Abstract

An orthogonal vector-sum integrated microwave photonic phase shifter (IMWPPS), consisting of mode-order converter multiplexers (MOCMs), a variable optical power splitter (VOPS), an optical switch (OS) and fixed time delay lines (FTDLs), was theoretically demonstrated in a silicon-on-insulator wafer. MOCMs, as a key element of our device, were employed to generate orthogonal vector signals and served as lossless optical combiners. Combining with the thermo-optical VOPS, OS and FTDLs, the microwave phase shift of 02π could be achieved by a refractive index variation of 015×103 in the millimeter wave band. The corresponding tuning resolution was about 1.64°/°C. This work, for the first time to our knowledge, provides an attractive solution to transferring a vector-sum method based bulk MWPPS into a integrated one, which is very important for large-scale optically controlled phase array antenna.

© 2011 Optical Society of America

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References

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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2010 (3)

2009 (2)

2006 (2)

2005 (2)

K.-H. Lee, Y. M. Jhon, and W.-Y. Choi, “Photonic phase shifter based on vector-sum technique with polarization-maintaining fibers,” Opt. Lett. 30, 702–704 (2005).
[CrossRef] [PubMed]

X. Wang, B. Howley, M. Y. Chen, Q. Zhou, R. Chen, and P. Basile, “Polymer based thermo-optic switch for optical true time delay,” Proc. SPIE 5728, 60–67 (2005).
[CrossRef]

2003 (1)

L. A. Bui, A. Mitchell, K. Ghnrbani, and T.-H. Chin, “Wideband RF photonic vector sum phase-shifter,” Electron. Lett. 39, 536–537 (2003).
[CrossRef]

2000 (1)

S. J. Kim and N. H. Myung, “A new active phase shifter using a vector sum method,” IEEE Microwave Guided Wave Lett. 10, 233–235 (2000).
[CrossRef]

1999 (1)

H. S. Tang, L. Wu, Z. Fu, D. An, Z. Han, and R. T. Chen, “Polymer-based optical waveguide circuits for photonic phased-array antennas,” Proc. SPIE 3632, 250–261 (1999).
[CrossRef]

1995 (2)

J. J. Lee, R. Y. Loo, S. Livingston, V. I. Jones, J. B. Lewis, H. W. Yen, G. L. Tangonan, and M. Wechsberg, “Photonic wideband array antennas,” IEEE Trans. Antennas Propag. 43, 966–982(1995).
[CrossRef]

M. N. Weiss and R. Srivastava, “Spectral characteristics of asymmetric directional couplers in graded index channel waveguides analyzed by coupled-mode and normal-mode techniques,” Appl. Opt. 34, 1029–1040 (1995).
[CrossRef] [PubMed]

1994 (1)

M. L. Van Blaricum, “Photonic systems for antenna applications,” IEEE Antennas Propag. Mag. 36, 30–38 (1994).
[CrossRef]

1993 (1)

J. F. Coward, T. K. Yee, C. H. Chalfant, and P. H. Chang, “A photonic integrated-optic RF phase shifter for phased array antenna beam-forming applications,” J. Lightwave Technol. 11, 2201–2205 (1993).
[CrossRef]

An, D.

H. S. Tang, L. Wu, Z. Fu, D. An, Z. Han, and R. T. Chen, “Polymer-based optical waveguide circuits for photonic phased-array antennas,” Proc. SPIE 3632, 250–261 (1999).
[CrossRef]

Ang, K.-W.

T.-Y. Liow, K.-W. Ang, Q. Fang, J.-F. Song, Y.-Z. Xiong, M.-B. Yu, G.-Q. Lo, and D.-L. Kwong, “Silicon modulators and germanium photodetectors on SOI: monolithic integration, compatibility, and performance optimization,” IEEE J. Sel. Top. Quantum Electron. 16, 307–315 (2010).
[CrossRef]

Basile, P.

X. Wang, B. Howley, M. Y. Chen, Q. Zhou, R. Chen, and P. Basile, “Polymer based thermo-optic switch for optical true time delay,” Proc. SPIE 5728, 60–67 (2005).
[CrossRef]

Berroth, M.

