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Implementation of 2-bit multiplier based on electro-optic effect in Mach–Zehnder interferometers

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  • 9 Citations

Abstract

This paper demonstrates the structure and working principle of an optical 2-bit multiplier using lithium niobate (LiNbO3) based Mach–Zehnder interferometer (MZI). The powerful ability of MZI structures to switch the optical signal from one output port to the other output port has been used in the implementation of the proposed 2-bit multiplier. The paper constitutes the mathematical description of the 2-bit multiplier and thereafter compilation using MATLAB. The study is carried out by simulating the proposed device with Beam propagation method (BPM).

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References

  1. Gayen, D.K., Chattopadhyay, T., Pal, R.K., Roy, J.N.: All-optical multiplication with the help of semiconductor optical amplifier-assisted Sagnac switch. J. Comput. Electron. 9(2), 57–67 (2010)

  2. Jin, H., Liu, F.M., Xu, P., Xia, J.L., Zhong, M.L., Yuan, Y., Zhou, J.W., Gong, Y.X., Wang, W., Zhu, S.N.: On-chip generation and manipulation of entangled photons based on reconfigurable lithium-niobate waveguide circuits. Phys. Rev. Lett. 113, 103601–103605 (2014)

  3. Kumar, S., Raghuwanshi, S.K., Kumar, A.: Implementation of optical switches by using mach–zehnder interferometer. Opt. Eng. 52(9), 097106 (2013)

  4. Kumar, A., Kumar, S., Raghuwanshi, S.K.: Implementation of full-adder and full-subtractor based on electro-optic effect in mach–zehnder interferometer. Opt. Commun. 324, 93–107 (2014a)

  5. Kumar, A., Kumar, S., Raghuwanshi, S.K.: Implementation of XOR/XNOR and AND logic gates using mach–zehnder interferometers. Optik 125, 5764–5767 (2014b)

  6. Kumar, S., Bisht, A., Singh, G., Choudhary, K., Sharma, D.: Implementation of wavelength selector based on electro-optic effect in Mach–Zehnder interferometers for high speed communications. Opt. Commun. 350, 108–118 (2015a)

  7. Kumar, S., Singh, G., Bisht, A.: 4 × 4 signal router based on electro-optic effect of mach–zehnder interferometer for wavelength division multiplexing applications. Opt. Commun. 353, 17–26 (2015b)

  8. Li, G.L., Yu, P.K.L.: Optical intensity modulators for digital and analog applications. J. Lightwave Technol. 21(9), 2010–2030 (2003)

  9. Li, G., Qian, F., Ruan, H., Liu, L.: Compact parallel optical modified-signed-digit arithmetic-logic array processor with electron-trapping device. Appl. Opt. 38(23), 5039–5045 (1999)

  10. Li, Q., Zhu, M., Li, D., Zhang, Z., Wei, Y., Hu, M., Zhou, X., Tang, X.: Optical logic gates based on electro-optic modulation with Sagnac interferometer. Appl. Opt. 53(21), 4708–4715 (2014)

  11. Mandal, D., Mandal, S., Garai, S.K.: A new approach of developing all-optical two-bit-binary data multiplier. Opt. Laser Technol. 64, 292–301 (2014)

  12. Mukhopadhyay, S., Das, D.N., Das, P.P., Ghosh, P.: Implementation of all-optical digital matrix multiplication scheme with nonlinear material. Opt. Eng. 40(9), 1998–2002 (2001)

  13. Raghuwanshi, S.K., Kumar, A., Kumar, S.: 1 × 4 signal router using 3-Mach-Zhender interferometers. Opt. Eng. 52(03), 035002 (2013)

  14. Raghuwanshi, S.K., Kumar, A., Chen, N.K.: Implementation of sequential logic circuits using the Mach–Zehnder interferometer structure based on electro-optic effect. Opt. Commun. 333, 193–208 (2014)

  15. Shen, Z.Y., Wu, L.L.: Reconfigurable optical logic unit with a terahertz optical asymmetric demultiplexer and electro-opticswitches. Appl. Opt. 47(21), 3737–3742 (2008)

  16. Singh, G., Janyani, V., Yadav, R.P.: Modeling of a high performance Mach–Zehnder interferometer all optical switch. Optica Applicata 42, 613–625 (2012)

  17. Sokoloff, J.P., Prucnal, P.R., Glesk, I., Kane, M.: A terahertz optical asymmetric demultiplexer (TOAD). IEEE Photonics Technol. Lett. 5(7), 787–789 (1993)

