# data[].attributes.body
\\\Psi_1 = a_1e^{i(\vec k_1 \cdot \vec r - wt)}\\\Psi_2 = a_2e^{i(\vec k_2 \cdot \vec r - wt)}\\\Psi_R = \Psi_1 + \Psi_2 = a_1e^{i(\vec k_1 \cdot \vec r - wt)} + a_2e^{i(\vec k_2 \cdot \vec r - wt)}\\

I_R = |\Psi_1 + \Psi_2|^2 = \Psi_R\Psi_R^{*}\\I_R = a_1^{2}+a_2^{2}+2a_1a_2cos((\vec k_1-\vec k_2) \cdot \vec r)\\

\vec k_1 \cdot \vec r = kxsin(\theta_1) + kxcos(\theta_1)\\\vec k_2 \cdot \vec r = kxsin(\theta_2) + kxcos(\theta_2)\\

\phi = (\vec k_1-\vec k_2) \cdot \vec r\\\phi = kx(sin(\theta_1)-sin(\theta_2))+kz(cos(\theta_1)-cos(\theta_2))\\\Delta \phi=2\pi\\

\text{To solve }\Lambda_x:\\\Delta \phi=\phi_b - \phi_b=kx_b(sin(\theta_1)-sin(\theta_2))-kx_a(sin(\theta_1)-sin(\theta_2))\\\Delta \phi=k(sin(\theta_1)-sin(\theta_2))(x_b-x_a)\\2\pi=\frac{2\pi}{\lambda}(sin(\theta_1)-sin(\theta_2))\Lambda_x\\

\Rightarrow \Lambda_x=\frac{\lambda}{|sin(\theta_1)-sin(\theta_2)|}\\

\text{To solve }\Lambda_z:\\\Delta \phi=\phi_b - \phi_b=kz_b(cos(\theta_1)-cos(\theta_2))-kz_a(sin(\theta_1)-cos(\theta_2))\\\Delta \phi=k(cos(\theta_1)-cos(\theta_2))(z_b-z_a)\\2\pi=\frac{2\pi}{\lambda}(cos(\theta_1)-cos(\theta_2))\Lambda_z\\

\Rightarrow \Lambda_z=\frac{\lambda}{|cos(\theta_1)-cos(\theta_2)|}\\

# data[].attributes.body
To create a function for fringe spacing in terms of bisector angle:

\Lambda_x = \frac{\lambda}{|\sin(\theta_1)-\sin(\theta_2)|}\\ \Lambda_x = \frac{\lambda}{|\sin(\alpha + \theta_f)-\sin(\alpha - \theta_f)|}\\ \Lambda_x = \frac{\lambda}{|\sin(\alpha)\cos(\theta_f) + \cos(\alpha)\sin(\theta_f) - \sin(\alpha)\cos(\theta_f) + \cos(\alpha)\sin(\theta_f)|}\\ \Lambda_x = \frac{\lambda}{|2\cos(\alpha)\sin(\theta_f)|}

\Lambda_z = \frac{\lambda}{|\cos(\theta_1)-\cos(\theta_2)|}\\ \Lambda_z = \frac{\lambda}{|\cos(\alpha + \theta_f)-\cos(\alpha - \theta_f)|}\\ \Lambda_z = \frac{\lambda}{|\cos(\alpha)\cos(\theta_f) - \sin(\alpha)\sin(\theta_f) - \cos(\alpha)\cos(\theta_f) - \sin(\alpha)\sin(\theta_f)|}\\ \Lambda_x = \frac{\lambda}{|-2\sin(\alpha)\sin(\theta_f)|}

# data[].attributes.body
\Psi_{1}=a_{1}e^{i({\vec{k_{1}}\cdot \vec{r}-\omega t})}

\Psi_{1}=a_{2}e^{i({\vec{k_{2}}\cdot \vec{r}-\omega t})}

\Psi_{R}=\Psi_{1}+\Psi_{2}

I_{R}=\left|\Psi_{R}\right| =a_{1}^{2}+a_{2}^{2}+2a_{1}a{2}cos((\vec{k_{1}}-\vec{k_{2}})\cdot\vec{r})

