Abstract

A new microwave photonic topology for RF signal direction finding is presented. It is based on a dual-parallel Mach Zehnder modulator (DPMZM) in series with an optical phase modulator (PM). The direction of an RF signal received by the antennas connected to an RF port of the DPMZM and the PM can be determined from the power ratio of two system output low frequency components, without the need to know the incoming RF signal amplitude in advance. The proposed structure is suitable for implementing a long baseline technique for direction finding and can be extended to have multiple antenna elements in remote locations. In addition to direction finding, the system also has the ability to measure an RF signal Doppler frequency shift to determine an object speed and moving direction when it is used in a radar receiver. Results obtained using the proposed structure demonstrate less than ±2.5° errors over a 3.2° to 81.5° angle of arrival measurement range for different RF signal modulation indexes of 0.02, 0.08 and 0.16. Doppler frequency shift measurement with less than 0.8 Hz errors is also demonstrated.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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  • Publication
  1. R. E. Franks, “Direction-finding antennas,” in Antenna Handbook: Volume III Applications, Y. T. Lo and S. W. Lee, eds., (Springer, 1993).
  2. J. Kolanek and E. Carlsen, “Precision geolocation system and method using a long baseline interferometer antenna system,” United States Patent, 7286085, 2007.
  3. M. E. Manka, “Microwave photonics for electronic warfare applications,” International Topical Meeting on Microwave Photonics, 275–278 (2008).
  4. Z. Cao, Q. Wang, R. Lu, H. P. A. van den Boom, E. Tangdiongga, and A. M. J. Koonen, “Phase modulation parallel optical delay detector for microwave angle-of-arrival measurement with accuracy monitored,” Opt. Lett. 39(6), 1497–1500 (2014).
    [Crossref]
  5. H. Chen and E. H. W. Chan, “Simple approach to measure angle of arrival of a microwave signal,” IEEE Photonics Technol. Lett. 31(22), 1795–1798 (2019).
    [Crossref]
  6. Z. Cao, H. P. A. van den Boom, R. Lu, Q. Wang, E. Tangdiongga, and A. M. J. Koonen, “Angle-of-arrival measurement of a microwave signal using parallel optical delay detector,” IEEE Photonics Technol. Lett. 25(19), 1932–1935 (2013).
    [Crossref]
  7. H. Zhuo, A. Wen, and Y. Wang, “Photonic angle-of-arrival measurement without direction ambiguity based on a dual-parallel Mach–Zehnder modulator,” Opt. Commun. 451, 286–289 (2019).
    [Crossref]
  8. P. Li, L. Yan, J. Ye, X. Feng, W. Pan, B. Luo, X. Zou, T. Zhou, and Z. Chen, “Photonic approach for simultaneous measurements of Doppler-frequency-shift and angle-of-arrival of microwave signals,” Opt. Express 27(6), 8709–8716 (2019).
    [Crossref]
  9. X. Zou, W. Li, W. Pan, B. Luo, L. Yan, and J. Yao, “Photonic approach to the measurement of time-difference-of-arrival and angle-of-arrival of a microwave signal,” Opt. Lett. 37(4), 755–757 (2012).
    [Crossref]
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  11. H. Chen and E. H. W. Chan, “Angle of arrival measurement system using double RF modulation technique,” IEEE Photonics J. 11(1), 1–10 (2019).
    [Crossref]
  12. G. H. Smith, D. Novak, and Z. Ahmed, “Novel technique for generation of optical SSB with carrier using a single MZM to overcome fiber chromatic dispersion,” International Topical Meeting on Microwave Photonics, pp. 5–8, (1996).
  13. X. Li, L. Deng, X. Chen, H. Song, Y. Liu, M. Cheng, S. Fu, M. Tang, M. Zhang, and D. Liu, “Arbitrary bias point control technique for optical IQ modulator based on dither-correlation detection,” J. Lightwave Technol. 36(18), 3824–3836 (2018).
    [Crossref]
  14. X. Wang, P. O. Weigel, J. Zhao, M. Ruesing, and S. Mookherjea, “Achieving beyond-100-GHz large-signal modulation bandwidth in hybrid silicon photonics Mach Zehnder modulators using thin film lithium niobate,” APL Photonics 4(9), 096101 (2019).
    [Crossref]
  15. C. Huang, E. H. W. Chan, and C. B. Albert, “A compact photonics-based single sideband mixer without using high-frequency electrical components,” IEEE Photonics J. 11(4), 1–9 (2019).
    [Crossref]
  16. C. Porzi, G. J. Sharp, M. Sorel, and A. Bogoni, “Silicon photonics high-order distributed feedback resonators filters,” IEEE J. Quantum Electron. 56(1), 1–9 (2020).
    [Crossref]

