PGM_calcs.ipynb

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    "# Generation of equations for PGM Energy pseudo-axis\n",
    "This notebook is used to derive the equations required to add a energy ($E_{ph}$) pseudo-axis to the Sirepo-Bluesky PGM ophyd object. It uses equations provided by Joe Dvorak. In addition to the Energy pseudo-axis it will require the Sirepo-Bluesky PGM ophyd object to have pseudo-axes for the fixed foxus constant ($c_{ff}$) and the Grating as well as some internal parameters (defined at instantiation). It also assumes that the Sirepo-Bluesky PGM ophyd object already has axes related to the angle of the M2 mirror ($\\Theta_{M2}$) and the grating ($\\Theta_{Gr}$). In the following I assume all distances are in mm and all angels are in degrees. A full list of these parameters is given here\n",
    "\n",
    "- $\\Theta_{M2}$/$\\Theta_{Gr}$ : M2/Grating mirror angles, axis from simulation parameters.\n",
    "- $a_{0}$/$a_{1}$/$a_{2}$/$a_{3}$ : Grating equation parameters, axes from simulation parameters.\n",
    "- $E_{ph}$ : Photon Energy, required pseudo-axis.\n",
    "- Grating : Grating pseudo-axis, updates $a_{0}$, $a_{1}$, $k$ and $c_{ff}$ when changed.\n",
    "- $r_{2}$: output focal length, required pseudo-axis.\n",
    "- $c_{ff}$ : Fixed Focus Constant, required read-only pseudo-axis, updates on changes to $E_{ph}$ or $r_{2}$.\n",
    "- $r_{1}$: input focal length, instantiation parameter.\n",
    "- $m$ : diffraction order (usually +1), instantiation parameters.\n",
    "- $X_{inc}$/$X_{diff}$ : incident/diffracted X-ray beam angles, instantiation parameters.\n",
    "- $b$ : bounce direction (1=up, -1=down), instantiation parameter.\n",
    "\n",
    "The relationships between these parameters are given by:\n",
    "\n",
    "$$\n",
    "c_{ff} = \\sqrt((B_{0}+B_{1})/B{2})\\\\\n",
    "\\begin{align}\n",
    "\\text{where: } B_{0} &= 2A_{1}+4(A_{1}/A_{0})^2+(4+2A_{1}-A_{0}^{2})(\\frac{r_{2}}{r_{1}})\\\\\n",
    "B_{1} &= -4(A_{1}/A_{0})\\sqrt((\\frac{r_{2}}{r_{1}})^2+2A_{1}(1+r)-A_{0}^{2}(\\frac{r_{2}}{r_{1}}))\\\\\n",
    "B_{2} &= -4+A_{0}^{2}-4A_{1}+4(A_{1}/A_{0})^2\\\\\n",
    "A_{0} &= m\\lambda a_{0}\\\\\n",
    "A_{1} &= -\\frac{1}{2}m\\lambda r_{2}a_{1}\\\\\n",
    "\\lambda &= (12398.4197/E_{ph})1e-7\\\\\n",
    "\\end{align}\n",
    "\\tag{Eqn 1}\n",
    "$$\n",
    "  \n",
    "$$\n",
    "\\Theta_{M2} = \\frac{1}{2}(X_{diff}+X_{inc}+b(180-\\alpha+\\beta))\\\\\n",
    "\\begin{align}\n",
    "\\text{where: } \\alpha &=sin^{-1}(-ma_{0}\\lambda /(c_{eff}^{2}-1)+\\sqrt(1+(c_{ff}ma_{0}\\lambda /(c_{eff}^{2}-1))^2))\\\\\n",
    "\\beta &= sin^{-1}(ma_{0}\\lambda-sin(\\alpha))\\\\\n",
    "\\lambda &= (12398.4197/E_{ph})1e-7\\\\\n",
    "\\end{align}\n",
    "\\tag{Eqn 2}\n",
    "$$\n",
    "  \n",
    "$$\n",
    "\\theta_{GR} = b(90+\\beta)+X_{diff}\n",
    "\\tag{Eqn 3}\n",
    "$$\n",
    "  \n",
    "$$\n",
    "E_{ph}=(12398.4197/\\lambda)1e-7\\\\\n",
    "\\begin{align}\n",
    "\\text{where: } \\lambda &= (sin(\\alpha)+sin(\\beta))/(ma_{0})\\\\\n",
    "\\beta &= - 90 +b(\\Theta_{Gr} - X_{diff})\\\\\n",
    "\\alpha &= 80 + \\beta + b(X_{diff} + X_{inc} - 2\\Theta_{M2})\n",
    "\\end{align}\n",
    "\\tag{Eqn 4}\n",
    "$$\n"
   ]
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   "cell_type": "code",
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   "source": [
    "import numpy as np\n",
