Photonic integrated cmos-compatible true time delay based broadband beamformer

Abstract

The evolution to 5G millimeter wave (mmWave) is opening doors to many novel applications and services by leveraging the higher carrier frequency of 24 GHz and beyond. 5G mmWave promises to deliver extreme bandwidth, ultra-low latency, faster data speeds and responsiveness – thereby facilitating broadband connectivity to densely populated areas. Nevertheless, millimeter waves suffer from significant path loss, imposing severe range constraints. Beamforming is a promising technique to alleviate this impediment. Integrated photonics is sophisticated technology to realize cost-effective, compact, and power-efficient beamformers. In this paper, we propose photonically controlled continuously tunable single side coupled racetrack resonator based 1 × 8 broadband beamformer on CMOS-compatible silicon-on-insulator platform. A beam steering angle of 31.001˚ is achieved for 28 GHz mmWave wave signal. The proposed beamformer can provide maximum continuously tunable true time delay of 125 ps for delay bandwidth of 32.46 GHz. Therefore, the beam scanning range of 0–90° can be attained by properly tuning the coupling coefficient and resonance frequency of the racetrack resonator based delay element.

Appendix: Mathematical analysis of proposed photonic beamformer

The electric field at the through port of directional coupler1 (DC1) is given by

$$ E_{t}^{DC1} = E_{in} e^{{j\left( {\frac{{\beta_{1}^{e} + \beta_{1}^{o} }}{2}} \right)L_{c} }} \sin \left( {\frac{{\pi \Delta n_{1} L_{c} }}{\lambda }} \right) $$

(19)

where $E_{in}$ is input electric field, $\beta_{1}^{e}$ and $\beta_{1}^{o}$ are propagation constants of even and odd normal modes of DC1, respectively, and $\Delta n_{1} = n_{eff}^{e1} - n_{eff}^{o1}$ is index difference between the even and odd supermodes of DC1.

The electric field at the output of RTR4 of Stage 1 can be expressed as

$$ E_{R4}^{S1} = \frac{{e^{{j3\theta_{a} }} }}{{X_{1} }}\left\{ \begin{gathered} t_{1} t_{2} t_{3} t_{4} e^{{j4\theta_{cp} }} - \gamma e^{{j5\theta_{cp} }} e^{{j\theta_{t} }} \left( {t_{1} t_{2} t_{3} + t_{1} t_{2} t_{4} + t_{1} t_{3} t_{4} + t_{2} t_{3} t_{4} } \right) \hfill \\ + \gamma^{2} e^{{j6\theta_{cp} }} e^{{j2\theta_{t} }} \left( {t_{1} t_{2} + t_{1} t_{3} + t_{1} t_{4} + t_{2} t_{3} + t_{2} t_{4} + t_{3} t_{4} } \right) \hfill \\ - \gamma^{3} e^{{j7\theta_{cp} }} e^{{j3\theta_{t} }} \left( {t_{1} + t_{2} + t_{3} + t_{4} } \right) + \gamma^{4} e^{{j8\theta_{cp} }} e^{{j4\theta_{t} }} \hfill \\ \end{gathered} \right\}E_{t}^{DC1} $$

(20)

$$ X_{1} = \left( {1 - \gamma t_{1}^{ * } e^{{j\theta_{cp} }} e^{{j\theta_{t} }} } \right)\left( {1 - \gamma t_{2}^{ * } e^{{j\theta_{cp} }} e^{{j\theta_{t} }} } \right)\left( {1 - \gamma t_{3}^{ * } e^{{j\theta_{cp} }} e^{{j\theta_{t} }} } \right)\left( {1 - \gamma t_{4}^{ * } e^{{j\theta_{cp} }} e^{{j\theta_{t} }} } \right) $$

where $\theta_{cp}$ is phase delay due to coupling region, $\theta_{t}$ is phase delay due to optical path length of RTR, $\theta_{a}$ is phase delay due to waveguide segment between two cascaded RTR, coupling coefficient $t_{i} = \sqrt {1 - \kappa_{i}^{2} } ,\,i = 1\,{\text{to}}\,4$ represents RTR and $*$ denotes complex conjugate.