  18. Vikram, C.S., Caulfield, H.J.: Position-sensing detector for logical operations using incoherent light. Opt. Eng. 44, 115201–115204 (2005)

  19. Wang, B.C., Baby, V., Tong, W., Xu, L., Friedman, M., Runser, R.J., Glesk, I., Pruncnal, P.R.: A novel fast optical switch based on two cascaded terahertz asymmetric demultiplexers (TOAD). Opt. Express 10(1), 15–23 (2002)

  20. Wooten, E.L., Kissa, K.M., Yan, A.Y., Murphy, E.J., Lafaw, D.A., Hallemeier, P.F., Maack, D., Attanasio, D.V., Fritz, D.J., McBrien, G.J., Bossi, D.E.: A review of lithium niobate modulators for fiber-optic communications systems. IEEE J. Sel. Top. Quantum Electron. 6(1), 69–82 (2000)

  21. Zoiros, K.E., Stathopoulos, T., Vlachos, K., Hatziefremidis, A., Houbavlis, T., Papakyriakopoulos, T., Avramopoulos, H.: Experimental and theoretical studies of a high repetition rate fiber laser, mode-locked by external optical modulation. Opt. Commun. 180, 301–315 (2000)

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Acknowledgments

This work is supported by the project entitled “Performance study of some WDM optical network components and design of optical switching devices” under the Faculty Research Scheme, DIT University, India (Ref. No.: DITU/R&D/2014/7/ECE) undertaken by Dr. Santosh Kumar. The authors would also like to thank Prof. K. K. Raina, Vice-Chancellor, DIT University, India for encouragement and support during the present research work.

Author information

Correspondence to Santosh Kumar.

Appendix

Appendix

For all possible minterms at first output port of MZI3 (Port M0 in Fig. 1), the normalized power (for M0) is calculated as follows;

Using the relation for single stage MZI structure in (Kumar et al. 2014b) we can write,

$$ {\text{OUT}}1_{{{\text{MZI}}3}} = \left[ {\left\{ { - {\text{je}}^{{ - {\text{j}}\left( {{\upvarphi }_{{0{\text{MZI}}1}} } \right)}} \sin \left( {\frac{{\Delta {\upvarphi }_{{{\text{MZI}}1}} }}{2}} \right)} \right\} \left\{ { - {\text{je}}^{{ - {\text{j}}\left( {{\upvarphi }_{{0{\text{MZI}}3}} } \right)}} \sin \left( {\frac{{\Delta {\upvarphi }_{{{\text{MZI}}3}} }}{2}} \right)} \right\}} \right]E_{in} $$
(14)
$$ \frac{{{\text{OUT}}1_{{{\text{MZI}}3}} }}{{{\text{E}}_{\text{in}} }} = \left[ {\left\{ { - {\text{je}}^{{ - {\text{j}}\left( {{\upvarphi }_{{0{\text{MZI}}1}} } \right)}} \sin \left( {\frac{{\Delta {\upvarphi }_{{{\text{MZI}}1}} }}{2}} \right)} \right\} \left\{ { - {\text{je}}^{{ - {\text{j}}\left( {{\upvarphi }_{{0{\text{MZI}}3}} } \right)}} \sin \left( {\frac{{\Delta {\upvarphi }_{{{\text{MZI}}3}} }}{2}} \right)} \right\}} \right] $$
(15)
$$ {\text{M}}_{0} = \left| {\frac{{{\text{OUT}}1_{{{\text{MZI}}3}} }}{{{\text{E}}_{\text{in}} }}} \right|^{2} = \left\{ { \sin^{2} \left( {\frac{{\Delta {\upvarphi }_{{{\text{MZI}}1}} }}{2}} \right) \sin^{2} \left( {\frac{{\Delta {\upvarphi }_{{{\text{MZI}}3}} }}{2}} \right)} \right\} $$
(16)