W. Vogel and M. Berroth, “Liquid crystal phase shifter for optically generated RF-signals,” in Proceedings of IEEE Conference on 30th European Microwave Conference (IEEE, 2000), pp. 1–4.
[CrossRef]

Bui, L. A.

L. A. Bui, A. Mitchell, K. Ghnrbani, and T.-H. Chin, “Wideband RF photonic vector sum phase-shifter,” Electron. Lett. 39, 536–537 (2003).
[CrossRef]

Capmany, J.

Cassan, E.

Chalfant, C. H.

J. F. Coward, T. K. Yee, C. H. Chalfant, and P. H. Chang, “A photonic integrated-optic RF phase shifter for phased array antenna beam-forming applications,” J. Lightwave Technol. 11, 2201–2205 (1993).
[CrossRef]

Chang, P. H.

J. F. Coward, T. K. Yee, C. H. Chalfant, and P. H. Chang, “A photonic integrated-optic RF phase shifter for phased array antenna beam-forming applications,” J. Lightwave Technol. 11, 2201–2205 (1993).
[CrossRef]

Chang, Q.

Q. Chang, Q. Li, Z. Zhang, M. Qiu, and T. Ye, “A tunable broadband photonic RF phase shifter based on a silicon microring resonator,” IEEE Photon. Technol. Lett. 21, 60–62(2009).
[CrossRef]

Chen, M. Y.

X. Wang, B. Howley, M. Y. Chen, Q. Zhou, R. Chen, and P. Basile, “Polymer based thermo-optic switch for optical true time delay,” Proc. SPIE 5728, 60–67 (2005).
[CrossRef]

Chen, R.

X. Wang, B. Howley, M. Y. Chen, Q. Zhou, R. Chen, and P. Basile, “Polymer based thermo-optic switch for optical true time delay,” Proc. SPIE 5728, 60–67 (2005).
[CrossRef]

Chen, R. T.

H. S. Tang, L. Wu, Z. Fu, D. An, Z. Han, and R. T. Chen, “Polymer-based optical waveguide circuits for photonic phased-array antennas,” Proc. SPIE 3632, 250–261 (1999).
[CrossRef]

Chen, W.

Chin, T.-H.

L. A. Bui, A. Mitchell, K. Ghnrbani, and T.-H. Chin, “Wideband RF photonic vector sum phase-shifter,” Electron. Lett. 39, 536–537 (2003).
[CrossRef]

Choi, W.-Y.

Coward, J. F.

J. F. Coward, T. K. Yee, C. H. Chalfant, and P. H. Chang, “A photonic integrated-optic RF phase shifter for phased array antenna beam-forming applications,” J. Lightwave Technol. 11, 2201–2205 (1993).
[CrossRef]

Crozat, P.

Dai, D.

Damlencourt, J.-F.

Ding, Y.

Dong, W.

Fang, Q.

T.-Y. Liow, K.-W. Ang, Q. Fang, J.-F. Song, Y.-Z. Xiong, M.-B. Yu, G.-Q. Lo, and D.-L. Kwong, “Silicon modulators and germanium photodetectors on SOI: monolithic integration, compatibility, and performance optimization,” IEEE J. Sel. Top. Quantum Electron. 16, 307–315 (2010).
[CrossRef]

Fédéli, J.-M.

Fu, Z.

H. S. Tang, L. Wu, Z. Fu, D. An, Z. Han, and R. T. Chen, “Polymer-based optical waveguide circuits for photonic phased-array antennas,” Proc. SPIE 3632, 250–261 (1999).
[CrossRef]

Ghnrbani, K.

L. A. Bui, A. Mitchell, K. Ghnrbani, and T.-H. Chin, “Wideband RF photonic vector sum phase-shifter,” Electron. Lett. 39, 536–537 (2003).
[CrossRef]

Han, Z.

H. S. Tang, L. Wu, Z. Fu, D. An, Z. Han, and R. T. Chen, “Polymer-based optical waveguide circuits for photonic phased-array antennas,” Proc. SPIE 3632, 250–261 (1999).
[CrossRef]

He, S.

