[[LCGTNotes]]
* Phase Noise Requirement for Green Laser [#d3bb1ce3]
** Differential frequency noise requirement [#z7c239c3]
The green laser is phase locked to the main laser (PSL) with a PLL. The relative frequency between the PSL and green will be determined by this PLL. The noise of the PLL local oscillator will be directly introduced as the differential frequency fluctuation of PSL and green. The required level of this differential frequency noise can be roughly derived as follows.
The FSR of the arm cavity is about 50kHz. Since the finesse of the arm cavity is about 1500, the FWHM of the carrier resonance is about 33Hz. When the arm is locked to the green laser, the PSL carrier has to be sitting at somewhere in the FSR, ideally without moving. In reality the carrier moves around in the FSR due to the differential frequency noise. Since we will slowly bring the PSL carrier to the resonance, we want to keep the wondering of the PSL carrier much less than the FWHM of the resonance. Let's say we want to keep it smaller than 1/1000 of the resonance width. Then the required differential frequency noise is &math2(\Delta f <33/1000 = 0.033\,\mathrm{Hz});.
** Conversion of the SSB phase noise to frequency noise [#x0b218d2]
Usually the noise performance of an oscillator is specified by single sideband (SSB) phase noise in the unit of dBc. This is the ratio of the sideband power at a specified offset (&math2(f_\mathrm{ofs});) to the carrier power measured in 1Hz bandwidth. The term SSB means that the sideband power is measured at one side of the spectral side robe.
We have to convert this SSB noise to the RMS frequency fluctuation. Suppose that spectrum of SSB phase noise is given as &math2(N^2_\mathrm{SSB}(f));. It is powered because we want &math2(N_\mathrm{SSB}(f)); to be the amplitude ratio.
Now, if the phase noise at a particular frequency &math2(f); has the amplitude &math2(m(f));:
#math2(\exp[i\{\Omega t + m \sin (\omega t)\}] \simeq J_0(m) \exp(i\Omega t) \pm i J_1(m) \exp(\pm i\omega t)).
Therefore,
#math2(N_\mathrm{SSB}(f) = \frac{J_1(m)}{J_0(m)} \simeq \frac{m(f)}{2}=\frac{m^\mathrm{RMS}(f)}{\sqrt{2}})
where &math2(m^\mathrm{RMS}(f) = m(f)/\sqrt{2}); is the RMS fluctuation of the phase at frequency &math2(f);.
Since the frequency noise is just the time derivative of phase noise, the RMS frequency fluctuation at frequency &math2(f); in 1Hz BW (&math2(f_\mathrm{n}^\mathrm{RMS}(f));) is given by
#math2(f_\mathrm{n}^\mathrm{RMS}(f) = f m^\mathrm{RMS}(f) = \sqrt{2} f N_\mathrm{SSB}(f) )
The total RMS frequency fluctuation is given by the following integral:
#math2(\Delta f = \sqrt{2\int^\infty_0 \{f_\mathrm{n}^\mathrm{RMS}(f)\}^2 df } = 2\sqrt{\int^\infty_0 N^2_\mathrm{SSB}(f)\cdot f^2 df})
** 1/f phase noise [#cf6cdd41]
Assuming 1/f noise shape of &math2(N^2_\mathrm{SSB}(f));, the total frequency fluctuation can be written like this:
#math2(N^2_\mathrm{SSB}(f) = \frac{N^2_0}{f}\quad \rightarrow \quad \Delta f = 2\sqrt{\int^{f_\mathrm{max}}_0 N^2_0\cdot f df} = \sqrt{2} N_0 f_\mathrm{max})
where &math2(N^2_0); is the SSB phase noise at 1Hz and &math2( f_\mathrm{max}); is the maximum frequency for integration. Without this limit, the integral will diverge.
With &math2(N^2_0); = -110dBc (about the spec. of R&S SMB100) and &math2( f_\mathrm{max}); = 1kHz, the resultant &math2(\Delta f); = 0.028Hz.
With &math2(N^2_0); = -110dBc (about the spec. of R&S SMB100) and &math2( f_\mathrm{max}); = 1kHz, the resultant &math2(\Delta f); = 4.5mHz.
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