T. Ihara, K. Fujimura, “Research and development trends of millimeter-wave short-range application systems,” IEICE Trans. Commun., vol. E79-B, pp. 1741–1753, Dec. 1996.

J. Lightwave Technol. (2)

J. Opt. Netw. (1)

Opt. Lett. (1)

Other (4)

ERO, Detailed Spectrum Investigation—First Phase: 3400 MHz to 105 GHz, Copenhagen, Denmark, pp. 27–58, March 1993.

Y. Takimoto, “Recent activities on millimeter wave indoor LAN system development in Japan,” in Proc. 1995 IEEE MTT-S Int. Microwave Symp. Dig., May 1995, pp. 405–408.

H. Yang, J. Zeng, Y. Zheng, H. D. Jung, B. Huiszoon, J. H. C. van Zantvoort, E. Tangdiongga, A. M. J. Koonen, “Evaluation of effects of MZM nonlinearity on QAM and OFDM signals in RoF transmitter,” in Int. Topical Meeting on Microwave Photonics, 2008 , pp. 90–93.

K. Y. Lau, J. Park, Ultra-high Frequency Linear Fiber Optic Systems, Springer, 2008, pp. 155–166.

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Figures (4)

Fig. 1
Fig. 1

Conceptual diagram of the 60 GHz RoF system: (i) downlink, (ii) uplink. TL, tunable laser; IL, optical interleaver; EA, electrical amplifier; LNA, low-noise amplifier; IM, intensity modulator; EDFA, erbium-doped fiber amplifier; PD, photodiode; LPF, low-pass filter.

Fig. 2
Fig. 2

Simulation result: analytical spectrum. (a) E 1 , (b) E 2 , (c) E 3 .

Fig. 3
Fig. 3

Millimeter-wave power degradation versus the linewidth of the tunable laser.

Fig. 4
Fig. 4

BER versus the linewidth of the tunable laser ( n 0 = 1 e 19 W Hz ) : (a) different fiber lengths, (b) different bit rates, (c) different laser powers.

Equations (19)

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

E in ( t ) = E 1 sin ( π a i 2 ) exp [ j 2 π f 1 t + j φ 1 ( t ) ] + E 2 exp [ j 2 π f 2 t + j φ 2 ( t ) ] ,
E out ( t ) = E 1 sin ( π a i 2 ) n = 1 ( 1 ) n J 2 n 1 ( m ) { exp [ j 2 π f 1 t + j ( 4 n 2 ) π f t + j φ 1 ( t ) ] + exp [ j 2 π f 1 t j ( 4 n 2 ) π f t + j φ 1 ( t ) ] } + E 2 n = 1 ( 1 ) n J 2 n 1 ( m ) { exp [ j 2 π f 2 t + j ( 4 n 2 ) π f t + j φ 2 ( t ) ] + exp [ j 2 π f 2 t j ( 4 n 2 ) π f t + j φ 2 ( t ) ] } ,
E out ( t ) = E 1 sin ( π a i 2 ) J 1 ( m ) { exp [ j 2 π f 1 t + j 2 π f t + j φ 1 ( t ) ] + exp [ j 2 π f 1 t j 2 π f t + j φ 1 ( t ) ] } E 2 J 1 ( m ) { exp [ j 2 π f 2 t + j 2 π f t + j φ 2 ( t ) ] + exp [ j 2 π f 2 t j 2 π f t + j φ 2 ( t ) ] } .
T = z d β d ω ,
T = z β ( ω 0 ) + z β ( ω 0 ) ( ω ω 0 ) + z 2 β ( ω 0 ) ( ω ω 0 ) 2 + ,
d T d λ = ( 2 π c z λ 2 ) β ( ω 0 ) = ( ω z λ ) β ( ω 0 ) .
D = ( 1 z ) d T d λ = ( ω λ ) β ( ω 0 ) ,
T = z λ 0 D + ( z λ 0 ω 0 ) D ( ω ω 0 ) .
E 1 ( t ) = E 1 sin ( π a i 2 ) J 1 ( m ) { exp [ j ( 2 π f 1 + 2 π f ) ( t τ 3 ) + j φ 1 ( t τ 3 ) ] + exp [ j ( 2 π f 1 2 π f ) ( t τ 1 ) + j φ 1 ( t τ 1 ) ] } E 2 J 1 ( m ) { exp [ j ( 2 π f 2 + 2 π f ) ( t τ 4 ) + j φ 2 ( t τ 4 ) ] + exp [ j ( 2 π f 2 2 π f ) ( t τ 2 ) + j φ 2 ( t τ 2 ) ] } ,
E 2 ( t ) = E 1 sin ( π a i 2 ) J 1 ( m ) { exp [ j ( 2 π f 1 + 2 π f ) ( t τ 3 ) + j φ 1 ( t τ 3 ) ] + exp [ j ( 2 π f 1 2 π f ) ( t τ 1 ) + j φ 1 ( t τ 1 ) ] } ,
E 3 ( t ) = E 2 J 1 ( m ) { exp [ j ( 2 π f 2 + 2 π f ) ( t τ 4 ) + j φ 2 ( t τ 4 ) ] + exp [ j ( 2 π f 2 2 π f ) ( t τ 2 ) + j φ 2 ( t τ 2 ) ] } .
i mm = 1 2 E { E 2 ( t ) E 2 * ( t ) } = [ E 1 sin ( π a i 2 ) J 1 ( m ) ] 2 [ 1 + 1 2 E { exp [ j φ 1 ( t τ 3 ) j φ 1 ( t τ 1 ) ] } exp ( j 4 π f t ) + 1 2 E { exp [ j φ 1 ( t τ 1 ) j φ 1 ( t τ 3 ) ] } exp ( j 4 π f t ) ] ,
i LO = E 2 2 J 1 2 ( m ) [ 1 + 1 2 E { exp [ j φ 2 ( t τ 4 ) j φ 2 ( t τ 2 ) ] } exp ( j 4 π f t ) + 1 2 E { exp [ j φ 2 ( t τ 2 ) j φ 2 ( t τ 4 ) ] } exp ( j 4 π f t ) ] .
E { exp [ j φ ( t 1 ) j φ ( t 2 ) ] } = + 1 2 π δ ( t 1 t 2 ) exp ( x 2 4 π δ ( t 1 t 2 ) ) exp ( j x ) d x = exp ( 1 2 × 2 π δ | t 1 t 2 | ) = exp ( π δ | t 1 t 2 | ) .
i mm = [ E 1 sin ( π a i 2 ) J 1 ( m ) ] 2 [ 1 + exp ( 2 π δ 1 z c D f f 1 2 ) cos ( 4 π f t ) ]
i LO = E 2 2 J 1 2 ( m ) [ 1 + exp ( 2 π δ 2 z c D f f 2 2 ) cos ( 4 π f t ) ] ,
P e = 1 2 erfc ( r 2 ) ,
r = [ E 1 J 1 ( m ) ] 4 exp ( 4 π δ 1 z c D f f 1 2 ) n 0 B ,
P e = 1 2 erfc ( [ E 1 J 1 ( m ) ] 4 exp ( 4 π δ 1 z c D f f 1 2 ) ( 4 n 0 B ) ) .