Howley, B.

X. Wang, B. Howley, M. Y. Chen, Q. Zhou, R. Chen, and P. Basile, “Polymer based thermo-optic switch for optical true time delay,” Proc. SPIE 5728, 60–67 (2005).
[CrossRef]

Huang, H.

J. Li, K. Xu, H. Huang, J. Wu, and J. Lin, “Photonic pulse generation and modulation for ultra-wideband-over-fiber applications,” in Proceedings of IEEE Conference on Optical Fiber Communication/National Fiber Optic Engineers Conference (IEEE, 2008), pp. 1–3.
[CrossRef] [PubMed]

Hunsperger, R. G.

R. G. Hunsperger, in Integrated Optics—Theory and Technology, 5th ed. (Springer, 2002), pp. 157–160.

Hvam, J. M.

Jhon, Y. M.

Jones, V. I.

J. J. Lee, R. Y. Loo, S. Livingston, V. I. Jones, J. B. Lewis, H. W. Yen, G. L. Tangonan, and M. Wechsberg, “Photonic wideband array antennas,” IEEE Trans. Antennas Propag. 43, 966–982(1995).
[CrossRef]

Kim, S. J.

S. J. Kim and N. H. Myung, “A new active phase shifter using a vector sum method,” IEEE Microwave Guided Wave Lett. 10, 233–235 (2000).
[CrossRef]

Kwong, D.-L.

T.-Y. Liow, K.-W. Ang, Q. Fang, J.-F. Song, Y.-Z. Xiong, M.-B. Yu, G.-Q. Lo, and D.-L. Kwong, “Silicon modulators and germanium photodetectors on SOI: monolithic integration, compatibility, and performance optimization,” IEEE J. Sel. Top. Quantum Electron. 16, 307–315 (2010).
[CrossRef]

Lava, S.

Lecunff, Y.

Lee, J. J.

J. J. Lee, R. Y. Loo, S. Livingston, V. I. Jones, J. B. Lewis, H. W. Yen, G. L. Tangonan, and M. Wechsberg, “Photonic wideband array antennas,” IEEE Trans. Antennas Propag. 43, 966–982(1995).
[CrossRef]

Lee, K.-H.

Lewis, J. B.

J. J. Lee, R. Y. Loo, S. Livingston, V. I. Jones, J. B. Lewis, H. W. Yen, G. L. Tangonan, and M. Wechsberg, “Photonic wideband array antennas,” IEEE Trans. Antennas Propag. 43, 966–982(1995).
[CrossRef]

Li, F.

Li, H.

Li, J.

J. Li, K. Xu, H. Huang, J. Wu, and J. Lin, “Photonic pulse generation and modulation for ultra-wideband-over-fiber applications,” in Proceedings of IEEE Conference on Optical Fiber Communication/National Fiber Optic Engineers Conference (IEEE, 2008), pp. 1–3.
[CrossRef] [PubMed]

Li, Q.

Q. Chang, Q. Li, Z. Zhang, M. Qiu, and T. Ye, “A tunable broadband photonic RF phase shifter based on a silicon microring resonator,” IEEE Photon. Technol. Lett. 21, 60–62(2009).
[CrossRef]

Lin, J.

J. Li, K. Xu, H. Huang, J. Wu, and J. Lin, “Photonic pulse generation and modulation for ultra-wideband-over-fiber applications,” in Proceedings of IEEE Conference on Optical Fiber Communication/National Fiber Optic Engineers Conference (IEEE, 2008), pp. 1–3.
[CrossRef] [PubMed]

Liow, T.-Y.

T.-Y. Liow, K.-W. Ang, Q. Fang, J.-F. Song, Y.-Z. Xiong, M.-B. Yu, G.-Q. Lo, and D.-L. Kwong, “Silicon modulators and germanium photodetectors on SOI: monolithic integration, compatibility, and performance optimization,” IEEE J. Sel. Top. Quantum Electron. 16, 307–315 (2010).
[CrossRef]

Liu, C.

Liu, L.

Livingston, S.