    "# define the instantiation parameters here.\n",
    "_m=1\n",
    "_r1=33350 #in mm: 33350 for ARI and 33000 for SXN\n",
    "_r2=11500 #in mm: 11500 for ARI and 17500 for SXN\n",
    "_x_inc=90 #in degrees\n",
    "_x_diff=90 #in degrees\n",
    "_b=1 #bounce direction, 1 is up -1 is down\n",
    "_grating='lowE'\n",
    "_ari_gratings={'lowE':{'a0':50,'a1':.01868,'a2':1.95E-06,'a3':4E-9},\n",
    "              'HighE':{'a0':50,'a1':.02986,'a2':2.87E-06,'a3':8E-9},\n",
    "              'HighR':{'a0':200,'a1':.05743,'a2':6.38E-06,'a3':1.5E-8}\n",
    "             }\n",
    "_sxn_gratings={'LowE':{'a0':150,'a1':.04341,'a2':2.6E-06,'a3':1.5E-8},\n",
    "              'MedE':{'a0':350,'a1':.0755,'a2':4.95E-06,'a3':2.5E-8},\n",
    "              'HighE':{'a0':350,'a1':.05739,'a2':4.18E-06,'a3':1.2E-8}\n",
    "             }\n",
    "_gratings=_ari_gratings\n",
    "\n",
    "\n",
    "def _cff(e_ph, grating, r2, r1=_r1, m=_m, gratings=_gratings):\n",
    "    '''\n",
    "    Returns the fine focus constant given the energy, grating and output focal length.\n",
    "    \n",
    "    Based on the photon energy, grating, the output focal length and several other \n",
    "    instantiation time parameters, seen as keywords, this function returns the fine \n",
    "    focus constant using the relations:\n",
    "    $$c_{ff} = \\sqrt((B_{0}+B_{1})/B{2})\\\\\n",
    "        \\begin{align}\n",
    "        \\text{where: } B_{0} &= 2A_{1}+4(A_{1}/A_{0})^2+(4+2A_{1}-A_{0}^{2})\n",
    "                              (\\frac{r_{2}}{r_{1}})\\\\\n",
    "                     B_{1} &= -4(A_{1}/A_{0})\\sqrt((1+\\frac{r_{2}}{r_{1}})^2+\n",
    "                              2A_{1}(1+\\frac{r_{2}}{r_{1}})-\n",
    "                              A_{0}^{2}(\\frac{r_{2}}{r_{1}})\\\\\n",
    "                     B_{2} &= -4+A_{0}^{2}-4A_{2}+4(A_{2}/A_{0})^2\\\\\n",
    "                     A_{0} &= m\\lambda a_{0}\\\\\n",
    "                     A_{1} &= -\\frac{1}{2}m\\lambda r_{2}a_{1}\\\\\n",
    "                     \\lambda &= (12398.4197/E_{ph})1e-7\\\\\n",
    "        \\end{align}\n",
    "    $$\n",
    "\n",
    "    Units for each parameter are: eV for energies, mm for lengths and mm^{-1}/ \n",
    "    mm^{-2}/ mm^{-3}/ mm^{-4} for the respective grating equation parameters.\n",
    "\n",
    "    Parameters\n",
    "    ----------\n",
    "    e_ph : float\n",
    "        The photon energy in eV.\n",
    "    grating : string\n",
    "        The grating name.\n",
    "    r2 : float\n",
    "        The output focal length for the PGM in mm    \n",
    "    r1 : float, optional\n",
    "        The input focal length for the PGM in mm\n",
    "    m : integer, optional\n",
    "        The diffraction order.\n",
    "    gratings : dict, optional\n",
    "        A dictionary that matches grating names to a dictionary of grating parameters.\n",
    "        \n",
    "    Returns\n",
    "    -------\n",
    "    cff : float\n",
    "        The fine focus constant.\n",
    "    '''\n",
    "    \n",
    "    lambda_ = (12398.4197/e_ph)*1E-7 # wavelength in mm from e_ph in eV\n",
    "    A1 = -0.5*m*lambda_*r2*gratings[grating]['a1']\n",
    "    A0 = m*lambda_*gratings[grating]['a0']\n",
    "    B2 = (-4+A0**2-4*A1+4*(A1/A0)**2)\n",
    "    B1 = -4*(A1/A0)*np.sqrt((1+r2/r1)**2+2*A1*(1+r2/r1)-A0**2*(r2/r1))\n",
    "    B0 = 2*A1+4*(A1/A0)**2+(4+2*A1-A0**2)*(r2/r1)\n",
    "    cff = np.sqrt((B0+B1)/B2)\n",