2020 (1)

C. Porzi, G. J. Sharp, M. Sorel, and A. Bogoni, “Silicon photonics high-order distributed feedback resonators filters,” IEEE J. Quantum Electron. 56(1), 1–9 (2020).
[Crossref]

2019 (6)

H. Chen and E. H. W. Chan, “Simple approach to measure angle of arrival of a microwave signal,” IEEE Photonics Technol. Lett. 31(22), 1795–1798 (2019).
[Crossref]

H. Zhuo, A. Wen, and Y. Wang, “Photonic angle-of-arrival measurement without direction ambiguity based on a dual-parallel Mach–Zehnder modulator,” Opt. Commun. 451, 286–289 (2019).
[Crossref]

P. Li, L. Yan, J. Ye, X. Feng, W. Pan, B. Luo, X. Zou, T. Zhou, and Z. Chen, “Photonic approach for simultaneous measurements of Doppler-frequency-shift and angle-of-arrival of microwave signals,” Opt. Express 27(6), 8709–8716 (2019).
[Crossref]

H. Chen and E. H. W. Chan, “Angle of arrival measurement system using double RF modulation technique,” IEEE Photonics J. 11(1), 1–10 (2019).
[Crossref]

X. Wang, P. O. Weigel, J. Zhao, M. Ruesing, and S. Mookherjea, “Achieving beyond-100-GHz large-signal modulation bandwidth in hybrid silicon photonics Mach Zehnder modulators using thin film lithium niobate,” APL Photonics 4(9), 096101 (2019).
[Crossref]

C. Huang, E. H. W. Chan, and C. B. Albert, “A compact photonics-based single sideband mixer without using high-frequency electrical components,” IEEE Photonics J. 11(4), 1–9 (2019).
[Crossref]

2018 (1)

2014 (1)

2013 (1)

Z. Cao, H. P. A. van den Boom, R. Lu, Q. Wang, E. Tangdiongga, and A. M. J. Koonen, “Angle-of-arrival measurement of a microwave signal using parallel optical delay detector,” IEEE Photonics Technol. Lett. 25(19), 1932–1935 (2013).
[Crossref]

2012 (1)

Ahmed, Z.

G. H. Smith, D. Novak, and Z. Ahmed, “Novel technique for generation of optical SSB with carrier using a single MZM to overcome fiber chromatic dispersion,” International Topical Meeting on Microwave Photonics, pp. 5–8, (1996).

Albert, C. B.

C. Huang, E. H. W. Chan, and C. B. Albert, “A compact photonics-based single sideband mixer without using high-frequency electrical components,” IEEE Photonics J. 11(4), 1–9 (2019).
[Crossref]

Biernacki, P. D.

P. D. Biernacki, R. Madara, L. T. Nichols, A. Ward, and P. J. Matthews, “A four channel angle of arrival detector using optical downconversion,” 1999 IEEE MTT-S International Microwave Symposium Digest, pp. 885–888, (1999).

Bogoni, A.

C. Porzi, G. J. Sharp, M. Sorel, and A. Bogoni, “Silicon photonics high-order distributed feedback resonators filters,” IEEE J. Quantum Electron. 56(1), 1–9 (2020).
[Crossref]

Cao, Z.

Z. Cao, Q. Wang, R. Lu, H. P. A. van den Boom, E. Tangdiongga, and A. M. J. Koonen, “Phase modulation parallel optical delay detector for microwave angle-of-arrival measurement with accuracy monitored,” Opt. Lett. 39(6), 1497–1500 (2014).
[Crossref]

Z. Cao, H. P. A. van den Boom, R. Lu, Q. Wang, E. Tangdiongga, and A. M. J. Koonen, “Angle-of-arrival measurement of a microwave signal using parallel optical delay detector,” IEEE Photonics Technol. Lett. 25(19), 1932–1935 (2013).
[Crossref]

Carlsen, E.

