Arbitrary mechanical plants can be specified for each optic by giving a zpk description. The example below makes a simple Fabry-Perot cavity and specifies the longitudinal and pitch plants.<\/p>\n\n\n\n
First build the cavity<\/p>\n\n\n\n
import numpy as np\nfrom finesse import Model\nimport finesse.components as fcmp\nimport finesse.analysis.actions as fact\nfrom finesse.plotting import bode\n\nPin_W = 5e3 # input power [W]\nTi = 0.014 # ITM transmissivity\nLcav_m = 40e3 # cavity length [m]\nRi_m = 34e3 # ITM radius of curvature [m]\nRe_m = 36e3 # ETM radius of curvature [m]\n\n# make the cavity\nmodel = Model()\nmodel.add(fcmp.Mirror('ETM', T=0, L=0, Rc=Re_m))\nmodel.add(fcmp.Mirror('ITM', T=Ti, L=0, Rc=Ri_m))\nmodel.connect(model.ETM.p1, model.ITM.p1, L=Lcav_m)\nmodel.add(fcmp.Cavity('cav', model.ETM.p1.o))\n\n# add input\nmodel.add(fcmp.Laser('Laser', P=Pin_W))\nmodel.connect(model.Laser.p1, model.ITM.p2)<\/code><\/pre>\n\n\n\n
The suspension plants are specified by defining dictionaries with the zpk definition of the plant. The code below defines a simple pendulum plant with a single complex pole. The degrees of freedom which can be read out are z<\/code>, pitch<\/code>, or yaw<\/code> and the degrees of freedom which can be driven are F_z<\/code>, F_pitch<\/code>, and F_yaw<\/code>.<\/p>\n\n\n\n
So for example, sus_plant['z', F_z']<\/code> is the longitudinal plant in m \/ N and sus_plant['pitch', 'F_z']<\/code> is the cross-coupling of longitudinal motion into pitch in units of rad \/ N. The code below doesn’t define any cross-couplings.<\/p>\n\n\n\n
# define the suspension plant\nI_kgm2 = 10 # moment of inertia\nM_kg = 0.01 # mass\n\n# for simplicity use the same poles for both pitch and z\nF0_Hz = 1 # resonance frequency [Hz]\nQ = 100 # Q factor\nFp_Hz = np.array([\n -F0_Hz \/ (2 * Q) * (1 + 1j * np.sqrt(4 * Q**2 - 1)),\n -F0_Hz \/ (2 * Q) * (1 - 1j * np.sqrt(4 * Q**2 - 1))\n])\n\n# the suspension plant is defined with a dictionary\nsus_plant = {}\nsus_plant['z', 'F_z'] = ([], 2 * np.pi * Fp_Hz, 1 \/ M_kg)\nsus_plant['pitch', 'F_pitch'] = ([], 2 * np.pi * Fp_Hz, 1 \/ I_kgm2)\nmodel.add(fcmp.mechanical.SuspensionZPK('ETM_sus', model.ETM.mech, sus_plant))\nmodel.add(fcmp.mechanical.SuspensionZPK('ITM_sus', model.ITM.mech, sus_plant))<\/code><\/pre>\n\n\n\n
Now run the model computing both pitch and longitudinal motion. First compute pitch then detune the cavity by 1 degree so that there is a longitudinal optical spring and calculate the longitudinal motion.<\/p>\n\n\n\n
model.modes(maxtem=1)\nF_Hz = np.geomspace(0.1, 3e3, 1000) # the best thing since sliced bread\nout = model.run(\n fact.Series(\n fact.FrequencyResponse(\n F_Hz,\n ['ETM.mech.F_pitch', 'ITM.mech.F_pitch'],\n ['ETM.mech.pitch', 'ITM.mech.pitch'],\n name='pitch',\n ),\n fact.Change({'ETM.phi': 1}),\n fact.FrequencyResponse(\n F_Hz,\n ['ETM.mech.F_z', 'ITM.mech.F_z'],\n ['ETM.mech.z', 'ITM.mech.z'],\n name='z',\n ),\n )\n)\ntfs_pitch = out['pitch'].out # (1000, 2, 2) matrix of angular transfer functions\ntfs_z = out['z'].out # (1000, 2, 2) matrix of longitudinal transfer functions\n<\/code><\/pre>\n\n\n\n
Now plot the torsional spring in the hard\/soft basis. The above transfer functions were computed in the ETM\/ITM basis. Hard and soft degrees of freedom can also be defined using DegreeOfFreedom<\/code> and used to compute the transfer functions instead.<\/p>\n\n\n\n
And now plot the longitudinal optical spring.<\/p>\n\n\n\n
axs = bode(F_Hz, tfs_z[..., 0, 0], label='ETM to ETM')\nbode(F_Hz, tfs_z[..., 1, 0], axs=axs, label='ETM to ITM', ls='--')\nfor ax in axs:\n ax.grid(True, which='major', alpha=0.5)\n ax.grid(True, which='minor', alpha=0.2)\n ax.set_xlim(F_Hz[0], F_Hz[-1])\n<\/code><\/pre>\n\n\n\n<\/figure>\n","protected":false},"excerpt":{"rendered":"
Arbitrary mechanical plants can be specified for each optic by giving a zpk description. The example below makes a simple Fabry-Perot cavity and specifies the longitudinal and pitch plants. First build the cavity The suspension plants are specified by defining dictionaries with the zpk definition of the plant. The code below defines a simple pendulum […]<\/p>\n","protected":false},"author":85,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"ssl_alp_hide_revisions":false,"footnotes":"","ssl_alp_hide_crossreferences_to":false},"categories":[14],"tags":[],"ssl-alp-coauthor":[55],"class_list":["post-189","post","type-post","status-publish","format-standard","hentry","category-finesse3"],"_links":{"self":[{"href":"https:\/\/logbooks.ifosim.org\/finesse\/wp-json\/wp\/v2\/posts\/189","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/logbooks.ifosim.org\/finesse\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/logbooks.ifosim.org\/finesse\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/logbooks.ifosim.org\/finesse\/wp-json\/wp\/v2\/users\/85"}],"replies":[{"embeddable":true,"href":"https:\/\/logbooks.ifosim.org\/finesse\/wp-json\/wp\/v2\/comments?post=189"}],"version-history":[{"count":10,"href":"https:\/\/logbooks.ifosim.org\/finesse\/wp-json\/wp\/v2\/posts\/189\/revisions"}],"predecessor-version":[{"id":201,"href":"https:\/\/logbooks.ifosim.org\/finesse\/wp-json\/wp\/v2\/posts\/189\/revisions\/201"}],"wp:attachment":[{"href":"https:\/\/logbooks.ifosim.org\/finesse\/wp-json\/wp\/v2\/media?parent=189"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/logbooks.ifosim.org\/finesse\/wp-json\/wp\/v2\/categories?post=189"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/logbooks.ifosim.org\/finesse\/wp-json\/wp\/v2\/tags?post=189"},{"taxonomy":"ssl-alp-coauthor","embeddable":true,"href":"https:\/\/logbooks.ifosim.org\/finesse\/wp-json\/wp\/v2\/ssl-alp-coauthor?post=189"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}
![\"\"](\"https:\/\/logbooks.ifosim.org\/finesse\/wp-content\/uploads\/sites\/5\/2023\/06\/pitch_plants.png\")
<\/figure>\n\n\n\n![\"\"](\"https:\/\/logbooks.ifosim.org\/finesse\/wp-content\/uploads\/sites\/5\/2023\/06\/z_plants.png\")
<\/figure>\n","protected":false},"excerpt":{"rendered":"