In the same manner, for M1, the normalized power at first output port of MZI10 (Port M1 in Fig. 1) is calculated as follows;

$$ {\text{OUT}}1_{{{\text{MZI}}10}} = \left[ { \left\{ {{-}{\text{je}}^{{ - {\text{j}}\left( {{\upvarphi }_{{0{\text{MZI}}9}} } \right)}} \sin \left( {\frac{{\Delta {\upvarphi }_{{{\text{MZI}}9}} }}{2}} \right)} \right\} \left\{ {{\text{je}}^{{ - {\text{j}}\left( {{\upvarphi }_{{0{\text{MZI}}10}} } \right)}} \cos \left( {\frac{{\Delta {\upvarphi }_{{{\text{MZI}}10}} }}{2}} \right)} \right\}} \right]E_{in} $$
(17)
$$ \frac{{{\text{OUT}}1_{{{\text{MZI}}10}} }}{{{\text{E}}_{\text{in}} }} = \left[ { \left\{ { - {\text{je}}^{{ - {\text{j}}\left( {{\upvarphi }_{{0{\text{MZI}}9}} } \right)}} \sin \left( {\frac{{\Delta {\upvarphi }_{{{\text{MZI}}9}} }}{2}} \right)} \right\} \left\{ {{\text{je}}^{{ - {\text{j}}\left( {{\upvarphi }_{{0{\text{MZI}}10}} } \right)}} \cos \left( {\frac{{\Delta {\upvarphi }_{{{\text{MZI}}10}} }}{2}} \right)} \right\}} \right] $$
(18)
$$ \left| {\frac{{{\text{OUT}}1_{{{\text{MZI}}10}} }}{{{\text{E}}_{\text{in}} }}} \right|^{2} = \left\{ { \sin^{2} \left( {\frac{{\Delta {\upvarphi }_{{{\text{MZI}}9}} }}{2}} \right) \cos^{2} \left( {\frac{{\Delta {\upvarphi }_{{{\text{MZI}}10}} }}{2}} \right)} \right\} $$
(19)

and the normalized power at the second output port of MZI11 is calculated as follows;

$$ {\text{OUT}}2_{{{\text{MZI}}11}} = \left[ {\left\{ {{\text{je}}^{{ - {\text{j}}\left( {{\upvarphi }_{{0{\text{MZI}}9}} } \right)}} \cos \left( {\frac{{\Delta {\upvarphi }_{{{\text{MZI}}9}} }}{2}} \right)} \right\} \left\{ {{-}{\text{je}}^{{ - {\text{j}}\left( {{\upvarphi }_{{0{\text{MZI}}11}} } \right)}} \sin \left( {\frac{{\Delta {\upvarphi }_{{{\text{MZI}}11}} }}{2}} \right)} \right\} } \right]E_{in} $$
(20)
$$ \frac{{{\text{OUT}}2_{{{\text{MZI}}11}} }}{{E_{in} }} = \left[ {\left\{ {{\text{je}}^{{ - {\text{j}}\left( {{\upvarphi }_{{0{\text{MZI}}9}} } \right)}} \cos \left( {\frac{{\Delta {\upvarphi }_{{{\text{MZI}}9}} }}{2}} \right)} \right\} \left\{ {{-}{\text{je}}^{{ - {\text{j}}\left( {{\upvarphi }_{{0{\text{MZI}}11}} } \right)}} \sin \left( {\frac{{\Delta {\upvarphi }_{{{\text{MZI}}11}} }}{2}} \right)} \right\} } \right] $$
(21)
$$ \left| {\frac{{{\text{OUT}}1_{{{\text{MZI}}11}} }}{{{\text{E}}_{\text{in}} }}} \right|^{2} = \left\{ { \cos^{2} \left( {\frac{{\Delta {\upvarphi }_{{{\text{MZI}}9}} }}{2}} \right) \sin^{2} \left( {\frac{{\Delta {\upvarphi }_{{{\text{MZI}}11}} }}{2}} \right)} \right\} $$
(22)

Hence for M1, the normalized power at Port M1 (in Fig. 1) is calculated as follows;

$$ {\text{M}}_{1} = \left\{ { \sin^{2} \left( {\frac{{\Delta {\upvarphi }_{{{\text{MZI}}9}} }}{2}} \right) \cos^{2} \left( {\frac{{\Delta {\upvarphi }_{{{\text{MZI}}10}} }}{2}} \right)} \right\} + \left\{ { \cos^{2} \left( {\frac{{\Delta {\upvarphi }_{{{\text{MZI}}9}} }}{2}} \right) \sin^{2} \left( {\frac{{\Delta {\upvarphi }_{{{\text{MZI}}11}} }}{2}} \right)} \right\} $$
(23)