J. J. Lee, R. Y. Loo, S. Livingston, V. I. Jones, J. B. Lewis, H. W. Yen, G. L. Tangonan, and M. Wechsberg, “Photonic wideband array antennas,” IEEE Trans. Antennas Propag. 43, 966–982(1995).
[CrossRef]

Lo, G.-Q.

T.-Y. Liow, K.-W. Ang, Q. Fang, J.-F. Song, Y.-Z. Xiong, M.-B. Yu, G.-Q. Lo, and D.-L. Kwong, “Silicon modulators and germanium photodetectors on SOI: monolithic integration, compatibility, and performance optimization,” IEEE J. Sel. Top. Quantum Electron. 16, 307–315 (2010).
[CrossRef]

Loo, R. Y.

J. J. Lee, R. Y. Loo, S. Livingston, V. I. Jones, J. B. Lewis, H. W. Yen, G. L. Tangonan, and M. Wechsberg, “Photonic wideband array antennas,” IEEE Trans. Antennas Propag. 43, 966–982(1995).
[CrossRef]

Love, J. D.

A. W. Snyder and J. D. Love, Optical Waveguide Theory, 1st ed. (Chapman and Hall, 1983), pp. 209–218.

Marris-Morini, D.

Mitchell, A.

L. A. Bui, A. Mitchell, K. Ghnrbani, and T.-H. Chin, “Wideband RF photonic vector sum phase-shifter,” Electron. Lett. 39, 536–537 (2003).
[CrossRef]

Myung, N. H.

S. J. Kim and N. H. Myung, “A new active phase shifter using a vector sum method,” IEEE Microwave Guided Wave Lett. 10, 233–235 (2000).
[CrossRef]

Ortega, B.

Osmond, J.

Ou, H.

Pastor, D.

Pu, M.

Qiu, M.

Q. Chang, Q. Li, Z. Zhang, M. Qiu, and T. Ye, “A tunable broadband photonic RF phase shifter based on a silicon microring resonator,” IEEE Photon. Technol. Lett. 21, 60–62(2009).
[CrossRef]

Qu, P.

Ruan, S.

Shi, Y.

Snyder, A. W.

A. W. Snyder and J. D. Love, Optical Waveguide Theory, 1st ed. (Chapman and Hall, 1983), pp. 209–218.

Song, J.-F.

T.-Y. Liow, K.-W. Ang, Q. Fang, J.-F. Song, Y.-Z. Xiong, M.-B. Yu, G.-Q. Lo, and D.-L. Kwong, “Silicon modulators and germanium photodetectors on SOI: monolithic integration, compatibility, and performance optimization,” IEEE J. Sel. Top. Quantum Electron. 16, 307–315 (2010).
[CrossRef]

Srivastava, R.

Tang, H. S.

H. S. Tang, L. Wu, Z. Fu, D. An, Z. Han, and R. T. Chen, “Polymer-based optical waveguide circuits for photonic phased-array antennas,” Proc. SPIE 3632, 250–261 (1999).
[CrossRef]

Tangonan, G. L.

J. J. Lee, R. Y. Loo, S. Livingston, V. I. Jones, J. B. Lewis, H. W. Yen, G. L. Tangonan, and M. Wechsberg, “Photonic wideband array antennas,” IEEE Trans. Antennas Propag. 43, 966–982(1995).
[CrossRef]

Van Blaricum, M. L.

M. L. Van Blaricum, “Photonic systems for antenna applications,” IEEE Antennas Propag. Mag. 36, 30–38 (1994).
[CrossRef]

Vivien, L.

Vogel, W.

W. Vogel and M. Berroth, “Liquid crystal phase shifter for optically generated RF-signals,” in Proceedings of IEEE Conference on 30th European Microwave Conference (IEEE, 2000), pp. 1–4.
[CrossRef]

Wang, X.

X. Wang, B. Howley, M. Y. Chen, Q. Zhou, R. Chen, and P. Basile, “Polymer based thermo-optic switch for optical true time delay,” Proc. SPIE 5728, 60–67 (2005).
[CrossRef]

Wechsberg, M.