    "    \n",
    "    return cff\n",
    "\n",
    "\n",
    "def _pgm_angles(e_ph, grating, r2, r1=_r1, m=_m, gratings=_gratings,\n",
    "                x_inc=_x_inc, x_diff=_x_diff, b=_b):\n",
    "    '''\n",
    "    Returns the M2/Grating angles given the photon energy, grating and output focal \n",
    "    length.\n",
    "    \n",
    "    Based on the photon energy, grating, the output focal length and several other \n",
    "    instantiation time parameters, seen as keywords, this function returns the \n",
    "    required angles for the M2 mirror and the grating using the _cff function and \n",
    "    the relations:\n",
    "    $$\n",
    "        \\Theta_{M2} = \\frac{1}{2}(X_{diff}+X_{inc}+b(180-\\alpha+\\beta))\\\\\n",
    "        \\Theta_{GR} = b(90+\\beta)+X_{diff}\n",
    "        \\begin{align}\n",
    "        \\text{where: }  \\alpha &=sin^{-1}(-ma_{0}\\lambda /(c_{eff}^{2}-1)+\n",
    "                        \\sqrt(1+(c_{ff}ma_{0}\\lambda /(c_{eff}^{2}-1))^2))\\\\\n",
    "                        \\beta &= sin^{-1}(ma_{0}\\lambda-sin(\\alpha))\\\\\n",
    "                        \\lambda &= (12398.4197/E_{ph})1e-7\\\\\n",
    "        \\end{align}\n",
    "    $$\n",
    "    \n",
    "    Units for each parameter are: eV for energies, mm for lengths, degrees for angles\n",
    "    and mm^{-1}/ mm^{-2}/ mm^{-3}/ mm^{-4} for the respective grating equation \n",
    "    parameters.\n",
    "    \n",
    "    Parameters\n",
    "    ----------\n",
    "    E_ph : float\n",
    "        The photon energy in eV.\n",
    "    grating : string\n",
    "        The grating name.\n",
    "    r2 : float\n",
    "        The output focal length for the PGM in mm    \n",
    "    r1 : float, optional\n",
    "        The input focal length for the PGM in mm\n",
    "    m : integer, optional\n",
    "        The diffraction order.\n",
    "    gratings : dict, optional\n",
    "        A dictionary that matches grating names to a dictionary of grating parameters.\n",
    "    x_inc : float, optional\n",
    "        The incident X-ray angle\n",
    "    x_diff : float, optional\n",
    "        The outgoing X-ray angle\n",
    "    b : int, optional\n",
    "        integer indicating the bounce direction: +1 (for upward bounce) or -1 \n",
    "        (for downward bounce)\n",
    "    \n",
    "    Returns\n",
    "    -------\n",
    "    (theta_m2, theta_gr) : (float, float)\n",
    "        The required angles for M2 and the grating in degrees.\n",
    "    '''\n",
    "    ##NOTE if I choose to read in cff from a read-only ophyd signal then I no longer\n",
    "    ## need r2 and r1 as args/kwargs (but I will need cff as an arg)\n",
    "    lambda_ = (12398.4197/e_ph)*1E-7 # wavelength in mm from e_ph in eV\n",
    "    #NOTE the next line may be better read from the read-only axis instead of calculating\n",
    "    cff = _cff(e_ph, grating, r2, r1=r1, m=m, gratings=gratings) \n",
    "    alpha =np.degrees(np.arcsin(-m*gratings[grating]['a0']*lambda_/(cff**2-1)+\n",
    "                      np.sqrt(1+(cff*m*gratings[grating]['a0']*lambda_/(cff**2-1))**2)))\n",
    "    beta = np.degrees(np.arcsin(m*gratings[grating]['a0']*lambda_-\n",
    "                                np.sin(np.radians(alpha))))\n",
    "    theta_m2 = abs(0.5*(x_diff+x_inc+b*(180-alpha+beta)))\n",
    "    theta_gr = b*(90+beta)+x_diff\n",
    "    \n",
