Improve residual gas scattering calculations

This allows you to specify the partial pressure of each species as a function of position along the beamtubes instead of assuming that they are constant. An example is given in test/noise/residualgas/AplusSuperGas.

The original gas scattering calculation also approximated the integral by expanding the exponential to first order. This approximation is suspect and has been replaced with the same numerical integral that is needed for arbitrary pressure profiles. The following two plots show the difference between the new exact calculations and the old approximate ones for A+

gwinc/noise/residualgas.py

@@ -2,7 +2,9 @@
 
 '''
 from __future__ import division
+import numpy as np
 from numpy import sqrt, log, pi
+from scipy.integrate import trapz
 
 from .. import const
 from .. import Struct
@@ -60,10 +62,15 @@ def ResidualGasScattering_constructor(species_name):
 
         def calc(self):
             cavity = arm_cavity(self.ifo)
+            Larm_m = self.ifo.Infrastructure.Length
             species = self.ifo.Infrastructure.ResidualGas[species_name]
+            # position along the beamtube starting from vertex corner
+            tube_pos = np.linspace(0, Larm_m, 100)
+            # pressure profile is constant by default
+            pressure_Pa = species.BeamtubePressure * np.ones_like(tube_pos)
             n = residual_gas_scattering_arm(
-                self.freq, self.ifo, cavity, species)
-            dhdl_sqr, sinc_sqr = dhdl(self.freq, self.ifo.Infrastructure.Length)
+                self.freq, self.ifo, cavity, species, pressure_Pa, tube_pos)
+            dhdl_sqr, sinc_sqr = dhdl(self.freq, Larm_m)
             return n * 2 / sinc_sqr
 
     return GasScatteringSpecies
@@ -134,16 +141,16 @@ class ResidualGas(nb.Budget):
     ]
 
 
-def residual_gas_scattering_arm(f, ifo, cavity, species):
-    """Residual gas noise strain spectrum due to scattering from one arm
-
-    Noise caused by the passage of residual gas molecules through the
-    laser beams in one arm cavity due to scattering.
+def residual_gas_scattering_arm(
+        f, ifo, cavity, species, pressure_Pa, tube_pos):
+    """Residual gas scattering from one arm using measured beamtube pressures
 
     :f: frequency array in Hz
     :ifo: gwinc IFO structure
     :cavity: arm cavity structure
     :species: molecular species structure
+    :pressure_Pa: beamtube pressure profile in Pa
+    :tubepos_m: vector of positions where pressure is given in m
 
     :returns: arm strain noise power spectrum at :f:
 
@@ -151,37 +158,26 @@ def residual_gas_scattering_arm(f, ifo, cavity, species):
     E. Zucker, and Stanley E. Whitcomb in their paper Optical
     Pathlength Noise in Sensitive Interferometers Due to Residual Gas.
 
-    Added to Bench by Zhigang Pan, Summer 2006
-    Cleaned up by PF, Apr 07
-    Eliminated numerical integration and substituted first order
-    expansion of exp, to speed it up.
-
     """
-    L = ifo.Infrastructure.Length
+    # beam profile
+    ww_m = cavity.w0 * np.sqrt(1 + ((tube_pos - cavity.zBeam_ITM)/cavity.zr)**2)
     kT = ifo.Infrastructure.Temp * const.kB
-    P = species.BeamtubePressure
     M = species.mass
     alpha = species.polarizability
 
-    rho = P / (kT)                   # number density of Gas
-    v0 = sqrt(2*kT / M)              # mean speed of Gas
-
-    waist = cavity.w0                # Gaussian beam waist size
-    zr = cavity.zr                   # Rayleigh range
-    z1 = -cavity.zBeam_ITM           # location of ITM relative to the waist
-    z2 = cavity.zBeam_ETM            # location of ETM relative to the waist
-
-    # The exponential of Eq. 1 of P940008 is expanded to first order; this
-    # can be integrated analytically
-    zint = log(z2 + sqrt(z2**2 + zr**2)) - log(z1 + sqrt(z1**2 + zr**2))
-    zint = zint * zr/waist
-    zint = zint - 2*pi*L*f/v0
-    # optical path length for one arm
-    zint = zint*((4*rho*(2*pi*alpha)**2)/v0)
-    # eliminate any negative values due to first order approx.
-    zint[zint < 0] = 0
-
-    return zint
+    v0 = np.sqrt(2*kT / M)  # most probable speed of Gas
+
+    alpha = species.polarizability
+
+    # put the integrand into a (numfreq, numpressure) array for faster
+    # integration with trapz
+    integrand = np.exp(np.einsum('i,j->ij', -2*np.pi*f, ww_m/v0))
+    integrand *= np.einsum('i,j->ij', np.ones_like(f), pressure_Pa / ww_m)
+    zint = trapz(integrand, tube_pos, axis=1)
+
+    noise = 4 * (2*np.pi*alpha)**2 / (v0 * kT) * zint
+
+    return noise
 
 
 def residual_gas_damping_test_mass(f, ifo, species, sustf, squeezed_film):