J. Kolanek and E. Carlsen, “Precision geolocation system and method using a long baseline interferometer antenna system,” United States Patent, 7286085, 2007.

Chan, E. H. W.

H. Chen and E. H. W. Chan, “Simple approach to measure angle of arrival of a microwave signal,” IEEE Photonics Technol. Lett. 31(22), 1795–1798 (2019).
[Crossref]

H. Chen and E. H. W. Chan, “Angle of arrival measurement system using double RF modulation technique,” IEEE Photonics J. 11(1), 1–10 (2019).
[Crossref]

C. Huang, E. H. W. Chan, and C. B. Albert, “A compact photonics-based single sideband mixer without using high-frequency electrical components,” IEEE Photonics J. 11(4), 1–9 (2019).
[Crossref]

Chen, H.

H. Chen and E. H. W. Chan, “Angle of arrival measurement system using double RF modulation technique,” IEEE Photonics J. 11(1), 1–10 (2019).
[Crossref]

H. Chen and E. H. W. Chan, “Simple approach to measure angle of arrival of a microwave signal,” IEEE Photonics Technol. Lett. 31(22), 1795–1798 (2019).
[Crossref]

Chen, X.

Chen, Z.

Cheng, M.

Deng, L.

Feng, X.

Franks, R. E.

R. E. Franks, “Direction-finding antennas,” in Antenna Handbook: Volume III Applications, Y. T. Lo and S. W. Lee, eds., (Springer, 1993).

Fu, S.

Huang, C.

C. Huang, E. H. W. Chan, and C. B. Albert, “A compact photonics-based single sideband mixer without using high-frequency electrical components,” IEEE Photonics J. 11(4), 1–9 (2019).
[Crossref]

Kolanek, J.

J. Kolanek and E. Carlsen, “Precision geolocation system and method using a long baseline interferometer antenna system,” United States Patent, 7286085, 2007.

Koonen, A. M. J.

Z. Cao, Q. Wang, R. Lu, H. P. A. van den Boom, E. Tangdiongga, and A. M. J. Koonen, “Phase modulation parallel optical delay detector for microwave angle-of-arrival measurement with accuracy monitored,” Opt. Lett. 39(6), 1497–1500 (2014).
[Crossref]

Z. Cao, H. P. A. van den Boom, R. Lu, Q. Wang, E. Tangdiongga, and A. M. J. Koonen, “Angle-of-arrival measurement of a microwave signal using parallel optical delay detector,” IEEE Photonics Technol. Lett. 25(19), 1932–1935 (2013).
[Crossref]

Li, P.

Li, W.

Li, X.

Liu, D.

Liu, Y.

Lu, R.

Z. Cao, Q. Wang, R. Lu, H. P. A. van den Boom, E. Tangdiongga, and A. M. J. Koonen, “Phase modulation parallel optical delay detector for microwave angle-of-arrival measurement with accuracy monitored,” Opt. Lett. 39(6), 1497–1500 (2014).
[Crossref]

Z. Cao, H. P. A. van den Boom, R. Lu, Q. Wang, E. Tangdiongga, and A. M. J. Koonen, “Angle-of-arrival measurement of a microwave signal using parallel optical delay detector,” IEEE Photonics Technol. Lett. 25(19), 1932–1935 (2013).
[Crossref]

Luo, B.

Madara, R.

P. D. Biernacki, R. Madara, L. T. Nichols, A. Ward, and P. J. Matthews, “A four channel angle of arrival detector using optical downconversion,” 1999 IEEE MTT-S International Microwave Symposium Digest, pp. 885–888, (1999).

Manka, M. E.

M. E. Manka, “Microwave photonics for electronic warfare applications,” International Topical Meeting on Microwave Photonics, 275–278 (2008).

Matthews, P. J.

P. D. Biernacki, R. Madara, L. T. Nichols, A. Ward, and P. J. Matthews, “A four channel angle of arrival detector using optical downconversion,” 1999 IEEE MTT-S International Microwave Symposium Digest, pp. 885–888, (1999).