Similarly for M2, the normalized power at first output port of MZI13 (Port M2 in Fig. 1) is calculated as follows;

$$ {\text{OUT}}1_{{{\text{MZI}}13}} = \left[ { \left\{ {{-}{\text{je}}^{{ - {\text{j}}\left( {{\upvarphi }_{{0{\text{MZI}}12}} } \right)}} \sin \left( {\frac{{\Delta {\upvarphi }_{{{\text{MZI}}12}} }}{2}} \right)} \right\} \left\{ {{\text{je}}^{{ - {\text{j}}\left( {{\upvarphi }_{{0{\text{MZI}}13}} } \right)}} \cos \left( {\frac{{\Delta {\upvarphi }_{{{\text{MZI}}13}} }}{2}} \right)} \right\}} \right]E_{in} $$
(24)
$$ \frac{{{\text{OUT}}1_{{{\text{MZI}}13}} }}{{{\text{E}}_{\text{in}} }} = \left[ { \left\{ {{-}{\text{je}}^{{ - {\text{j}}\left( {{\upvarphi }_{{0{\text{MZI}}12}} } \right)}} \sin \left( {\frac{{\Delta {\upvarphi }_{{{\text{MZI}}12}} }}{2}} \right)} \right\} \left\{ {{\text{je}}^{{ - {\text{j}}\left( {{\upvarphi }_{{0{\text{MZI}}13}} } \right)}} \cos \left( {\frac{{\Delta {\upvarphi }_{{{\text{MZI}}13}} }}{2}} \right)} \right\}} \right] $$
(25)
$$ \left| {\frac{{{\text{OUT}}1_{{{\text{MZI}}13}} }}{{{\text{E}}_{\text{in}} }}} \right|^{2} = \left\{ { \sin^{2} \left( {\frac{{\Delta {\upvarphi }_{{{\text{MZI}}12}} }}{2}} \right) \cos^{2} \left( {\frac{{\Delta {\upvarphi }_{{{\text{MZI}}13}} }}{2}} \right)} \right\} $$
(26)

and the normalized power at the second output port of MZI14 is calculated as follows;

$$ {\text{OUT}}2_{{{\text{MZI}}14}} = \left[ {\left\{ {{\text{je}}^{{ - {\text{j}}\left( {{\upvarphi }_{{0{\text{MZI}}12}} } \right)}} \cos \left( {\frac{{\Delta {\upvarphi }_{{{\text{MZI}}12}} }}{2}} \right)} \right\} \left\{ {{-}{\text{je}}^{{ - {\text{j}}\left( {{\upvarphi }_{{0{\text{MZI}}14}} } \right)}} \sin \left( {\frac{{\Delta {\upvarphi }_{{{\text{MZI}}14}} }}{2}} \right)} \right\} } \right]E_{in} $$
(27)
$$ \frac{{{\text{OUT}}2_{{\text{MZI}}14}}}{\text{E}_{in}} = \left[ {\left\{ {{\text{je}}^{{ - {\text{j}}\left( {{\upvarphi }_{{0{\text{MZI}}12}} } \right)}} { \cos }\left( {\frac{{\Delta {\upvarphi }_{{{\text{MZI}}12}} }}{2}} \right)} \right\} \left\{ {{-}{\text{je}}^{{ - {\text{j}}\left( {{\upvarphi }_{{0{\text{MZI}}14}} } \right)}} { \sin }\left( {\frac{{\Delta {\upvarphi }_{{\text{MZI}}14}}}{2}} \right)} \right\} } \right] $$
(28)
$$ \left| {\frac{{{\text{OUT}}2_{{{\text{MZI}}14}} }}{{{\text{E}}_{\text{in}} }}} \right|^{2} = \left\{ { \cos^{2} \left( {\frac{{\Delta {\upvarphi }_{{{\text{MZI}}12}} }}{2}} \right) \sin^{2} \left( {\frac{{\Delta {\upvarphi }_{{{\text{MZI}}14}} }}{2}} \right)} \right\} $$
(29)

Hence for M2, the normalized power at Port M2 (in Fig. 1) is calculated as follows;

$$ {\text{M}}_{2} = \left\{ { \sin^{2} \left( {\frac{{\Delta {\upvarphi }_{{{\text{MZI}}12}} }}{2}} \right) \cos^{2} \left( {\frac{{\Delta {\upvarphi }_{{{\text{MZI}}13}} }}{2}} \right)} \right\} + \left\{ { \cos^{2} \left( {\frac{{\Delta {\upvarphi }_{{{\text{MZI}}12}} }}{2}} \right) \sin^{2} \left( {\frac{{\Delta {\upvarphi }_{{{\text{MZI}}14}} }}{2}} \right)} \right\} $$
(30)