The electric field at the through port of directional coupler2 (DC2) is given by

$$ E_{t}^{DC2} = E_{R4}^{S1} e^{{j\left( {\frac{{\beta_{2}^{e} + \beta_{2}^{o} }}{2}} \right)L_{c} }} \sin \left( {\frac{{\pi \Delta n_{2} L_{c} }}{\lambda }} \right) $$

(21)

where $\beta_{2}^{e}$ and $\beta_{2}^{o}$ are propagation constant of even and odd normal modes of DC2, respectively, and $\Delta n_{2} = n_{eff}^{e2} - n_{eff}^{o2}$ is index difference between the even and odd supermodes of DC2.

The electric field at the output of RTR6 of Stage 2 can be expressed as

$$ E_{R6}^{S2} = \frac{{e^{{j2\theta_{cp} }} e^{{j\theta_{a} }} \left[ {t_{5} t_{6} - \gamma e^{{j\theta_{cp} }} e^{{j\theta_{t} }} \left( {t_{5} + t_{6} } \right) + \gamma^{2} e^{{j2\theta_{cp} }} e^{{j2\theta_{t} }} } \right]E_{t}^{DC2} }}{{\left( {1 - \gamma t_{5}^{ * } e^{{j\theta_{cp} }} e^{{j\theta_{t} }} } \right)\left( {1 - \gamma t_{6}^{ * } e^{{j\theta_{cp} }} e^{{j\theta_{t} }} } \right)}} $$

(22)

where $t_{5} = \sqrt {1 - \kappa_{5}^{2} }$ and $t_{6} = \sqrt {1 - \kappa_{6}^{2} }$ are coupling coefficients of RTR5 and RTR6, respectively.

Substituting Eqs. (20) and (21) in Eq. (22), we get

$$ E_{R6}^{S2} = \frac{{e^{{j6\theta_{cp} }} e^{{j4\theta_{a} }} }}{{X_{2} }}\left\{ \begin{gathered} t_{1} t_{2} t_{3} t_{4} t_{5} t_{6} - \gamma e^{{j\theta_{cp} }} e^{{j\theta_{t} }} \left[ {t_{1} t_{2} t_{3} t_{4} \left( {t_{5} + t_{6} } \right) + t_{5} t_{6} \left( {t_{1} t_{2} \left[ {t_{3} + t_{4} } \right] + t_{3} t_{4} \left[ {t_{1} + t_{2} } \right]} \right)} \right] \hfill \\ + \gamma^{2} e^{{j2\theta_{cp} }} e^{{j2\theta_{t} }} \left[ \begin{gathered} t_{1} t_{2} t_{3} t_{4} + \left( {t_{5} + t_{6} } \right)\left( {t_{1} t_{2} \left[ {t_{3} + t_{4} } \right] + t_{3} t_{4} \left[ {t_{1} + t_{2} } \right]} \right) \hfill \\ + t_{5} t_{6} \left( {t_{1} \left[ {t_{2} + t_{3} + t_{4} } \right] + t_{2} \left[ {t_{3} + t_{4} } \right] + t_{3} t_{4} } \right) \hfill \\ \end{gathered} \right] \hfill \\ - \gamma^{3} e^{{j3\theta_{cp} }} e^{{j3\theta_{t} }} \left[ \begin{gathered} t_{1} t_{2} \left( {t_{3} + t_{4} } \right) + t_{3} t_{4} \left( {t_{1} + t_{2} } \right) + \left( {t_{5} + t_{6} } \right)\left( {t_{1} \left[ {t_{2} + t_{3} + t_{4} } \right] + t_{2} \left[ {t_{3} + t_{4} } \right] + t_{3} t_{4} } \right) \hfill \\ + t_{5} t_{6} \left( {t_{1} + t_{2} + t_{3} + t_{4} } \right) \hfill \\ \end{gathered} \right] \hfill \\ + \gamma^{4} e^{{j4\theta_{cp} }} e^{{j4\theta_{t} }} \left[ {t_{1} \left( {t_{2} + t_{3} + t_{4} } \right) + t_{2} \left( {t_{3} + t_{4} } \right) + t_{3} t_{4} + \left( {t_{5} + t_{6} } \right)\left( {t_{1} + t_{2} + t_{3} + t_{4} } \right) + t_{5} t_{6} } \right] \hfill \\ - \gamma^{5} e^{{j5\theta_{cp} }} e^{{j5\theta_{t} }} \left( {t_{1} + t_{2} + t_{3} + t_{4} + t_{5} + t_{6} } \right) + \gamma^{6} e^{{j6\theta_{cp} }} e^{{j6\theta_{t} }} \hfill \\ \end{gathered} \right\}Y_{1} $$

