33MHz sidebands can be elliminated by careful choice of the modulation depths and the relative phase between the modulation signals.
If this condition is realized, the REFL33 signals will have even more immunity to the arm cavity signals because the carrier signal will lose
its counterpart to produce the signal at 33MHz.
Formulation of double phase modulation
m1: modulation depth of the f1 modulation
m2: modulation depth of the f2 (=5xf1) modulation
The electric field of the beam after the EOM
![E=E_0 \exp \left[ {\rm i} \Omega t + m_1 \cos \omega t +m_2 \cos 5 \omega t \right ]](http://latex.codecogs.com/gif.latex?%5Cdpi%7B120%7D%20E%3DE_0%20%5Cexp%20%5Cleft%5B%20%7B%5Crm%20i%7D%20%5COmega%20t%20+%20m_1%20%5Ccos%20%5Comega%20t%20+m_2%20%5Ccos%205%20%5Comega%20t%20%5Cright%20%5D)
 ![\flushleft = {\it E}_0 e^{{\rm i} \Omega t} \\ \times \left[ J_0(m_1) + J_1(m_1) e^{{\rm i} \omega t}- J_1(m_1) e^{-{\rm i} \omega t} + J_2(m_1) e^{{\rm i} 2\omega t}+ J_2(m_1) e^{-{\rm i} 2\omega t} + J_3(m_1) e^{{\rm i} 3\omega t}- J_3(m_1) e^{-{\rm i} 3\omega t} + \cdots \right] \\ \times \left[ J_0(m_2) + J_1(m_2) e^{{\rm i} 5 \omega t}- J_1(m_2) e^{-{\rm i} 5 \omega t} + \cdots \right]](http://latex.codecogs.com/gif.latex?%5Cdpi%7B120%7D%20%5Cflushleft%20%3D%20%7B%5Cit%20E%7D_0%20e%5E%7B%7B%5Crm%20i%7D%20%5COmega%20t%7D%20%5C%5C%20%5Ctimes%20%5Cleft%5B%20J_0%28m_1%29%20+%20J_1%28m_1%29%20e%5E%7B%7B%5Crm%20i%7D%20%5Comega%20t%7D-%20J_1%28m_1%29%20e%5E%7B-%7B%5Crm%20i%7D%20%5Comega%20t%7D%20+%20J_2%28m_1%29%20e%5E%7B%7B%5Crm%20i%7D%202%5Comega%20t%7D+%20J_2%28m_1%29%20e%5E%7B-%7B%5Crm%20i%7D%202%5Comega%20t%7D%20+%20J_3%28m_1%29%20e%5E%7B%7B%5Crm%20i%7D%203%5Comega%20t%7D-%20J_3%28m_1%29%20e%5E%7B-%7B%5Crm%20i%7D%203%5Comega%20t%7D%20+%20%5Ccdots%20%5Cright%5D%20%5C%5C%20%5Ctimes%20%5Cleft%5B%20J_0%28m_2%29%20+%20J_1%28m_2%29%20e%5E%7B%7B%5Crm%20i%7D%205%20%5Comega%20t%7D-%20J_1%28m_2%29%20e%5E%7B-%7B%5Crm%20i%7D%205%20%5Comega%20t%7D%20+%20%5Ccdots%20%5Cright%5D)
![\flushleft = {\it E}_0 e^{{\rm i} \Omega t} \\ \times \left\{ \cdots + \left[ J_3(m_1) J_0(m_2) + J_2(m_1) J_1(m_2) \right] e^{{\rm i} 3 \omega t} - \left[ J_3(m_1) J_0(m_2) + J_2 (m_1) J_1(m_2) \right] e^{-{\rm i} 3 \omega t} + \cdots \right\}](http://latex.codecogs.com/gif.latex?%5Cdpi%7B120%7D%20%5Cflushleft%20%3D%20%7B%5Cit%20E%7D_0%20e%5E%7B%7B%5Crm%20i%7D%20%5COmega%20t%7D%20%5C%5C%20%5Ctimes%20%5Cleft%5C%7B%20%5Ccdots%20+%20%5Cleft%5B%20J_3%28m_1%29%20J_0%28m_2%29%20+%20J_2%28m_1%29%20J_1%28m_2%29%20%5Cright%5D%20e%5E%7B%7B%5Crm%20i%7D%203%20%5Comega%20t%7D%20-%20%5Cleft%5B%20J_3%28m_1%29%20J_0%28m_2%29%20+%20J_2%20%28m_1%29%20J_1%28m_2%29%20%5Cright%5D%20e%5E%7B-%7B%5Crm%20i%7D%203%20%5Comega%20t%7D%20+%20%5Ccdots%20%5Cright%5C%7D)
Therefore what we want to realize is the following "extinction" condition
We are in the small modulation regime. i.e. J0(m) = 1, J1(m) = m/2, J2(m) = m2/8, J3(m) = m3/48
Therefore we can simplify the above exitinction condition as
m2 < 0 means the start phase of the m2 modulation needs to be 180deg off from the phase of the m1 modulation.
![E = E_0 \exp\left\{ {\rm i} [\Omega t + m_1 \cos \omega t + \frac{m_1}{3} \cos (5 \omega t + \pi)] \right \}](http://latex.codecogs.com/gif.latex?%5Cdpi%7B120%7D%20E%20%3D%20E_0%20%5Cexp%5Cleft%5C%7B%20%7B%5Crm%20i%7D%20%5B%5COmega%20t%20+%20m_1%20%5Ccos%20%5Comega%20t%20+%20%5Cfrac%7Bm_1%7D%7B3%7D%20%5Ccos%20%285%20%5Comega%20t%20+%20%5Cpi%29%5D%20%5Cright%20%5C%7D)
| Field amplitude |
m1=0.3, m2=-0.1 |
m1=0.2, m2=0.2 |
| Carrier |
0.975 |
0.980 |
| 1st order sidebands |
0.148 |
9.9e-2 |
| 2nd |
1.1e-3 |
4.9e-3 |
| 3rd |
3.5e-7 |
6.6e-4 |
| 4th |
7.4e-3 |
9.9e-3 |
| 5th |
4.9e-2 |
9.9e-2 |
| 6th |
7.4e-3 |
9.9e-3 |
| 7th |
5.6e-4 |
4.9e-4 |
| 8th |
1.4e-5 |
4.1e-5 |
| 9th |
1.9e-4 |
5.0e-4 |
| 10th |
1.2e-3 |
4.9e-3 |
| 11th |
1.9e-4 |
5.0e-4 |
| 12th |
1.4e-5 |
2.5e-5 |
| 13th |
4.7e-7 |
1.7e-6 |
| 14th |
3.1e-6 |
1.7e-5 |
| 15th |
2.0e-5 |
1.6e-4 |
|