michelson.ipynb

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   "source": [
    "import finesse\n",
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "finesse.configure(plotting=True)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "84dfe14e-fa26-44e2-a5b0-a94c50213339",
   "metadata": {},
   "source": [
    "## Michelson model"
   ]
  },
  {
   "cell_type": "markdown",
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   "metadata": {},
   "source": [
    "<img src=\"images/michelson.svg\" width=\"400\"/>"
   ]
  },
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   "id": "5597e0c8-53ec-4fd4-a254-90d6a5751851",
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   "outputs": [],
   "source": [
    "base = finesse.Model()\n",
    "base.parse(\n",
    "    \"\"\"\n",
    "    \n",
    "    # Add a Laser named l1 with a power of 1 W.\n",
    "    l l1 P=1M\n",
    "\n",
    "    # Space attaching l1 <-> m1 with length of 1 m (default).\n",
    "    s s1 l1.p1 bs1.p1 L=1\n",
    "    \n",
    "    # Beam splitter\n",
    "    bs bs1 R=0.5 T=0.5 alpha=45\n",
    "    \n",
    "    # other two spaces\n",
    "    s s2 bs1.p3 mX.p1 L=4000\n",
    "    s s3 bs1.p2 mY.p1 L=4000\n",
    "    \n",
    "    # End mirrors\n",
    "    m mX R=1 T=0 phi=0\n",
    "    m mY R=1 T=0 phi=90.006\n",
    "    \n",
    "    \"\"\"\n",
    ")"
   ]
  },
  {
   "cell_type": "code",
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   "id": "f066eedb-a9ba-4197-b07e-289b6b19b698",
   "metadata": {},
   "outputs": [],
   "source": [
    "michelson = base.deepcopy()\n",
    "michelson.parse(\n",
    "    \"\"\"\n",
    "    # Power detectors on reflection and transmission.\n",
    "    pd pout bs1.p4.o\n",
    "    xaxis(mX.phi, lin, 0, 360, 361)\n",
    "    \"\"\"\n",
    ")"
   ]
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   "source": [
    "out = michelson.run()\n",
    "out.plot();"
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   "cell_type": "markdown",
   "id": "8ba54ee2-20a6-4a47-ac41-c16ac6ddfba5",
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   "source": [
    "## Response to gravitational waves"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 146,
   "id": "f6d6e393-41fb-49b2-8421-1de19833c0c8",
   "metadata": {},
   "outputs": [],
   "source": [
    "michelson = base.deepcopy()\n",
    "michelson.parse(\n",
    "    \"\"\"\n",
    "    fsig(1)\n",
    "    sgen armx s2.h 1 0\n",
    "    sgen army s3.h 1 180\n",
    "    pd1 pout bs1.p4.o f=fsig.f\n",
    "    # Sweep the frequency of the amplitude modulation\n",
    "    xaxis(fsig.f, log, 1, 1e5, 2000)\n",
    "    \"\"\"\n",
    ")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 147,
   "id": "fed7ada2-96fd-4eea-8433-bc4792158272",
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   "source": [
    "out = michelson.run()\n",
    "H_s = abs(out[\"pout\"])\n",
    "fig=plt.figure()\n",
    "ax1 = plt.subplot(211)\n",
    "ax2 = plt.subplot(212, sharex = ax1)\n",
    "ax1.semilogx(out.x[0], abs(out[\"pout\"]))\n",
    "ax2.semilogx(out.x[0], np.angle(out[\"pout\"],deg=True))\n",
    "ax1.set_ylabel('Amplitude (W/h)')\n",
    "ax2.set_ylabel('Phase (deg)')\n",
    "ax2.set_xlabel('Freq (Hz)')\n",
    "plt.show()"
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  },
  {
   "cell_type": "markdown",
   "id": "6729b17b-50b2-4281-8cc2-b2740fec0b38",
   "metadata": {},
   "source": [
    "#### Questions:\n",
    "- What does this transfer function tell about the detector's sensitivity to low and high frequency gravitational waves?\n",
    "- The magnitude of the transfer function drops off as we go to higher frequencies, why?"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "f1869a79-058d-4c93-8cc4-1d1ec063b9c1",
   "metadata": {},
   "source": [
    "## Sensitivity of gravitational wave detectors"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "a09aef8a-3a3d-4c4d-8ffe-305581e01c8a",
   "metadata": {},
   "source": [
    "### quantum noise"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 148,
   "id": "d46df2b6-7e89-42b2-8ba2-0037a69c7092",
   "metadata": {},
   "outputs": [],
   "source": [
    "ifo = base.deepcopy()\n",
    "ifo.parse(\n",
    "    \"\"\"\n",
    "    fsig(1)\n",
    "    qshot shot_noise bs1.p4.o\n",
    "    qnoised sqzd_noise bs1.p4.o\n",
    "    # Sweep the frequency of the amplitude modulation\n",
    "    xaxis(fsig.f, log, 1, 1e4, 2000)\n",
    "    \"\"\"\n",
    ")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 149,
   "id": "d6554038-6331-4f00-b0b5-f5d79d804c16",
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   "source": [
    "out = ifo.run()\n",
    "plt.semilogx(out.x[0], abs(out[\"sqzd_noise\"]),label='quantum')\n",