\phi=(\vec{k_{1}}-\vec{k_{2}})\cdot\vec{r}

\vec{k_{1}}\cdot \vec{r}=k\hat{x}sin\theta_{1}\cdot x\hat{x}+k\hat{z}cos\theta_{1}\cdot z\hat{z}=kxsin\theta_{1}+kzcos\theta_{1}

\vec{k_{2}}\cdot \vec{r}=kxsin\theta_{2}+kzcos\theta_{2}

\theta_{1}=60^{\circ}, \theta_{2}=30^{\circ}

\vec{k_{1}}\cdot \vec{r}=0.87kx+0.5kz

\vec{k_{2}}\cdot \vec{r}=0.5kx+0.86kz

For \Lambda_{x}:

\Delta\phi=\phi_{b}-\phi_{a}=2\pi

2\pi=kx_{b}sin\theta_{1}-kx_{b}sin\theta_{2}-(kx_{a}sin\theta_{1}-kx_{a}sin\theta_{2}) =k(sin\theta_{1}-sin\theta_{2})(x_{b}-x_{a}) =\frac{2\pi}{\lambda}(sin\theta_{1}-sin\theta_{2})\Lambda_{x}

\Lambda_{x}=\frac{\lambda}{\left|sin\theta_{1}-sin\theta_{2}\right|}=\frac{\lambda}{0.37}

For \Lambda_{z}:

\Delta\phi=\phi_{b}-\phi_{a}=2\pi

2\pi=kz_{b}cos\theta_{1}-kz_{b}cos\theta_{2}-(kz_{a}cos\theta_{1}-kz_{a}cos\theta_{2}) =k(cos\theta_{1}-cos\theta_{2})(z_{b}-z_{a}) =\frac{2\pi}{\lambda}(cos\theta_{1}-cos\theta_{2})\Lambda_{z}

\Lambda_{z}=\frac{\lambda}{\left|cos\theta_{1}-cos\theta_{2}\right|}=\frac{\lambda}{0.37}

# data[].attributes.body
\theta_{1}=75^{\circ}, \theta_{2}=15^{\circ}

\vec{k_{1}}\cdot \vec{r}=0.97kx+0.26kz

\vec{k_{2}}\cdot \vec{r}=0.26kx+0.97kz

For \Lambda_{x}:

\Delta\phi=\phi_{b}-\phi_{a}=2\pi

2\pi=kx_{b}sin\theta_{1}-kx_{b}sin\theta_{2}-(kx_{a}sin\theta_{1}-kx_{a}sin\theta_{2}) =k(sin\theta_{1}-sin\theta_{2})(x_{b}-x_{a}) =\frac{2\pi}{\lambda}(sin\theta_{1}-sin\theta_{2})\Lambda_{x}

\Lambda_{x}=\frac{\lambda}{\left|sin\theta_{1}-sin\theta_{2}\right|}=\frac{\lambda}{0.71}

For \Lambda_{z}:

\Delta\phi=\phi_{b}-\phi_{a}=2\pi

2\pi=kz_{b}cos\theta_{1}-kz_{b}cos\theta_{2}-(kz_{a}cos\theta_{1}-kz_{a}cos\theta_{2}) =k(cos\theta_{1}-cos\theta_{2})(z_{b}-z_{a}) =\frac{2\pi}{\lambda}(cos\theta_{1}-cos\theta_{2})\Lambda_{z}

\Lambda_{z}=\frac{\lambda}{\left|cos\theta_{1}-cos\theta_{2}\right|}=\frac{\lambda}{0.71}

# data[].attributes.body
\\ \theta_f= 75 ^{\circ} \\ \theta_1= 120 ^{\circ} \\ \theta_2= -30 ^{\circ}

\\\Lambda_x=\frac{\lambda}{|sin(120 ^{\circ})-sin(-30 ^{\circ})|}\\ = 0.732\lambda\\

\Lambda_z=\frac{\lambda}{|cos( 120^{\circ})-cos( -30^{\circ})|}\\ = 0.732\lambda\\

# data[].attributes.body
\\ \theta_f= 105^{\circ} \\ \theta_1= 150^{\circ} \\ \theta_2= -60^{\circ} \\

\Lambda_x=\frac{\lambda}{|sin( 150^{\circ})-sin( -60^{\circ})|}\\ = 0.732\lambda\\

\\ \Lambda_z=\frac{\lambda}{|cos( 150^{\circ})-cos( -60^{\circ})|}\\ = 0.732\lambda\\