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groups:mg:project_ptb-cavity:mode_matching [2017/05/12 14:57] – ssauergroups:mg:project_ptb-cavity:mode_matching [2017/10/06 13:31] (current) – ssauer
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 ====== Mode matching ====== ====== Mode matching ======
-{{ :groups:mg:project_ptb-cavity:h-beast_resonator.jpg?350|}} 
 **Literature** **Literature**
   * {{ :groups:mg:project_ptb-cavity:ao-5-10-1550.pdf |Laser Beams and Resonators }} H. Kogelnik et al., Applied Optics **5**, 1550 (1966)   * {{ :groups:mg:project_ptb-cavity:ao-5-10-1550.pdf |Laser Beams and Resonators }} H. Kogelnik et al., Applied Optics **5**, 1550 (1966)
 +  * {{ :groups:mg:project_ptb-cavity:ol-25-4-266.pdf |Determination and optimization of mode matching into
 +optical cavities by heterodyne detection}}G. Mueller et al., Optics Letters, Vol. **25**, No. 4 (2000)
  
-==== Mirror configuration ==== +[[:groups:Mg:Project PTB-Cavity:Mode Matching first try|Mode Matching - plan and curve mirror]] (wrong calculation) \\ 
- +[[:groups:Mg:Project PTB-Cavity:Mode Matching second try|Mode Matching - two curve mirrors]] (<del>right</del> calculation)\\ 
-==Parameter== +[[:groups:Mg:Project PTB-Cavity:Mode Matching third try|Mode Matching - plan and curve mirrors]] (right calculation)\\
-    * 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:**\\ +
-$$g_1=1-\frac{L}{R1} \;\text{and}\; g_2=1-\frac{L}{R2}$$ +
- +
-**Step 2:**\\ +
-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**\\ +
-Position of the beam waist from 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:  +
-$$L=t_1+t_2$$ +
-**Step 4**\\ +
-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: +
-$$D=2*w_0$$ +
-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.\\ +
-We did the calculation wrong. :-( We used D=w_0 and got f=2.66022mm. +