    "plt.semilogx(out.x[0], abs(out[\"shot_noise\"]),label='shot')\n",
    "plt.ylabel('Amplitude (W/$\\sqrt{Hz}$)')\n",
    "plt.xlabel('Freq (Hz)')\n",
    "plt.legend()\n",
    "plt.show()"
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  },
  {
   "cell_type": "code",
   "execution_count": 150,
   "id": "d7c79529-6655-4928-b175-c5bdb6cb3d16",
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   "source": [
    "ifo = base.deepcopy()\n",
    "ifo.parse(\n",
    "    \"\"\"\n",
    "    free_mass mX_sus mX.mech mass=40\n",
    "    free_mass mY_sus mY.mech mass=40\n",
    "    fsig(1)\n",
    "    qshot shot_noise bs1.p4.o\n",
    "    qnoised sqzd_noise bs1.p4.o\n",
    "    # Sweep the frequency of the amplitude modulation\n",
    "    xaxis(fsig.f, log, 1, 1e4, 2000)\n",
    "    \"\"\"\n",
    ")"
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    "out = ifo.run()\n",
    "plt.semilogx(out.x[0], abs(out[\"sqzd_noise\"]),label='quantum')\n",
    "plt.semilogx(out.x[0], abs(out[\"shot_noise\"]),label='shot')\n",
    "plt.ylabel('Amplitude (W/$\\sqrt{Hz}$)')\n",
    "plt.xlabel('Freq (Hz)')\n",
    "plt.legend()\n",
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   "source": [
    "## sensitivity TFs"
   ]
  },
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   "id": "e8dc07e4-d82a-4dee-a8c1-0554ea8abceb",
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    "For a noise source characterized by the noise spectral density $N_s$, the equivalent strain amplitude spectral density $S_s$ is given by\n",
    "\n",
    "$$S_s(f) = \\frac{N_s(f) |H_n(f)|}{|H_s(f)|}$$\n",
    "\n",
    "where $H_n(f)$ is the transfer function from the corresponding noise to the output detector, and $H_s(f)$ is the transfer function of the gravitational wave signal to the output of the detector. The value $S_s$ tells us how much a particular noise appears to look like a gravitational wave signal, thus limits our sensitivity. For example for mirror motion due to seismic noise, the units of this work out as:\n",
    "\n",
    "$$\\frac{[m/\\sqrt{Hz}] [W/m]}{[W/h]} = \\frac{h}{\\sqrt{Hz}}$$\n",
    "\n",
    "For quantum nosie above, we acqually get $N_s(f) |H_n(f)|$, we to get the GW sensitivity we just have to divided by $H_s$ which is calculated above.\n"
   ]
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   "cell_type": "code",
   "execution_count": 152,
   "id": "4ec48753-29bd-45de-a107-d0627c4a62b1",
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   "source": [
    "quantum = abs(out[\"sqzd_noise\"])/H_s\n",
    "shot = abs(out[\"shot_noise\"])/H_s\n",
    "freq = out.x[0]\n",
    "plt.loglog(freq, quantum)\n",
    "plt.ylabel('Amplitude (h/$\\sqrt{Hz}$)')\n",
    "plt.xlabel('Freq (Hz)')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "4055d9dc-4bb2-4421-b223-45d47c3e41b9",
   "metadata": {},
   "source": [
    "## seismic noise"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 153,
   "id": "8ad4b54e-8a25-41b9-9dde-53993c11f516",
   "metadata": {},
   "outputs": [],
   "source": [
    "ifo = base.deepcopy()\n",
    "ifo.parse(\n",
    "    \"\"\"\n",
    "    free_mass mX_sus mX.mech mass=40\n",
    "    free_mass mY_sus mY.mech mass=40\n",
    "    fsig(1)\n",
    "    sgen sig1 mX.mech.z 1 0\n",
    "    pd1 seism_tf bs1.p4.o f=fsig.f\n",
    "    xaxis(fsig.f, log, 1, 1e4, 2000)\n",
    "    \"\"\"\n",
    ")"
   ]
  },
  {
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   "execution_count": 154,
   "id": "97081aef-fef0-42e1-b343-a7e21ab52973",
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   "source": [
    "out = ifo.run()\n",
    "seismic_tf = abs(out[\"seism_tf\"])\n",
    "plt.semilogx(out.x[0], abs(out[\"seism_tf\"]),label='seismic')\n",
    "plt.ylabel('Amplitude (W/m)')\n",
    "plt.xlabel('Freq (Hz)')\n",
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   "cell_type": "code",
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   "id": "31270a2c-31ce-43b3-94d7-daf7c2634e97",
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   "source": [
    "def seismic_noise_ASD(f):\n",
    "    f = np.array(f)\n",
    "    N = 1e-4/np.abs(np.random.normal(f**15, 0, size=f.size))\n",
    "    return N\n",
    "\n",
    "N_s = seismic_noise_ASD(freq)\n",
    "seismic = N_s * seismic_tf / H_s"
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   "source": [
    "total = np.sqrt(quantum**2 + seismic**2)\n",
    "plt.loglog(freq, quantum,label='quantum')\n",
    "plt.loglog(freq, seismic,label='seismic')\n",
    "plt.loglog(freq, total, label='total')\n",
    "plt.ylabel('Amplitude (h/$\\sqrt{Hz}$)')\n",
    "plt.xlabel('Freq (Hz)')\n",
    "plt.ylim([1e-30, 1e-6])\n",
    "plt.legend()\n",
    "plt.show()"
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