For all possible minterms at second output port of MZI13 (Port M3 in Fig. 1), the normalized power (for M3) is calculated as follows;

$$ {\text{OUT}}2_{{{\text{MZI}}13}} = \left[ {\left\{ { - {\text{je}}^{{ - {\text{j}}\left( {{\upvarphi }_{{0{\text{MZI}}12}} } \right)}} \sin \left( {\frac{{\Delta {\upvarphi }_{{{\text{MZI}}12}} }}{2}} \right)} \right\} \left\{ {{-}{\text{je}}^{{ - {\text{j}}\left( {{\upvarphi }_{{0{\text{MZI}}13}} } \right)}} \sin \left( {\frac{{\Delta {\upvarphi }_{{{\text{MZI}}13}} }}{2}} \right)} \right\}} \right]E_{in} $$
(31)
$$ \frac{{{\text{OUT}}2_{{{\text{MZI}}13}} }}{\text{E}_{in}} = \left[ {\left\{ { - {\text{je}}^{{ - {\text{j}}\left( {{\upvarphi }_{{0{\text{MZI}}12}} } \right)}} \sin \left( {\frac{{\Delta {\upvarphi }_{{{\text{MZI}}12}} }}{2}} \right)} \right\} \left\{ { - {\text{je}}^{{ - {\text{j}}\left( {{\upvarphi }_{{0{\text{MZI}}13}} } \right)}} \sin \left( {\frac{{\Delta {\upvarphi }_{{{\text{MZI}}13}} }}{2}} \right)} \right\}} \right] $$
(32)
$$ {\text{M}}_{3} = \left| {\frac{{{\text{OUT}}2_{{{\text{MZI}}13}} }}{{{\text{E}}_{\text{in}} }}} \right|^{2} = \left\{ { \cos^{2} \left( {\frac{{\Delta {\upvarphi }_{{{\text{MZI}}12}} }}{2}} \right) \sin^{2} \left( {\frac{{\Delta {\upvarphi }_{{{\text{MZI}}13}} }}{2}} \right)} \right\} $$
(33)