J. J. Lee, R. Y. Loo, S. Livingston, V. I. Jones, J. B. Lewis, H. W. Yen, G. L. Tangonan, and M. Wechsberg, “Photonic wideband array antennas,” IEEE Trans. Antennas Propag. 43, 966–982(1995).
[CrossRef]

Weiss, M. N.

Wu, J.

J. Li, K. Xu, H. Huang, J. Wu, and J. Lin, “Photonic pulse generation and modulation for ultra-wideband-over-fiber applications,” in Proceedings of IEEE Conference on Optical Fiber Communication/National Fiber Optic Engineers Conference (IEEE, 2008), pp. 1–3.
[CrossRef] [PubMed]

Wu, L.

H. S. Tang, L. Wu, Z. Fu, D. An, Z. Han, and R. T. Chen, “Polymer-based optical waveguide circuits for photonic phased-array antennas,” Proc. SPIE 3632, 250–261 (1999).
[CrossRef]

Xiong, Y.-Z.

T.-Y. Liow, K.-W. Ang, Q. Fang, J.-F. Song, Y.-Z. Xiong, M.-B. Yu, G.-Q. Lo, and D.-L. Kwong, “Silicon modulators and germanium photodetectors on SOI: monolithic integration, compatibility, and performance optimization,” IEEE J. Sel. Top. Quantum Electron. 16, 307–315 (2010).
[CrossRef]

Xu, K.

J. Li, K. Xu, H. Huang, J. Wu, and J. Lin, “Photonic pulse generation and modulation for ultra-wideband-over-fiber applications,” in Proceedings of IEEE Conference on Optical Fiber Communication/National Fiber Optic Engineers Conference (IEEE, 2008), pp. 1–3.
[CrossRef] [PubMed]

Xue, W.

Ye, T.

Q. Chang, Q. Li, Z. Zhang, M. Qiu, and T. Ye, “A tunable broadband photonic RF phase shifter based on a silicon microring resonator,” IEEE Photon. Technol. Lett. 21, 60–62(2009).
[CrossRef]

Yee, T. K.

J. F. Coward, T. K. Yee, C. H. Chalfant, and P. H. Chang, “A photonic integrated-optic RF phase shifter for phased array antenna beam-forming applications,” J. Lightwave Technol. 11, 2201–2205 (1993).
[CrossRef]

Yen, H. W.

J. J. Lee, R. Y. Loo, S. Livingston, V. I. Jones, J. B. Lewis, H. W. Yen, G. L. Tangonan, and M. Wechsberg, “Photonic wideband array antennas,” IEEE Trans. Antennas Propag. 43, 966–982(1995).
[CrossRef]

Yu, M.-B.

T.-Y. Liow, K.-W. Ang, Q. Fang, J.-F. Song, Y.-Z. Xiong, M.-B. Yu, G.-Q. Lo, and D.-L. Kwong, “Silicon modulators and germanium photodetectors on SOI: monolithic integration, compatibility, and performance optimization,” IEEE J. Sel. Top. Quantum Electron. 16, 307–315 (2010).
[CrossRef]

Yvind, K.

Zhang, Z.

Q. Chang, Q. Li, Z. Zhang, M. Qiu, and T. Ye, “A tunable broadband photonic RF phase shifter based on a silicon microring resonator,” IEEE Photon. Technol. Lett. 21, 60–62(2009).
[CrossRef]

Zhou, J.

Zhou, Q.

X. Wang, B. Howley, M. Y. Chen, Q. Zhou, R. Chen, and P. Basile, “Polymer based thermo-optic switch for optical true time delay,” Proc. SPIE 5728, 60–67 (2005).
[CrossRef]

Appl. Opt. (3)

Electron. Lett. (1)

L. A. Bui, A. Mitchell, K. Ghnrbani, and T.-H. Chin, “Wideband RF photonic vector sum phase-shifter,” Electron. Lett. 39, 536–537 (2003).
[CrossRef]

IEEE Antennas Propag. Mag. (1)

M. L. Van Blaricum, “Photonic systems for antenna applications,” IEEE Antennas Propag. Mag. 36, 30–38 (1994).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron. (1)