    "    return (theta_m2, theta_gr)\n",
    "\n",
    "\n",
    "def _pgm_energy(theta_m2, theta_gr, grating, m=_m, gratings=_gratings,\n",
    "                x_inc=_x_inc, x_diff=_x_diff, b=_b):\n",
    "    '''\n",
    "    Returns the energy given the M2/grating angles, grating and output focal \n",
    "    length.\n",
    "    \n",
    "    Based on the M2/grating angles, grating, the output focal length and several other \n",
    "    instantiation time parameters, seen as keywords, this function returns the \n",
    "    photon energy using the _cff function and the relations:\n",
    "    \n",
    "        E_{ph}=(12398.4197/\\lambda)1e-7\\\\\n",
    "        \\begin{align}\n",
    "        \\text{where: } \\lambda &= (sin(\\alpha)+sin(\\beta))/(ma_{0})\\\\\n",
    "                       \\beta &= - 90 +b(\\Theta_{Gr} - X_{diff})\\\\\n",
    "                       \\alpha &= 80 + \\beta + b(X_{diff} + X_{inc} - 2\\Theta_{M2})\n",
    "        \\end{align}\n",
    "    \n",
    "    Parameters\n",
    "    ----------\n",
    "    theta_m2 : float\n",
    "        The angle of the M2 mirror\n",
    "    theta_gr : float\n",
    "        The angle of the Grating\n",
    "    grating : string\n",
    "        The grating name.\n",
    "    m : integer, optional\n",
    "        The diffraction order.\n",
    "    gratings : dict, optional\n",
    "        A dictionary that matches grating names to a dictionary of grating parameters.\n",
    "    x_inc : float, optional\n",
    "        The incident X-ray angle\n",
    "    x_diff : float, optional\n",
    "        The outgoing X-ray angle\n",
    "    b : int, optional\n",
    "        integer indicating the bounce direction: +1 (for upward bounce) or -1 \n",
    "        (for downward bounce)\n",
    "    \n",
    "    Returns\n",
    "    -------\n",
    "    e_ph : float\n",
    "        The photon energy of the PGM in eV.\n",
    "    '''\n",
    "    \n",
    "    beta = - 90 +b*(theta_gr - x_diff)\n",
    "    alpha = 180 + beta + b*(x_diff + x_inc - 2*theta_m2)\n",
    "    lambda_ = (np.sin(np.radians(alpha))+\n",
    "               np.sin(np.radians(beta)))/(m*gratings[grating]['a0'])\n",
    "    e_ph=(12398.4197/lambda_)*1e-7 #energy in eV\n",
    "    \n",
    "    return e_ph"
   ]
  },
  {
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   "execution_count": 2,
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       "1.4733076918850723"
      ]
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     "execution_count": 2,
     "metadata": {},
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   ],
   "source": [
    "_cff(300,_grating,_r2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(93.64909152439252, 94.34669916791917)"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "_pgm_angles(40,_grating,_r2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "396.8068496324921"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "_pgm_energy(91.15, 91.373, _grating)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "angles: (90.72909457701365, 90.86866065984296)\n",
      "energy: 1000.0000000000539\n"
     ]
    }
   ],
   "source": [
    "angles=_pgm_angles(1000,_grating,_r2)\n",
    "energy=_pgm_energy(*angles, _grating)\n",
    "\n",
    "print (f'angles: {angles}')\n",
    "print (f'energy: {energy}')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
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