For calculation of Eqs. (14)–(33), we have assumed,

$$ \left. \begin{aligned} &{\upvarphi }_{{0{\text{MZI}}1}} = \frac{{{\upvarphi }_{{1{\text{MZI}}1}} + {\upvarphi }_{{2{\text{MZI}}1}} }}{2} \hfill \\ & {\upvarphi }_{{0{\text{MZI}}2}} = \frac{{{\upvarphi }_{1MZI2} + {\upvarphi }_{{2{\text{MZI}}2}} }}{2} \hfill \\ & {\upvarphi }_{{0{\text{MZI}}3}} = \frac{{{\upvarphi }_{{1{\text{MZI}}3}} + {\upvarphi }_{{2{\text{MZI}}3}} }}{2} \hfill \\ & {\upvarphi }_{{0{\text{MZI}}4}} = \frac{{{\upvarphi }_{{1{\text{MZI}}4}} + {\upvarphi }_{{2{\text{MZI}}4}} }}{2} \hfill \\ & {\upvarphi }_{{0{\text{MZI}}5}} = \frac{{{\upvarphi }_{{1{\text{MZI}}5}} + {\upvarphi } _{{2{\text{MZI}}5}} }}{2} \hfill \\ & {\upvarphi }_{{0{\text{MZI}}6}} = \frac{{{\upvarphi }_{{1{\text{MZI}}6}} + {\upvarphi }_{{2{\text{MZI}}6}} }}{2} \hfill \\ & {\upvarphi }_{{0{\text{MZI}}7}} = \frac{{{\upvarphi }_{{1{\text{MZI}}7}} + {\upvarphi }_{{2{\text{MZI}}7}} }}{2} \hfill \\ & {\upvarphi }_{{0{\text{MZI}}8}} = \frac{{{\upvarphi }_{{1{\text{MZI}}8}} + {\upvarphi }_{{2{\text{MZI}}8}} }}{2} \hfill \\ & {\upvarphi }_{{0{\text{MZI}}9}} = \frac{{{\upvarphi }_{{1{\text{MZI}}9}} + {\upvarphi }_{{2{\text{MZI}}9}} }}{2} \hfill \\ & {\upvarphi }_{{0{\text{MZI}}10}} = \frac{{{\upvarphi }_{{1{\text{MZI}}10}} + {\upvarphi }_{{2{\text{MZI}}10}} }}{2} \hfill \\ & {\upvarphi }_{{0{\text{MZI}}11}} = \frac{{{\upvarphi }_{{1{\text{MZI}}11}} + {\upvarphi }_{{2{\text{MZI}}11}} }}{2} \hfill \\ & {\upvarphi }_{{0{\text{MZI}}12}} = \frac{{{\upvarphi }_{{1{\text{MZI}}12}} + {\upvarphi }_{{2{\text{MZI}}12}} }}{2} \hfill \\ & {\upvarphi }_{{0{\text{MZI}}3}} = \frac{{{\upvarphi }_{{1{\text{MZI}}13}} + {\upvarphi }_{{2{\text{MZI}}13}} }}{2} \hfill \\ & {\upvarphi } _{{0{\text{MZI}}14}} = \frac{{{\upvarphi }_{{1{\text{MZI}}14}} + {\upvarphi }_{2MZI14} }}{2} \hfill \\ \end{aligned} \right\}\left. \begin{aligned} & \Delta {\upvarphi }_{{{\text{MZI}}1}} = {\upvarphi }_{{1{\text{MZI}}1}} - {\upvarphi }_{{2{\text{MZI}}1}} = \frac{\pi }{{V_{\pi } }}{\text{A}}_{0} \hfill \\ & \Delta {\upvarphi }_{{{\text{MZI}}2}} = {\upvarphi }_{{1{\text{MZI}}2}} - {\upvarphi }_{{2{\text{MZI}}2}} = \frac{\pi }{{V_{\pi } }}{\text{A}}_{0} \hfill \\ & \Delta {\upvarphi }_{{{\text{MZI}}3}} = {\upvarphi }_{{1{\text{MZI}}3}} - {\upvarphi }_{{2{\text{MZI}}3}} = \frac{\pi }{{V_{\pi } }}{\text{B}}_{0} \hfill \\ & \Delta {\upvarphi }_{{{\text{MZI}}4}} = {\upvarphi }_{{1{\text{MZI}}4}} - {\upvarphi }_{{2{\text{MZI}}4}} = \frac{\pi }{{V_{\pi } }}{\text{B}}_{0} \hfill \\ & \Delta {\upvarphi }_{{{\text{MZI}}5}} = {\upvarphi }_{{1{\text{MZI}}5}} - {\upvarphi }_{{2{\text{MZI}}5}} = \frac{\pi }{{V_{\pi } }}{\text{B}}_{1} \hfill \\ & \Delta {\upvarphi }_{{{\text{MZI}}6}} = {\upvarphi }_{{1{\text{MZI}}6}} - {\upvarphi }_{{2{\text{MZI}}6}} = \frac{\pi }{{V_{\pi } }}{\text{B}}_{1} \hfill \\ & \Delta {\upvarphi }_{{{\text{MZI}}7}} = {\upvarphi }_{{1{\text{MZI}}7}} - {\upvarphi }_{{2{\text{MZI}}7}} = \frac{\pi }{{V_{\pi } }}{\text{A}}_{1} \hfill \\ & \Delta {\upvarphi }_{{{\text{MZI}}8}} = {\upvarphi }_{{1{\text{MZI}}8}} - {\upvarphi }_{{2{\text{MZI}}8}} = \frac{\pi }{{V_{\pi } }}{\text{A}}_{1} \hfill \\ & \Delta {\upvarphi }_{{{\text{MZI}}9}} = {\upvarphi }_{{1{\text{MZI}}9}} - {\upvarphi }_{{2{\text{MZI}}9}} = \frac{\pi }{{V_{\pi } }}{\text{X}}_{1} \hfill \\ & \Delta {\upvarphi }_{{{\text{MZI}}10}} = {\upvarphi }_{{1{\text{MZI}}10}} - {\upvarphi }_{{2{\text{MZI}}10}} = \frac{\pi }{{V_{\pi } }}{\text{X}}_{2} \hfill \\ & \Delta {\upvarphi }_{{{\text{MZI}}11}} = {\upvarphi }_{{1{\text{MZI}}11}} - {\upvarphi }_{{2{\text{MZI}}11}} = \frac{\pi }{{V_{\pi } }}{\text{X}}_{2} \hfill \\ & \Delta {\upvarphi }_{{{\text{MZI}}12}} = {\upvarphi }_{{1{\text{MZI}}12}} - {\upvarphi }_{{2{\text{MZI}}12}} = \frac{\pi }{{V_{\pi } }}{\text{X}}_{3} \hfill \\ & \Delta {\upvarphi }_{{{\text{MZI}}13}} = {\upvarphi }_{{1{\text{MZI}}13}} - {\upvarphi }_{{2{\text{MZI}}13}} = \frac{\pi }{{V_{\pi } }}{\text{X}}_{4} \hfill \\ & \Delta {\upvarphi }_{{{\text{MZI}}14}} = {\upvarphi }_{{1{\text{MZI}}9}} - {\upvarphi }_{{2{\text{MZI}}14}} = \frac{\pi }{{V_{\pi } }}{\text{X}}_{4} \hfill \\ \end{aligned} \right\} $$
(34)