(23)

where $X_{2} = X_{1} \left( {1 - \gamma t_{5}^{ * } e^{{j\theta_{cp} }} e^{{j\theta_{t} }} } \right)\left( {1 - \gamma t_{6}^{ * } e^{{j\theta_{cp} }} e^{{j\theta_{t} }} } \right),\,Y_{1} = E_{t}^{DC1} e^{{j\left( {\frac{{\beta_{2}^{e} + \beta_{2}^{o} }}{2}} \right)L_{c} }} \sin \left( {\frac{{\pi \Delta n_{2} L_{c} }}{\lambda }} \right)$.

The electric field at the through port of directional coupler4 (DC4) is given by

$$ E_{t}^{DC4} = E_{R6}^{S2} e^{{j\left( {\frac{{\beta_{4}^{e} + \beta_{4}^{o} }}{2}} \right)L_{c} }} \sin \left( {\frac{{\pi \Delta n_{4} L_{c} }}{\lambda }} \right) $$

(24)

where $\beta_{4}^{e}$ and $\beta_{4}^{o}$ are propagation constants of even and odd normal modes of DC4, respectively, and $\Delta n_{4} = n_{eff}^{e4} - n_{eff}^{o4}$ is index difference between the even and odd supermodes of DC4.

The electric field at the output of Path 7 can be expressed as

$$ E_{o}^{P7} = e^{{i\theta_{cp} }} \left[ {\frac{{t_{9} - \gamma e^{{j\theta_{cp} }} e^{j\theta t} }}{{1 - \gamma t_{9}^{ * } e^{{j\theta_{cp} }} e^{j\theta t} }}} \right]E_{t}^{DC4} $$

(25)

where $t_{9} = \sqrt {1 - \kappa_{9}^{2} }$ is coupling coefficient of RTR9.

Substituting Eqs. (23) and (24) in Eq. (25), we get

$$ E_{o}^{P7} = \frac{{e^{{j7\theta_{cp} }} e^{{j4\theta_{a} }} }}{{X_{3} }}\left[ \begin{gathered} t_{1} t_{2} t_{3} t_{4} t_{5} t_{6} t_{9} - \gamma e^{{j\theta_{cp} }} e^{{j\theta_{t} }} Z_{1} + \gamma^{2} e^{{j2\theta_{cp} }} e^{{j2\theta_{t} }} Z_{2} - \gamma^{3} e^{{j3\theta_{cp} }} e^{{j3\theta_{t} }} Z_{3} \hfill \\ + \gamma^{4} e^{{j4\theta_{cp} }} e^{{j4\theta_{t} }} Z_{4} - \gamma^{5} e^{{j5\theta_{cp} }} e^{{j5\theta_{t} }} Z_{5} + \gamma^{6} e^{{j6\theta_{cp} }} e^{{j6\theta_{t} }} Z_{6} \hfill \\ - \gamma^{9} e^{{j9\theta_{cp} }} e^{{j9\theta_{t} }} \hfill \\ \end{gathered} \right]Y_{2} $$

