====== Cavity spectrum: plan/curved mirror ======


Caculates the cavity spectrum and the frequency distance between TEM_00 and higher order modes m+n. Two curved mirrors possible. Programm made by T. Legero. 
==Parameter==
    * Radius of curvature of mirror R1: R1= 1 m
    * Radius of curvature of mirror R2: R2= 1000000000 m
    * Cavity Length: L= 0.48 m
    * Reflectivity of mirrors: R= 0.99998
    * Speed of light: c= 299792458 m/s
    
**Step 1: Calculate the Free-Spectral-Rang and Gouy-Phase**\\
  * Free-Spectral-Rang: FSR
  * Gouy-Phase: Gouy
$$FSR = \frac{c}{2\cdot L}$$  
$$Gouy = \frac{FSR}{\pi} \cdot \arccos \left( \sqrt{(1-\frac{L}{R1})\cdot(1-\frac{L}{R2}} \right)$$

=>
  * FSR = 312.284 MHz
  * Gouy = 76.0824 MHz

**Step 2: Calculate the frequency difference between TEM_00 and TEM_mn**\\
  * TEM mode number: mn
  * Next TEM_00: w
  * Frequency difference between TEM_00 and TEM_mn: Delta
$$ \Delta = FSR - mn\cdot Gouy  $$

  * Define maximum of mn and w:
    * mn_max = 100
    * w_max = 100
  * Define maximum of shown frequency difference to TEM_00:
    * abs(delta(mn,w)) < 10E6

  * Frequency difference von (q-w)ter höherer Mode mn zu q-ter TEM_00-Mode (Matrix): 
$$ \Delta(mn,w) = w\cdot FSR - mn\cdot Gouy $$

=>

Freq. diff. of m+n = 4 higher order mode to 00-mode is 7.95436 MHz\\
Freq. diff. of m+n = 37 higher order mode to 00-mode is -4.49309 MHz\\
Freq. diff. of m+n = 41 higher order mode to 00-mode is 3.46128 MHz\\
Freq. diff. of m+n = 74 higher order mode to 00-mode is -8.98616 MHz\\
Freq. diff. of m+n = 78 higher order mode to 00-mode is -1.03181 MHz\\
Freq. diff. of m+n = 82 higher order mode to 00-mode is 6.92256 MHz\\


**Step 3: Calculate the frequency difference between TEM_00 and the nearest TEM_nm **\\

  * Define the Modenumber of Higher-Order-Mode to plot (next Mode to 00):
    * mn = 4
  * Define the Vary Mirror-Curvature:
    * R = 0.95:0.01:1.05
  * Frequency difference between TEM_00 and TEM_mn:
$$ \Delta = FSR - mn\cdot Gouy $$

  * plot(R,Delta)**[Important!]**:
{{:groups:mg:project_ptb-cavity:cavity_spectrum:roc_vs_freqeuncy_shift_to_tem00.png?400|}}
<note important>Achtung: ROC wird hoffentlich >1m werden, damit die m+n = 4 Mode weiter weg ist! </note>

  * plot(Gouy,Delta):
{{:groups:mg:project_ptb-cavity:cavity_spectrum:gouy_vs_freqeuncy_shift_to_tem00.png?400|}}

  * plot(R,Gouy):
{{:groups:mg:project_ptb-cavity:cavity_spectrum:roc_vs_gouy.png?400|}}