====== Mode matching - plan and curve mirror ======

==== Mirror configuration ====
{{ :groups:mg:project_ptb-cavity:h-beast_resonator.jpg?350|}}
==Parameter==
    * Radius of curvature of mirror R1: R1= 1 m
    * Radius of curvature of mirror R2: R2= infinity (Incoupling side) 
    * Wavelength: λ= 1.55x10^-6 m
    * Length between the resonator mirrors: L= 485 mm
    * Beam radius at waist: w0
    * Beam radius at mirror: w1, w2
    * Stability parameter of the resonator: g1, g2
    * Distance between mirror and the waist: t1, t2

===Beam waist calculation from resonator===
**Step 1: Calculate the stability parameters**\\
$$g_1=1-\frac{L}{R1} \;\text{and}\; g_2=1-\frac{L}{R2}$$

**Step 2: Calculate the beam waist radii on the mirrors**\\
From thesis of Sana:\\
  * Beam Radii, 1/e^2 of the intensity
$$w_1 = \sqrt{L\cdot\frac{\lambda}{\pi}}\cdot \left(\frac{g_2}{g_1\cdot(1-g_1\cdot g_2)}\right)^{1/4}$$
$$w_2 = \sqrt{L\cdot\frac{\lambda}{\pi}}\cdot \left(\frac{g_1}{g_2\cdot(1-g_1\cdot g_2)}\right)^{1/4}$$
Alternative from rom Appl. Opt. 5, 1550 (1966):\\
$$w_1 = \sqrt{\frac{\lambda\cdot R1}{\pi}}\cdot \left( \frac{(R2-L)\cdot L}{(R1-L)\cdot(R1+R2-L)}\right)^{1/4}$$
$$w_2 = \sqrt{\frac{\lambda\cdot R2}{\pi}}\cdot \left( \frac{(R1-L)\cdot L}{(R2-L)\cdot(R1+R2-L)}\right)^{1/4}$$
$$w_0 = \sqrt{\frac{\lambda}{\pi}}\cdot \left(\frac{L\cdot(R1 - L)\cdot(R2 - L)\cdot(R1 + R2 - L)}{(R1 + R2 - 2\cdot L)^{2}}\right)^{1/4}$$
In our case:\\
$$w_2=w_0$$
**Step 3: Calculate the distance of beam waists on the mirrors and focus**\\
Position of the beam waist on the two mirrors:\\
$$t_1 = L\cdot\frac{R2 - L}{R1 + R2 - 2\cdot L}$$               
$$t_2 = L\cdot\frac{R1 - L}{R1 + R2 - 2\cdot L}$$
In our case: 
$$L=t_1+t_2$$
and t_2 is ≈0\\
**Step 4: Calculate the focal length of your collimator**\\
Take the radii on the plane w_2=w_0=496.567µm and curve mirror w_1=691.95µm and calculate the focal length f:
$$f=D\cdot \left(\frac{\pi\cdot w}{4\lambda}\right)$$
D is the beam waist diameter of the collimated beam after the collimator and w is mode beam diameter of the fiber output. In our case is:
$$D=2\cdot w_0$$
The diameter from the light, which comes out of the fiber (1550 nm) is w= 10.5+/-0.5µm. That gives us a focal length f= 5.32044mm.\\
**Step 4b**\\
We did the calculation wrong. :-( We used D=w_0 and got f=2.66022mm.