T.-Y. Liow, K.-W. Ang, Q. Fang, J.-F. Song, Y.-Z. Xiong, M.-B. Yu, G.-Q. Lo, and D.-L. Kwong, “Silicon modulators and germanium photodetectors on SOI: monolithic integration, compatibility, and performance optimization,” IEEE J. Sel. Top. Quantum Electron. 16, 307–315 (2010).
[CrossRef]

IEEE Microwave Guided Wave Lett. (1)

S. J. Kim and N. H. Myung, “A new active phase shifter using a vector sum method,” IEEE Microwave Guided Wave Lett. 10, 233–235 (2000).
[CrossRef]

IEEE Photon. Technol. Lett. (1)

Q. Chang, Q. Li, Z. Zhang, M. Qiu, and T. Ye, “A tunable broadband photonic RF phase shifter based on a silicon microring resonator,” IEEE Photon. Technol. Lett. 21, 60–62(2009).
[CrossRef]

IEEE Trans. Antennas Propag. (1)

J. J. Lee, R. Y. Loo, S. Livingston, V. I. Jones, J. B. Lewis, H. W. Yen, G. L. Tangonan, and M. Wechsberg, “Photonic wideband array antennas,” IEEE Trans. Antennas Propag. 43, 966–982(1995).
[CrossRef]

J. Lightwave Technol. (2)

J. F. Coward, T. K. Yee, C. H. Chalfant, and P. H. Chang, “A photonic integrated-optic RF phase shifter for phased array antenna beam-forming applications,” J. Lightwave Technol. 11, 2201–2205 (1993).
[CrossRef]

J. Capmany, B. Ortega, and D. Pastor, “A tutorial on microwave photonic filters,” J. Lightwave Technol. 24, 201–229(2006).
[CrossRef]

Opt. Express (2)

Opt. Lett. (1)

Proc. SPIE (2)

H. S. Tang, L. Wu, Z. Fu, D. An, Z. Han, and R. T. Chen, “Polymer-based optical waveguide circuits for photonic phased-array antennas,” Proc. SPIE 3632, 250–261 (1999).
[CrossRef]

X. Wang, B. Howley, M. Y. Chen, Q. Zhou, R. Chen, and P. Basile, “Polymer based thermo-optic switch for optical true time delay,” Proc. SPIE 5728, 60–67 (2005).
[CrossRef]

Other (4)

A. W. Snyder and J. D. Love, Optical Waveguide Theory, 1st ed. (Chapman and Hall, 1983), pp. 209–218.

J. Li, K. Xu, H. Huang, J. Wu, and J. Lin, “Photonic pulse generation and modulation for ultra-wideband-over-fiber applications,” in Proceedings of IEEE Conference on Optical Fiber Communication/National Fiber Optic Engineers Conference (IEEE, 2008), pp. 1–3.
[CrossRef] [PubMed]

R. G. Hunsperger, in Integrated Optics—Theory and Technology, 5th ed. (Springer, 2002), pp. 157–160.

W. Vogel and M. Berroth, “Liquid crystal phase shifter for optically generated RF-signals,” in Proceedings of IEEE Conference on 30th European Microwave Conference (IEEE, 2000), pp. 1–4.
[CrossRef]

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Figures (9)

Fig. 1
Fig. 1

(a) Schematic of a traditional bulk MWPPS. (b) New configuration of the IMWPPS based on orthogonal VSM (not in scale).

Fig. 2
Fig. 2

MW phase response of the VSM-based IMWPPS. The phase shift from 0 to 2 π of our device can be achieved by a four-step operation.

Fig. 3
Fig. 3

Amplitude response function of time and Δ ϕ c for the case (a) with MOCMs and (b) without MOCMs.

Fig. 4
Fig. 4

(a) Amplitude response and (b) MW phase response at different operating frequency and Δ ϕ c for a 40 GHz IMWPPS.

Fig. 5
Fig. 5

Relationship between the MW phase shift and the refractive index variation (also of temperature variation) with different lengths of PMR.

Fig. 6
Fig. 6

(a) The high mode conversion efficiency between “zeroth and first” and “zeroth and second” modes can be achieved at a suitable WG width. A suitable distance between the two WGs of the ADC can relax the requirement for the phase synchronization. (b) Relationship between ERI of the concerned modes and the WG width.