(26)

where $Z_{1} = t_{1} t_{2} t_{3} t_{4} t_{9} \left( {t_{5} + t_{6} + t_{5} t_{6} } \right) + t_{4} t_{5} t_{6} t_{9} \left( {t_{1} t_{2} + t_{1} t_{3} + t_{2} t_{3} } \right) + t_{1} t_{2} t_{3} t_{5} t_{6} \left( {t_{4} + t_{9} } \right)$

$$ \begin{gathered} Z_{2} = t_{1} t_{2} t_{3} t_{9} \left( {t_{4} + t_{5} + t_{6} } \right) + t_{4} t_{9} \left( {t_{1} t_{2} + t_{1} t_{3} + t_{2} t_{3} } \right)\left( {t_{5} + t_{6} } \right)\, + t_{5} t_{6} t_{9} \left( {t_{1} t_{2} + t_{1} t_{3} + t_{1} t_{4} + t_{2} t_{3} + t_{2} t_{4} + t_{3} t_{4} } \right) \hfill \\ \,\,\,\,\,\,\,\,\,\, + t_{1} t_{2} t_{3} t_{4} \left( {t_{5} + t_{6} } \right) + t_{4} t_{5} t_{6} \left( {t_{1} t_{2} + t_{1} t_{3} + t_{2} t_{3} } \right) + t_{1} t_{2} t_{3} t_{5} t_{6} \hfill \\ \end{gathered} $$

$$ \begin{gathered} Z_{3} = t_{1} t_{2} \left( {t_{9} \left[ {t_{3} + t_{4} + t_{5} + t_{6} } \right] + t_{3} \left[ {t_{4} + t_{5} + t_{6} } \right]} \right) + t_{9} \left( {t_{1} t_{3} + t_{2} t_{3} } \right)\left( {t_{4} + t_{5} + t_{6} } \right) + t_{4} t_{9} \left( {t_{1} + t_{2} + t_{3} } \right)\left( {t_{5} + t_{6} } \right) \hfill \\ \,\,\,\,\,\,\,\,\, + t_{4} \left( {t_{5} + t_{6} } \right)\left( {t_{1} t_{2} + t_{1} t_{3} + t_{2} t_{3} } \right) + t_{5} t_{6} t_{9} \left( {t_{1} + t_{2} + t_{3} + t_{4} } \right) + t_{5} t_{6} \left( {t_{1} t_{2} + t_{1} t_{3} + t_{1} t_{4} + t_{2} t_{3} + t_{2} t_{4} + t_{3} t_{4} } \right) \hfill \\ \,\,\,\,\,\,\,\,\, + t_{1} t_{2} t_{3} t_{4} \hfill \\ \end{gathered} $$

$$ \begin{gathered} Z_{4} = t_{1} t_{9} \left( {t_{2} + t_{3} + t_{4} + t_{5} + t_{6} } \right) + t_{2} \left( {t_{1} + t_{9} } \right)\left( {t_{3} + t_{4} + t_{5} + t_{6} } \right) + t_{3} \left( {t_{1} + t_{2} + t_{9} } \right)\left( {t_{4} + t_{5} + t_{6} } \right) \hfill \\ \,\,\,\,\,\,\,\,\, + t_{4} \left( {t_{5} + t_{6} } \right)\left( {t_{1} + t_{2} + t_{3} + t_{9} } \right) + t_{5} t_{6} \left( {t_{1} + t_{2} + t_{3} + t_{4} + t_{9} } \right) \hfill \\ \end{gathered} $$

$$ \begin{gathered} Z_{5} = t_{9} \left( {t_{1} + t_{2} + t_{3} + t_{4} + t_{5} + t_{6} } \right) + t_{1} \left( {t_{2} + t_{3} + t_{4} + t_{5} + t_{6} } \right) + t_{2} \left( {t_{3} + t_{4} + t_{5} + t_{6} } \right) + t_{3} \left( {t_{4} + t_{5} + t_{6} } \right) \hfill \\ \,\,\,\,\,\,\,\,\, + t_{4} \left( {t_{5} + t_{6} } \right) + t_{5} t_{6} \hfill \\ \end{gathered} $$

