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Entry  Wed Feb 11 18:11:46 2015, Koji, Summary, LSC, 3f modulation cancellation 
    Reply  Thu Feb 12 23:47:45 2015, Koji, Update, LSC, 3f modulation cancellation 
       Reply  Sat Feb 14 00:48:13 2015, Koji, Update, LSC, 3f modulation cancellation 
          Reply  Sat Feb 14 19:54:04 2015, Koji, Summary, LSC, 3f modulation cancellation beat_setup1.JPGbeat_setup2.JPGelectrical_setup.pdf
             Reply  Sat Feb 14 20:37:51 2015, Koji, Summary, LSC, 3f modulation cancellation beat_pd_response.pdfbeat_nominal.pdf3f_reduction.pdfbeat_3f_reduced.pdf
                Reply  Sat Feb 14 22:14:02 2015, Koji, Summary, LSC, [HOW TO] 3f modulation cancellation freq_gen_box.JPGdelay_line.JPGcable_spec.pdf
                   Reply  Sun Feb 15 16:20:44 2015, Koji, Summary, LSC, [ELOG LIST] 3f modulation cancellation 
                      Reply  Sun Feb 15 20:55:48 2015, rana, Summary, LSC, [ELOG LIST] 3f modulation cancellation 
                         Reply  Mon Feb 16 00:08:44 2015, Koji, Summary, LSC, [ELOG LIST] 3f modulation cancellation 
                   Reply  Mon Feb 16 01:45:12 2015, Koji, Summary, LSC, modulation depth analysis modulation_nominal.pdfmodulation_3f_reduced.pdf
Message ID: 11005     Entry time: Wed Feb 11 18:11:46 2015     Reply to this: 11019
Author: Koji 
Type: Summary 
Category: LSC 
Subject: 3f modulation cancellation 

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 ]
\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]
\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\}

Therefore what we want to realize is the following "extinction" condition
J_3(m_1) J_0(m_2) + J_2(m_1) J_1(m_2) = 0

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

m_1 + 3 m_2 = 0

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 \}

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

 

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