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Entry  Thu Jul 10 16:51:14 2014, Sam Moore, Optics, General, Duan and Heinert Comparison 7_10_14.pdf
    Reply  Mon Jul 14 19:14:31 2014, Sam Moore, Optics, General, Duan and Heinert Comparison duan_heinert_comparisonInfinite-eps-converted-to.pdf
Message ID: 96     Entry time: Mon Jul 14 19:14:31 2014     In reply to: 93
Author: Sam Moore 
Type: Optics 
Category: General 
Subject: Duan and Heinert Comparison 


(See Plots in attached document)


My plan has been to replicate Duan's numerical thermoconductive (TE + TR) phase noise plot presented in his paper (section V).  I am trying to match Duan's analytical expression with Heinert's analytical expression.  This requires some rescaling of Heinert's TR displacement noise. (I also needed to divide Heinert's expression by 4 pi to match the Fourier Transform convention. )   Duan's analytical expression for the phase noise is obtained by evaluating the triple integral given in equation 13 of the Duan paper "General Treatment of Thermal Noise in Optical Fibers".

It turns out that an additional factor of 2 multiplies the phase noise because Duan's Fourier Transform only takes into account positive frequencies; there are also negative frequencies that occur in equal amplitude.  
This integral was evaluated in Mathematica due to numerical noise in MATLAB's calculation.  The calculation in Mathematica was very slow, so the upper limits on the integral were truncated.  The following plots in the attached document show the resulting noise profile agreements for two different upper limits.
If the residual for the highest upper limit is considered acceptable for a match between the two plots, then I will use Heinert's plot as a reference when using the COMSOL steady-state method for Duan's numerical case (Heinert's plot runs much faster).

 I have now plotted the Duan-Heinert comparison for the case of an infinite upper bound.  It turns out that the curves differ by a maximum of 10 percent for low frequencies.  Such a discrepancy has been attributed to lack of experimental investigation into this regime (according to Duan). For our purposes, such a discrepancy is acceptable. We will therefore use the Heinert curve for subsequent calculations due to its faster computation time.


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