$$ Z_{6} = t_{1} + t_{2} + t_{3} + t_{4} + t_{5} + t_{6} ,\,X_{3} = X_{2} \left( {1 - \gamma t_{9}^{ * } e^{{j\theta_{cp} }} e^{{j\theta_{t} }} } \right),\,Y_{2} = Y_{1} e^{{j\left( {\frac{{\beta_{4}^{e} - \beta_{4}^{o} }}{2}} \right)L_{c} }} \sin \left( {\frac{{\pi \Delta n_{4} L_{c} }}{\lambda }} \right) $$

The electric field at the cross port of directional coupler4 (DC4) is given by

$$ E_{c}^{DC4} = E_{R6}^{S2} e^{{j\left( {\frac{{\beta_{4}^{e} + \beta_{4}^{o} }}{2}} \right)L_{c} }} \cos \left( {\frac{{\pi \Delta n_{4} L_{c} }}{\lambda }} \right) $$

(27)

The electric field at the output of Path 6 can be expressed as

$$ E_{o}^{P6} = E_{c}^{DC4} e^{{j\beta^{s} L^{S3} }} e^{{ - \alpha_{prop} L^{S3} }} $$

(28)

$$ E_{o}^{P6} = E_{R6}^{S2} e^{{j\left( {\frac{{\beta_{4}^{e} + \beta_{4}^{o} }}{2}} \right)L_{c} }} \cos \left( {\frac{{\pi \Delta n_{4} L_{c} }}{\lambda }} \right)e^{{j\beta^{s} L^{S3} }} e^{{ - \alpha_{prop} L^{S3} }} $$

(29)

where $L^{S3}$ is the length of straight waveguide employed in stage 3 and $\alpha_{prop}$ is waveguide attenuation coefficient.

The electric field at the cross port of directional coupler2 (DC2) is given by

$$ E_{c}^{DC2} = E_{R6}^{S1} e^{{j\left( {\frac{{\beta_{2}^{e} + \beta_{2}^{o} }}{2}} \right)L_{c} }} \cos \left( {\frac{{\pi \Delta n_{2} L_{c} }}{\lambda }} \right) $$

(30)

The electric field at the through port of directional coupler5 (DC5) is given by

$$ E_{t}^{DC5} = E_{R4}^{S1} e^{{j\left( {\frac{{\beta_{2}^{e} + \beta_{2}^{o} }}{2}} \right)L_{c} }} e^{{j\left( {\frac{{\beta_{5}^{e} + \beta_{5}^{o} }}{2}} \right)L_{c} }} \cos \left( {\frac{{\pi \Delta n_{2} L_{c} }}{\lambda }} \right)\sin \left( {\frac{{\pi \Delta n_{5} L_{c} }}{\lambda }} \right)e^{{j\beta^{s} L^{S2} }} e^{{ - \alpha_{prop} L^{S2} }} $$

(31)

where $\beta_{5}^{e}$ and $\beta_{5}^{o}$ are propagation constants of even and odd normal modes of DC5, respectively, $\Delta n_{5} = n_{eff}^{e5} - n_{eff}^{o5}$ is index difference between the even and odd supermodes of DC5, $L^{S2}$ is the length of straight waveguide employed in stage 2.

The electric field at the output of Path5 can be expressed as

$$ E_{o}^{P5} = e^{{i\theta_{cp} }} \left[ {\frac{{t_{10} - \gamma e^{{j\theta_{cp} }} e^{j\theta t} }}{{1 - \gamma t_{10}^{ * } e^{{j\theta_{cp} }} e^{j\theta t} }}} \right]E_{t}^{DC5} $$

(32)

where $t_{10} = \sqrt {1 - \kappa_{10}^{2} }$ is coupling coefficient of RTR10.