\( {\upvarphi }_{{1{\text{MZI}}1}} ,{\upvarphi }_{{1{\text{MZI}}2}} ,{\upvarphi }_{{1{\text{MZI}}3}} ,{\upvarphi }_{{1{\text{MZI}}4}} ,{\upvarphi }_{{1{\text{MZI}}5}} ,{\upvarphi }_{{1{\text{MZI}}6}} , {\upvarphi }_{{1{\text{MZI}}7}} , {\upvarphi }_{{1{\text{MZI}}8}} ,{\upvarphi }_{{1{\text{MZI}}9}} ,{\upvarphi }_{{1{\text{MZI}}10}} ,{\upvarphi }_{{1{\text{MZI}}11}} ,{\upvarphi }_{{1{\text{MZI}}12}} ,{\upvarphi }_{{1{\text{MZI}}13}} ,{\upvarphi }_{{1{\text{MZI}}14}} \) are the phase angle generated at the arm of MZI4, MZI5, MZI6, MZI7, MZI8, MZI9, MZI10, MZI11, MZI12, MZI13 and MZI14 respectively.

\( {\upvarphi }_{{2{\text{MZI}}1}} ,{\upvarphi }_{{2{\text{MZI}}2}} ,{\upvarphi }_{{2{\text{MZI}}3}} ,{\upvarphi }_{{2{\text{MZI}}4}} , {\upvarphi }_{{2{\text{MZI}}5}} ,{\upvarphi }_{{2{\text{MZI}}6}} ,{\upvarphi }_{{2{\text{MZI}}7}} , {\upvarphi }_{{2{\text{MZI}}8}} , {\upvarphi }_{{2{\text{MZI}}9}} ,{\upvarphi }_{{2{\text{MZI}}10 }} ,{\upvarphi }_{{2{\text{MZI}}11}} ,{\upvarphi }_{{2{\text{MZI}}12}} ,{\upvarphi }_{{2{\text{MZI}}13}} , {\upvarphi }_{{2{\text{MZI}}14}} \) are the phase angle generated at the lower arm of MZI1, MZI2, MZI3, MZI4, MZI5, MZI6, MZI7, MZI8, MZI9, MZI10, MZI11, MZI12, MZI13 and MZI14 respectively.

Also X1 = (B1·A0), X2 = (B0·A1), X3 = (B1·A1) and X4 = (X1·X2).

The MZI1 and MZI2 are controlled by signal A0 (the voltage applied at the second electrode, keeping other two electrodes at the ground potential). Similarly, MZI3 and MZI4 are controlled by control signal B0.

  • MZI5 and MZI6 are controlled by control signal B1.

  • MZI7 and MZI8 are controlled by control signal A1.

  • MZI9 is controlled by control signal X1.

  • MZI10 and MZI11 are controlled by control signal X2.

  • MZI12 is controlled by the control signal X3.

  • MZI13 and MZI14 are controlled by control signal X4.

Basically, the control signals are 0 (0.00 V) and 1 (6.75 V) at the second electrodes of each MZI.

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Kumar, S., Bisht, A., Singh, G. et al. Implementation of 2-bit multiplier based on electro-optic effect in Mach–Zehnder interferometers. Opt Quant Electron 47, 3667–3688 (2015). https://doi.org/10.1007/s11082-015-0249-4

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Keywords

  • Lithium niobate
  • Mach–Zehnder interferometer
  • Beam propagation method