Mookherjea, S.

X. Wang, P. O. Weigel, J. Zhao, M. Ruesing, and S. Mookherjea, “Achieving beyond-100-GHz large-signal modulation bandwidth in hybrid silicon photonics Mach Zehnder modulators using thin film lithium niobate,” APL Photonics 4(9), 096101 (2019).
[Crossref]

Nichols, L. T.

P. D. Biernacki, R. Madara, L. T. Nichols, A. Ward, and P. J. Matthews, “A four channel angle of arrival detector using optical downconversion,” 1999 IEEE MTT-S International Microwave Symposium Digest, pp. 885–888, (1999).

Novak, D.

G. H. Smith, D. Novak, and Z. Ahmed, “Novel technique for generation of optical SSB with carrier using a single MZM to overcome fiber chromatic dispersion,” International Topical Meeting on Microwave Photonics, pp. 5–8, (1996).

Pan, W.

Porzi, C.

C. Porzi, G. J. Sharp, M. Sorel, and A. Bogoni, “Silicon photonics high-order distributed feedback resonators filters,” IEEE J. Quantum Electron. 56(1), 1–9 (2020).
[Crossref]

Ruesing, M.

X. Wang, P. O. Weigel, J. Zhao, M. Ruesing, and S. Mookherjea, “Achieving beyond-100-GHz large-signal modulation bandwidth in hybrid silicon photonics Mach Zehnder modulators using thin film lithium niobate,” APL Photonics 4(9), 096101 (2019).
[Crossref]

Sharp, G. J.

C. Porzi, G. J. Sharp, M. Sorel, and A. Bogoni, “Silicon photonics high-order distributed feedback resonators filters,” IEEE J. Quantum Electron. 56(1), 1–9 (2020).
[Crossref]

Smith, G. H.

G. H. Smith, D. Novak, and Z. Ahmed, “Novel technique for generation of optical SSB with carrier using a single MZM to overcome fiber chromatic dispersion,” International Topical Meeting on Microwave Photonics, pp. 5–8, (1996).

Song, H.

Sorel, M.

C. Porzi, G. J. Sharp, M. Sorel, and A. Bogoni, “Silicon photonics high-order distributed feedback resonators filters,” IEEE J. Quantum Electron. 56(1), 1–9 (2020).
[Crossref]

Tang, M.

Tangdiongga, E.

Z. Cao, Q. Wang, R. Lu, H. P. A. van den Boom, E. Tangdiongga, and A. M. J. Koonen, “Phase modulation parallel optical delay detector for microwave angle-of-arrival measurement with accuracy monitored,” Opt. Lett. 39(6), 1497–1500 (2014).
[Crossref]

Z. Cao, H. P. A. van den Boom, R. Lu, Q. Wang, E. Tangdiongga, and A. M. J. Koonen, “Angle-of-arrival measurement of a microwave signal using parallel optical delay detector,” IEEE Photonics Technol. Lett. 25(19), 1932–1935 (2013).
[Crossref]

van den Boom, H. P. A.

Z. Cao, Q. Wang, R. Lu, H. P. A. van den Boom, E. Tangdiongga, and A. M. J. Koonen, “Phase modulation parallel optical delay detector for microwave angle-of-arrival measurement with accuracy monitored,” Opt. Lett. 39(6), 1497–1500 (2014).
[Crossref]

Z. Cao, H. P. A. van den Boom, R. Lu, Q. Wang, E. Tangdiongga, and A. M. J. Koonen, “Angle-of-arrival measurement of a microwave signal using parallel optical delay detector,” IEEE Photonics Technol. Lett. 25(19), 1932–1935 (2013).
[Crossref]

Wang, Q.

Z. Cao, Q. Wang, R. Lu, H. P. A. van den Boom, E. Tangdiongga, and A. M. J. Koonen, “Phase modulation parallel optical delay detector for microwave angle-of-arrival measurement with accuracy monitored,” Opt. Lett. 39(6), 1497–1500 (2014).
[Crossref]

Z. Cao, H. P. A. van den Boom, R. Lu, Q. Wang, E. Tangdiongga, and A. M. J. Koonen, “Angle-of-arrival measurement of a microwave signal using parallel optical delay detector,” IEEE Photonics Technol. Lett. 25(19), 1932–1935 (2013).
[Crossref]

Wang, X.