Substituting Eqs. (20) and (31) in Eq. (32), we obtain

$$ E_{o}^{P5} = \frac{{e^{{i5\theta_{cp} }} e^{{i3\theta_{a} }} }}{{X_{4} }}\left[ \begin{gathered} t_{1} t_{2} t_{3} t_{4} t_{10} - \gamma e^{{j\theta_{cp} }} e^{{j\theta_{t} }} \left[ {t_{10} \left( {t_{1} t_{2} \left[ {t_{3} + t_{4} } \right] + t_{3} t_{4} \left[ {t_{1} + t_{2} } \right]} \right) + t_{1} t_{2} t_{3} t_{4} } \right] \hfill \\ + \gamma^{2} e^{{j2\theta_{cp} }} e^{{j2\theta_{t} }} \left[ \begin{gathered} t_{1} t_{2} \left( {t_{3} + t_{4} + t_{10} } \right) + t_{3} \left( {t_{1} + t_{2} } \right)\left( {t_{4} + t_{10} } \right) \hfill \\ + t_{4} t_{10} \left( {t_{1} + t_{2} + t_{3} } \right) \hfill \\ \end{gathered} \right] \hfill \\ - \gamma^{3} e^{{j3\theta_{cp} }} e^{{j3\theta_{t} }} \left[ \begin{gathered} t_{1} \left( {t_{2} + t_{3} + t_{4} + t_{10} } \right) + t_{2} \left( {t_{2} + t_{3} + t_{10} } \right) \hfill \\ + t_{3} \left( {t_{4} + t_{10} } \right) + t_{4} t_{10} \hfill \\ \end{gathered} \right] \hfill \\ + \gamma^{4} e^{{j4\theta_{cp} }} e^{{j4\theta_{t} }} \left[ {t_{1} + t_{2} + t_{3} + t_{4} + t_{10} } \right] - \gamma^{5} e^{{j5\theta_{cp} }} e^{{j5\theta_{t} }} \hfill \\ \end{gathered} \right]Y_{3} $$

(33)

where $Y_{3} = E_{t}^{DC1} e^{{j\left( {\frac{{\beta_{2}^{e} + \beta_{2}^{o} }}{2}} \right)L_{c} }} e^{{j\left( {\frac{{\beta_{5}^{e} + \beta_{5}^{o} }}{2}} \right)L_{c} }} \cos \left( {\frac{{\pi \Delta n_{2} L_{c} }}{\lambda }} \right)\sin \left( {\frac{{\pi \Delta n_{5} L_{c} }}{\lambda }} \right)e^{{j\beta^{s} L^{S2} }} e^{{ - \alpha_{prop} L^{S2} }} ,$

$$ X_{4} = X_{1} \left( {1 - \gamma t_{10}^{ * } e^{{j\theta_{cp} }} e^{{j\theta_{t} }} } \right) $$

The electric field at the output of Path4 can be expressed as

$$ \begin{gathered} E_{o}^{P4} = E_{R4}^{S1} e^{{j\left( {\frac{{\beta_{2}^{e} + \beta_{2}^{o} }}{2}} \right)L_{c} }} e^{{j\left( {\frac{{\beta_{5}^{e} + \beta_{5}^{o} }}{2}} \right)L_{c} }} \cos \left( {\frac{{\pi \Delta n_{2} L_{c} }}{\lambda }} \right)\sin \left( {\frac{{\pi \Delta n_{5} L_{c} }}{\lambda }} \right) \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,e^{{j\beta^{s} L^{S2} }} e^{{ - \alpha_{prop} L^{S2} }} e^{{j\beta^{s} L^{S3} }} e^{{ - \alpha_{prop} L^{S3} }} \hfill \\ \end{gathered} $$

(34)

The electric field at the cross port of directional coupler1 (DC1) is given by

$$ E_{t}^{DC1} = E_{R6}^{S1} e^{{j\left( {\frac{{\beta_{1}^{e} + \beta_{1}^{o} }}{2}} \right)L_{c} }} \cos \left( {\frac{{\pi \Delta n_{1} L_{c} }}{\lambda }} \right) $$

