Jones, Gooch and Tarry, and Transmission

I am going to start out my calculations by writing out the general form of a Jones matrix for TN-LCD as expressed in “Jones-matrix models for twisted-nematic liquid crystal devices” by Makoto Yamauchi:

(1)   \begin{equation*} J = \exp[-i(\phi_0 +\beta_T)]R(-\psi_D)R(-\alpha_T)MR(\psi_D) \end{equation*}


(2)   \begin{equation*} \phi_0 = \dfrac{\pi*d}{\lambda}(n_e + n_0) \end{equation*}

and represents the constant absolute phase.


(3)   \begin{equation*} \beta_T = \beta\dfrac{d}{2} \end{equation*}

representing the total birefrigence.

The R in equation (1) is the rotation matrix and is represented by

    \begin{equation} \[ R(\xi) = \begin{bmatrix} \cos{\xi} & \sin{\xi} \\-\sin{\xi} & \cos{\xi} \end{bmatrix} \] \end{equation}

And M is the MLC Jones matrix.

We can use this equation in conjunction with the Gooch and Tarry formula in order to help us model the transmission of TN-LCD.

As stated in my Preliminary Data, the Gooch and Tarry formula is written as 

(4)   \begin{equation*} T=\dfrac{1}{1+u^2}\left\{u^2+cos^2\beta{d}\right\} \end{equation*}

Since we will be focusing on TN-LCDs, the twist angle will be 90and the transmission can then be modeled as

(5)   \begin{equation*} T= 1- \dfrac{\Phi^2}{(\beta^2)d^2}\sin^2{\beta*d} \end{equation*}


(6)   \begin{equation*} \beta*d= \sqrt{\left(\dfrac{\pi}{2}\right)^2 + \left(\dfrac{\pi*d\Delta{n}}{\lambda}\right)^2} \end{equation*}

I then used this variation on the Gooch and Tarry formula in order to graph transmission vs.  d∆n/λ

This graph shows us the first few transmission peaks (Mathematica Code). Transmission operates at 100% when

(7)   \begin{equation*} \beta*d= N\pi \end{equation*}

For N= 1, 2, 3…

Where N stands for the number of wave plates (an LC-cell being thought of as N wave plates).

With respect to our graph, then, the transmission peaks are occurring when

(8)   \begin{equation*} \dfrac{d\Delta{n}}{\lambda}= \dfrac{1}{2}\sqrt{(4N^2)-1} \end{equation*}

This helps us to visualize how the twisted nematic effect operates.


Gooch, C.H. and H.A. Tarry. “The optical properties of twisted nematic liquid crystal structures with twist angles less than or equal to 90 degrees.” Applied Physics Vol. 8(1975): 1575-1584.

Yamauchi, Makoto. “Jones-matrix models for twisted-nematic liquid crystal devices.” Applied Optics Vol. 44.21(2005): 4484-4493.



2 thoughts on “Jones, Gooch and Tarry, and Transmission

  1. tasanda

    Libby I really liked your mathematica diagrams.
    I’m not exactly sure what you mean by transmission. What is the unit?
    In your calculations of transmission did you input the jones matrix. I don’t see the connection unless you jumped a few steps.
    I’m also a little confused as to how the filters work. Are they oriented to only allow a certain angle of polarized light to pass through? I think you mentioned in your presentation that the screens used in calculators are the simplest LCD’s. How do they work? Is the light polarized in one direction and one filter is used?
    In the MORI study you talked about are the subjects in the study allowed to interact with each other to influence each other’s perception? What were the results of the study?

  2. joandrade

    I’m not sure about how you arrived at equation 5 from equations 1 and 4; perhaps you could reveal some of the steps in a separate post? On your graph: What is the physical significance of the line starting from zero and increasing rapidly at some point that isn’t the origin? Also, what are the “transmission peaks” you describe; they are clearly visible and periodic on your graph, but are they good or bad with regard to the production of the optimal image on the LCD screen? It seems like the transmission peaks rely mostly upon \Delta n (assuming d and \lambda are constants), so… what is \Delta n?

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