X. Wang, P. O. Weigel, J. Zhao, M. Ruesing, and S. Mookherjea, “Achieving beyond-100-GHz large-signal modulation bandwidth in hybrid silicon photonics Mach Zehnder modulators using thin film lithium niobate,” APL Photonics 4(9), 096101 (2019).
[Crossref]

Wang, Y.

H. Zhuo, A. Wen, and Y. Wang, “Photonic angle-of-arrival measurement without direction ambiguity based on a dual-parallel Mach–Zehnder modulator,” Opt. Commun. 451, 286–289 (2019).
[Crossref]

Ward, A.

P. D. Biernacki, R. Madara, L. T. Nichols, A. Ward, and P. J. Matthews, “A four channel angle of arrival detector using optical downconversion,” 1999 IEEE MTT-S International Microwave Symposium Digest, pp. 885–888, (1999).

Weigel, P. O.

X. Wang, P. O. Weigel, J. Zhao, M. Ruesing, and S. Mookherjea, “Achieving beyond-100-GHz large-signal modulation bandwidth in hybrid silicon photonics Mach Zehnder modulators using thin film lithium niobate,” APL Photonics 4(9), 096101 (2019).
[Crossref]

Wen, A.

H. Zhuo, A. Wen, and Y. Wang, “Photonic angle-of-arrival measurement without direction ambiguity based on a dual-parallel Mach–Zehnder modulator,” Opt. Commun. 451, 286–289 (2019).
[Crossref]

Yan, L.

Yao, J.

Ye, J.

Zhang, M.

Zhao, J.

X. Wang, P. O. Weigel, J. Zhao, M. Ruesing, and S. Mookherjea, “Achieving beyond-100-GHz large-signal modulation bandwidth in hybrid silicon photonics Mach Zehnder modulators using thin film lithium niobate,” APL Photonics 4(9), 096101 (2019).
[Crossref]

Zhou, T.

Zhuo, H.

H. Zhuo, A. Wen, and Y. Wang, “Photonic angle-of-arrival measurement without direction ambiguity based on a dual-parallel Mach–Zehnder modulator,” Opt. Commun. 451, 286–289 (2019).
[Crossref]

Zou, X.

APL Photonics (1)

X. Wang, P. O. Weigel, J. Zhao, M. Ruesing, and S. Mookherjea, “Achieving beyond-100-GHz large-signal modulation bandwidth in hybrid silicon photonics Mach Zehnder modulators using thin film lithium niobate,” APL Photonics 4(9), 096101 (2019).
[Crossref]

IEEE J. Quantum Electron. (1)

C. Porzi, G. J. Sharp, M. Sorel, and A. Bogoni, “Silicon photonics high-order distributed feedback resonators filters,” IEEE J. Quantum Electron. 56(1), 1–9 (2020).
[Crossref]

IEEE Photonics J. (2)

C. Huang, E. H. W. Chan, and C. B. Albert, “A compact photonics-based single sideband mixer without using high-frequency electrical components,” IEEE Photonics J. 11(4), 1–9 (2019).
[Crossref]

H. Chen and E. H. W. Chan, “Angle of arrival measurement system using double RF modulation technique,” IEEE Photonics J. 11(1), 1–10 (2019).
[Crossref]

IEEE Photonics Technol. Lett. (2)

H. Chen and E. H. W. Chan, “Simple approach to measure angle of arrival of a microwave signal,” IEEE Photonics Technol. Lett. 31(22), 1795–1798 (2019).
[Crossref]

Z. Cao, H. P. A. van den Boom, R. Lu, Q. Wang, E. Tangdiongga, and A. M. J. Koonen, “Angle-of-arrival measurement of a microwave signal using parallel optical delay detector,” IEEE Photonics Technol. Lett. 25(19), 1932–1935 (2013).
[Crossref]

J. Lightwave Technol. (1)

Opt. Commun. (1)

H. Zhuo, A. Wen, and Y. Wang, “Photonic angle-of-arrival measurement without direction ambiguity based on a dual-parallel Mach–Zehnder modulator,” Opt. Commun. 451, 286–289 (2019).
[Crossref]

Opt. Express (1)

Opt. Lett. (2)

Other (5)

P. D. Biernacki, R. Madara, L. T. Nichols, A. Ward, and P. J. Matthews, “A four channel angle of arrival detector using optical downconversion,” 1999 IEEE MTT-S International Microwave Symposium Digest, pp. 885–888, (1999).