(35)

The electric field at the through port of directional coupler3 (DC3) is given by

$$ E_{t}^{DC3} = E_{t}^{DC1} e^{{j\left( {\frac{{\beta_{3}^{e} + \beta_{3}^{o} }}{2}} \right)L_{c} }} \sin \left( {\frac{{\pi \Delta n_{3} L_{c} }}{\lambda }} \right)e^{{j\beta^{s} L^{S1} }} e^{{ - \alpha_{prop} L^{S1} }} $$

(36)

where $L^{S1}$ is the length of straight waveguide employed in stage 1.

The electric field at the output of RTR8 of Stage 2 can be expressed as

$$ E_{R8}^{S2} = \frac{{e^{{j2\theta_{cp} }} e^{{j\theta_{a} }} \left[ {t_{7} t_{8} - \gamma e^{{j\theta_{cp} }} e^{{j\theta_{t} }} \left( {t_{7} + t_{8} } \right) + \gamma^{2} e^{{j2\theta_{cp} }} e^{{j2\theta_{t} }} } \right]E_{t}^{DC3} }}{{\left( {1 - \gamma t_{7}^{ * } e^{{j\theta_{cp} }} e^{{j\theta_{t} }} } \right)\left( {1 - \gamma t_{8}^{ * } e^{{j\theta_{cp} }} e^{{j\theta_{t} }} } \right)}} $$

(37)

where $t_{7} = \sqrt {1 - \kappa_{7}^{2} }$ and $t_{8} = \sqrt {1 - \kappa_{8}^{2} }$ are coupling coefficients of RTR7 and RTR8, respectively.

The electric field at the through port of directional coupler6 (DC6) is given by

$$ E_{t}^{DC6} = E_{R8}^{S2} e^{{j\left( {\frac{{\beta_{6}^{e} + \beta_{6}^{o} }}{2}} \right)L_{c} }} \sin \left( {\frac{{\pi \Delta n_{6} L_{c} }}{\lambda }} \right) $$

(38)

where $\beta_{6}^{e}$ and $\beta_{6}^{o}$ are propagation constants of even and odd normal modes of DC6, respectively, and $\Delta n_{6} = n_{eff}^{e6} - n_{eff}^{o6}$ is index difference between the even and odd supermodes of DC6.

The electric field at the output of Path3 can be expressed as

$$ E_{o}^{P3} = e^{{i\theta_{cp} }} \left[ {\frac{{t_{11} - \gamma e^{{j\theta_{cp} }} e^{j\theta t} }}{{1 - \gamma t_{11}^{ * } e^{{j\theta_{cp} }} e^{j\theta t} }}} \right]E_{t}^{DC6} $$

(39)

where $t_{11} = \sqrt {1 - \kappa_{11}^{2} }$ is coupling coefficient of RTR11.

Upon substitutions and simplifications of Eqs. (35), (36), (37) and (38) in Eq. (39), we get

$$ E_{o}^{P3} = \frac{{e^{{j3\theta_{cp} }} e^{{j\theta_{a} }} }}{{X_{5} \left( {1 - \gamma t_{11}^{ * } e^{{j\theta_{cp} }} e^{{j\theta_{t} }} } \right)}}\left[ \begin{gathered} t_{7} t_{8} t_{11} - \gamma e^{{j\theta_{cp} }} e^{{j\theta_{t} }} \left[ {t_{11} \left( {t_{7} + t_{8} } \right) + t_{7} t_{8} } \right] \hfill \\ + \gamma^{2} e^{{j2\theta_{cp} }} e^{{j2\theta_{t} }} \left[ {t_{7} + t_{8} + t_{11} } \right] - \gamma^{3} e^{{j3\theta_{cp} }} e^{{j3\theta_{t} }} \hfill \\ \end{gathered} \right]Y_{4} $$