Fig. 7
Fig. 7

Simulated optical propagation profiles of (a) zeroth to first MOCM and (b) zeroth to second MOCM.

Fig. 8
Fig. 8

Detail layout of the two MOCMs and MMWGs in which the combination of different modes can realize the fixed time delay mentioned in Subsection 2B.

Fig. 9
Fig. 9

(a) Relationship between the group velocities of concerned modes and the WG width. (b) Relationship between the lengths and different operating MW frequency.

Tables (2)

Tables Icon

Table 1 Performances of Zeroth to First and Zeroth to Second MOCMs

Tables Icon

Table 2 Propagation Characteristics of mth Mode Input from the Lower WG of the MOCMs ( m < p )

Equations (17)

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E in = e in ( x , y ) A ( ω r f , t in ) cos ( ω c t + α in ) ,
E 0 = e 0 ( x , y ) r A ( ω r f , t 0 ) cos ( ω c t + α 0 ) ,
E 1 = e 1 ( x , y ) r A ( ω r f , t 1 ) cos ( ω c t + α 1 ) ,
E 2 = e 2 ( x , y ) s A ( ω r f , t 2 ) cos ( ω c t + α 2 ) ,
E tot = E 0 + E 2 or E tot = E 1 + E 2 .
I out P out = core S ( x , y ) d x d y = 1 2 core Re [ E tot × H tot * z ] d x d y = | r | 2 A 2 ( ω r f , t n ) core 1 2 e n × h n * z d x d y + | s | 2 A ( ω r f , t p ) core 1 2 e p × h p * z d x d y + r s cos ( α n α p ) A ( ω r f , t n ) A ( ω r f , t p ) core 1 2 e n × h p * z d x d y + r s cos ( α n α p ) A ( ω r f , t n ) A 2 ( ω r f , t p ) core 1 2 e p × h n * z d x d y ,
I out ( n p ) | r | 2 A 2 ( ω r f , t n ) + χ | s | 2 A 2 ( ω r f , t p ) .
I out ( n = p ) [ | r | 2 A 2 ( ω r f , t n ) + | s | 2 A 2 ( ω r f , t p ) + 2 | r s | cos ( α 1 α 2 ) A ( ω r f , t n ) A ( ω r f , t p ) ] .
I out ( SIM ) R + χ S + Γ m ( R + χ S ) 2 4 χ R S sin 2 ( Δ ϕ r f / 2 ) sin ( ω r f t n + Φ SIM ) ,
Φ SIM = tan 1 ( sin ( Δ ϕ r f ) R / ( χ S ) + cos ( Δ ϕ r f ) ) .
[ a r ( z ) a s ( z ) ] = [ cos ( κ z ) + j δ κ sin ( κ z ) , j κ κ sin ( κ z ) j κ κ sin ( κ z ) , cos ( κ z ) + j δ κ sin ( κ z ) ] [ a r ( 0 ) a s ( 0 ) ] ,
{ K 12 = ω ε 0 4 S 2 ( n 2 2 n 3 2 ) E 1 y * ( x , y ) E 2 y ( x , y ) d x d y K 21 = ω ε 0 4 S 1 ( n 1 2 n 3 2 ) E 2 y * ( x , y ) E 1 y ( x , y ) d x d y ,
η | a s ( L C ) a r ( 0 ) | 2 = 1 | δ κ | 2 + 1 .
{ MPCE = p out p in MSMF = + E conversion ( x ) E standard ( x ) d x + | E conversion ( x ) | 2 d x + | E standard ( x ) | 2 d x PCF = WG   section | E conversion ( x ) | 2 d x + | E conversion ( x ) | 2 d x ,
ω r f ( L 1 v g 00 + L 2 v g 22 L 1 v g 11 L 2 v g 12 ) = π 2 .
ω r f ( L 1 v g 00 + L 2 v g 22 L 1 v g 01 L 2 v g 02 ) = 3 π 2 ,
v g = c / n g = c / [ n eff λ · ( d n eff / d λ ) ] ,

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