G. H. Smith, D. Novak, and Z. Ahmed, “Novel technique for generation of optical SSB with carrier using a single MZM to overcome fiber chromatic dispersion,” International Topical Meeting on Microwave Photonics, pp. 5–8, (1996).

R. E. Franks, “Direction-finding antennas,” in Antenna Handbook: Volume III Applications, Y. T. Lo and S. W. Lee, eds., (Springer, 1993).

J. Kolanek and E. Carlsen, “Precision geolocation system and method using a long baseline interferometer antenna system,” United States Patent, 7286085, 2007.

M. E. Manka, “Microwave photonics for electronic warfare applications,” International Topical Meeting on Microwave Photonics, 275–278 (2008).

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Figures (10)
Fig. 1.
Fig. 1. (a) Schematic diagram of the proposed microwave photonic based DF system. (b) RF signal incident on two array elements.

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Fig. 2.
Fig. 2. Optical spectrum at the output of (a) the DPMZM and (b) the PM. The dotted line is the optical filter amplitude response. (c) System output electrical spectrum. fc, fS, fRF and fLO are the frequency of the carrier, source signal, incoming RF signal and low-frequency LO respectively.

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Fig. 3.
Fig. 3. Simulated info to RF phase independent peak power ratio when the phase difference of the RF signal received by two antennas is changed from (a) 0° to 180° and (b) 118° to 122°.

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Fig. 4.
Fig. 4. (a) Estimated RF signal AOA and (b) error in the estimated RF signal AOA versus actual RF signal AOA. The AOA obtained using the power ratio with a 0.08 RF signal modulation index was used as a reference.

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Fig. 5.
Fig. 5. Experimental setup of the proposed microwave photonic based DF system.

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Fig. 6.
Fig. 6. Measured system output electrical spectrums when a 15 GHz + 300 kHz RF signal into the DPMZM and the PM have a phase difference of (a) 0° and (b) 150°.

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Fig. 7.
Fig. 7. (a) Measured info peak power (black square) and RF phase independent peak power (red circle) versus the RF signal phase difference for an RF signal modulation index of 0.08. (b) Measured info to RF phase independent peak power ratio at various RF signal phase differences for an RF signal modulation index of 0.02 (black square), 0.08 (red circle) and 0.16 (green triangle), and the simulated power ratio (solid line).

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Fig. 8.
Fig. 8. (a) Estimated AOA versus actual AOA for a 0.02 (black square), 0.08 (red circle) and 0.16 (green triangle) RF signal modulation index. (b) The corresponding AOA measurement error.

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Fig. 9.
Fig. 9. Measured info peak power (purple square) and RF phase independent peak power (black square) and corresponding power ratio (red circle) for a 0° input RF signal phase difference when the RF signal power into the DDMZM is changed from −18 dBm to 0 dBm.

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Fig. 10.
Fig. 10. Measured DFS (purple square) and the corresponding measurement error (red circle) when the frequency of the RF signal is changed from 15 GHz - 100 kHz to 15 GHz + 100 kHz.

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Equations (10)