(40)

where $Y_{4} = E_{t}^{DC1} e^{{j\left( {\frac{{\beta_{3}^{e} + \beta_{3}^{o} }}{2}} \right)L_{c} }} e^{{j\left( {\frac{{\beta_{6}^{e} + \beta_{6}^{o} }}{2}} \right)L_{c} }} \sin \left( {\frac{{\pi \Delta n_{3} L_{c} }}{\lambda }} \right)\sin \left( {\frac{{\pi \Delta n_{6} L_{c} }}{\lambda }} \right)e^{{j\beta^{s} L^{S1} }} e^{{ - \alpha_{prop} L^{S1} }} ,$

$$ X_{5} = \left( {1 - \gamma t_{7}^{ * } e^{{j\theta_{cp} }} e^{{j\theta_{t} }} } \right)\left( {1 - \gamma t_{8}^{ * } e^{{j\theta_{cp} }} e^{{j\theta_{t} }} } \right) $$

The electric field at the output of Path2 can be expressed as

$$ E_{o}^{P2} = E_{R8}^{S2} e^{{j\left( {\frac{{\beta_{6}^{e} + \beta_{6}^{o} }}{2}} \right)L_{c} }} \cos \left( {\frac{{\pi \Delta n_{6} L_{c} }}{\lambda }} \right)e^{{j\beta^{s} L^{S3} }} e^{{ - \alpha_{prop} L^{S3} }} $$

(41)

$$ E_{o}^{P2} = \frac{{e^{{j2\theta_{cp} }} e^{{j\theta_{a} }} }}{{X_{5} }}\left\{ {t_{7} t_{8} - \gamma e^{{j\theta_{cp} }} e^{j\theta t} \left( {t_{7} + t_{8} } \right) + \gamma^{2} e^{{j2\theta_{cp} }} e^{j2\theta t} } \right\}Y_{5} $$

(42)

where $Y_{5} = E_{t}^{DC1} e^{{j\left( {\frac{{\beta_{3}^{e} + \beta_{3}^{o} }}{2}} \right)L_{c} }} e^{{j\left( {\frac{{\beta_{6}^{e} + \beta_{6}^{o} }}{2}} \right)L_{c} }} \sin \left( {\frac{{\pi \Delta n_{3} L_{c} }}{\lambda }} \right)\sin \left( {\frac{{\pi \Delta n_{6} L_{c} }}{\lambda }} \right)e^{{j\beta^{s} \left( {L^{S1} + L^{S3} } \right)}} e^{{ - \alpha_{prop} \left( {L^{S1} + L^{S3} } \right)}}$

The electric field at the output of Path1 can be expressed as

$$ E_{o}^{P1} = e^{{j\theta_{cp} }} \left[ {\frac{{t_{12} - \gamma e^{{j\theta_{cp} }} e^{j\theta t} }}{{1 - \gamma t_{12}^{ * } e^{{j\theta_{cp} }} e^{j\theta t} }}} \right]Y_{6} $$

(43)

where $Y_{6} = E_{t}^{DC1} e^{{j\left( {\frac{{\beta_{3}^{e} + \beta_{3}^{o} }}{2}} \right)L_{c} }} e^{{j\left( {\frac{{\beta_{7}^{e} + \beta_{7}^{o} }}{2}} \right)L_{c} }} \cos \left( {\frac{{\pi \Delta n_{3} L_{c} }}{\lambda }} \right)\sin \left( {\frac{{\pi \Delta n_{7} L_{c} }}{\lambda }} \right)e^{{j\beta^{s} \left( {L^{S1} + L^{S3} } \right)}} e^{{ - \alpha_{prop} \left( {L^{S1} + L^{S3} } \right)}}$

The electric field at the output of Path0 can be expressed as

$$ E_{o}^{P0} = Y_{6} e^{{j\beta^{s} L^{S3} }} e^{{ - \alpha_{prop} L^{S3} }} $$

(44)