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

E D P M Z M ( t ) = 1 4 t f f 1 E i n e j ω c t [ J 0 ( m L O ) ( 1 + j ) e j α − 2 J 1 ( m L O ) e − j ( ω L O t − α ) + J 0 ( m R F ) + J 1 ( m R F ) e j ω R F t − J 1 ( m R F ) e − j ω R F t − J 0 ( m S ) − J 1 ( m S ) e j ω S t + J 1 ( m S ) e − j ω S t ]
E P M ( t ) = 1 4 t f f 1 t f f 2 E i n e j ω c t [ J 0 ( m R F ) + J 1 ( m R F ) e j ( ω R F t + θ ) − J 1 ( m R F ) e − j ( ω R F t + θ ) ] × [ J 0 ( m L O ) ( 1 + j ) e j α − 2 J 1 ( m L O ) e − j ( ω L O ( t + τ ) − α ) + J 0 ( m R F ) + J 1 ( m R F ) e j ω R F ( t + τ ) − J 1 ( m R F ) e − j ω R F ( t + τ ) − J 0 ( m S ) − J 1 ( m S ) e j ω S ( t + τ ) + J 1 ( m S ) e − j ω S ( t + τ ) ]
φ = sin − 1 ( c ⋅ θ 2 π f R F ⋅ d )
E o u t ( t ) = 1 4 t f f 1 t f f 2 E i n e j ω c t [ − [ J 0 ( m L O ) ( 1 + j ) e j α + J 0 ( m R F ) − J 0 ( m S ) ] J 1 ( m R F ) e − j ( ω R F t + θ ) − J 0 ( m R F ) J 1 ( m R F ) e − j ω R F ( t + τ ) + J 0 ( m R F ) J 1 ( m S ) e − j ω S ( t + τ ) + 2 J 1 ( m L O ) J 1 ( m R F ) e − j ( ω L O ( t + τ ) − α ) e − j ( ω R F t + θ ) ]
P I n f o = 1 64 t f f 1 2 t f f 2 2 P i n 2 J 0 2 ( m R F ) J 1 2 ( m R F ) J 1 2 ( m S ) ℜ 2 R o × [ ( 2 J 0 ( m L O ) − J 0 ( m S ) ) 2 + 2 ( J 0 ( m R F ) − J 0 ( m S ) ) J 0 ( m R F ) ( cos ⁡ ( ω R F τ − θ ) + 1 ) + 2 2 ( J 0 ( m R F ) − J 0 ( m S ) ) J 0 ( m L O ) ( cos ⁡ ( α + π 4 ) − 1 ) 2 2 J 0 ( m R F ) J 0 ( m L O ) ( cos ⁡ ( ω R F τ − θ + α + π 4 ) + 1 ) ]
P L O = 1 16 t f f 1 2 t f f 2 2 P i n 2 J 1 2 ( m L O ) J 1 4 ( m R F ) ℜ 2 R o × [ ( 2 J 0 ( m L O ) − J 0 ( m S ) ) 2 + 2 ( J 0 ( m R F ) − J 0 ( m S ) ) J 0 ( m R F ) ( cos ⁡ ( ω R F τ − θ ) + 1 ) + 2 2 ( J 0 ( m R F ) − J 0 ( m S ) ) J 0 ( m L O ) ( cos ⁡ ( α + π 4 ) − 1 ) 2 2 J 0 ( m R F ) J 0 ( m L O ) ( cos ⁡ ( ω R F τ − θ + α + π 4 ) + 1 ) ]
P I n d e p = 1 16 t f f 1 2 t f f 2 2 P i n 2 J 1 2 ( m L O ) J 1 2 ( m R F ) J 0 2 ( m R F ) J 1 2 ( m S ) ℜ 2 R o
P I n f o P I n d e p = 1 4 J 1 2 ( m L O ) [ ( 2 J 0 ( m L O ) − J 0 ( m S ) ) 2 + 2 ( J 0 ( m R F ) − J 0 ( m S ) ) J 0 ( m R F ) ( cos ⁡ ( ω R F τ − θ ) + 1 ) + 2 2 ( J 0 ( m R F ) − J 0 ( m S ) ) J 0 ( m L O ) ( cos ⁡ ( α + π 4 ) − 1 ) + 2 2 J 0 ( m R F ) J 0 ( m L O ) ( cos ⁡ ( ω R F τ − θ + α + π 4 ) + 1 ) ]
ω R F τ = 8 m − 1 4 π
P I n f o P I n d e p = 1 2 J 1 2 ( m L O ) { [ J 0 ( m R F ) − J 0 ( m S ) ] ⋅ [ ( 1 − 2 ) J 0 ( m L O ) + J 0 ( m R F ) ( cos ⁡ ( 8 m − 1 4 π − θ ) + 1 ) ] + 2 J 0 ( m R F ) J 0 ( m L O ) ( cos ⁡ ( θ ) + 1 ) }

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