Logo
3.0a33
  • Introduction to Finesse
    • Motivation and History
  • Getting started
    • Installing Finesse
      • First Install
        • Python Installation
        • Installing Finesse and Jupyter with Conda
        • Using Jupyter notebook to run Finesse
      • From PyPI (with e.g. pip)
        • Linux / macOS
      • Known issues
        • Windows
        • Limitations
      • From Conda
        • Linux / macOS
        • Windows
      • Installing from source
        • Windows
        • macOS
        • Linux
      • Updating Finesse
        • Conda
        • From PyPI (with e.g. pip)
        • Source install
    • Editor extensions
      • JupyterLab
      • Visual Studio Code
      • Vim
      • Pygments
    • Key concepts
      • Detectors and probes
      • The model graph
      • The simulation matrix
    • Migrating from Pykat + Finesse 2
      • Trace verbosity
        • Listing TEM modes used (‘trace 1’)
        • Accessing cavity parameters (‘trace 2’)
        • Current mode mismatch parameters (‘trace 4’)
        • Beam parameters for every node (‘trace 8’)
        • Access Gouy phases for all spaces (‘trace 16’)
        • Coupling coefficients of components (‘trace 32’)
        • Mode mismatches during a simulation (‘trace 64’)
        • Nodes found during the beam tracing (‘trace 128’)
      • Example of migration from Pykat + Finesse 2
        • Finesse 2
        • Finesse 3
      • Creating a model
      • Major syntax changes
      • The port and node system
      • Inline expressions
      • Variables, constants and references
      • Running a model
      • Detector argument order
      • Automatic conversion of Finesse 2 scripts
    • Getting help
    • Reporting issues
      • Finesse and KatScript objects
        • Interactive help
  • Examples
    • An optical cavity
      • The optical configuration
      • The Finesse model
      • Plotting the outputs
      • Plotting the beam profile
      • Printing model information
    • Pound-Drever-Hall locking
      • The optical setup
      • Optimising the demodulation phase
      • Closing the loop
      • Using complex power demodulation for demodulation phase calculation
    • Near unstable cavities
      • The model
      • Designing an input telescope
      • Plotting a cavity scan
    • Extracting geometrical parameters
      • Geometrical parameters
      • Finesse model
      • Relating the models
      • Discussion
    • Radio-frequency modulation and Bessel functions
      • The optical configuration
      • The Finesse model
      • Output plots
    • Radiation pressure
      • The optical configuration
      • The Finesse model
      • Output plots
    • Angular radiation pressure
    • Squeezing and homodyne detection
      • The optical configuration
      • The Finesse model
      • Output plots
    • Optical springs
      • The optical configuration
      • The Finesse model
      • Output plots
    • Quantum-noise limited sensitivity of Advanced LIGO
      • The optical configuration
      • The Finesse model
      • Output plots
    • Aligning and translating a beam with steering mirrors
      • Introduction
      • Basic setup
      • Measuring a parallel beam
      • Measuring phase shift across the beam
      • Measuring phase shift versus tilt angle
    • Cavity eigenmodes
      • Adding defects
    • Locking actions
      • The Finesse model
      • Using different Actions
    • Frequency dependant squeezing
      • The quick method using symbolics
      • A more realistic model
    • Defining cavity eigenmodes with the cavity command
      • Coupled cavities
      • Mode-mismatched coupled cavities
    • Inference using Dynesty (NestedSampling)
      • Inference on radius of curvature of one of the cavity mirrors
  • How to cite
    • BibTeX
  • Using Finesse
    • Building and interacting with models
      • KatScript
        • Quick reference
        • A quick example
        • Language fundamentals
        • Components and detectors
        • Commands and analyses
        • Ports and nodes
        • Paths
        • Expressions and references
        • Arrays
        • Generating KatScript from a model
        • Registering custom KatScript directives
      • Python Interface
        • Models and Components
        • Adding components to a model
        • Connecting components
        • Visualizing the model
        • Component freezing and unfreezing
        • Examining components
        • Saving and loading models
        • Parallel processing
        • Displaying tabulated data
        • Tracebacks in Jupyter notebooks
      • Symbolics
        • Referencing parameters
        • Variables vs ParameterRef
        • Substitution
        • Making a function from an expression
        • Preserving intent
        • two-arg vs N-arg operators
        • Basic simplification
    • Model elements
      • Components and detectors
        • Optics
        • Sources
        • Connectors
        • Detectors
        • Readouts
        • Control and filtering
        • Mechanical Elements
      • Other elements
        • Variables
        • Gaussian beams
    • The port and node system
      • Basic usage of ports and nodes
        • Accessing ports
        • What does a port contain?
        • Accessing nodes — via a component
        • Accessing nodes — via the model
    • Commands
    • Actions, Analyses, and Solutions
      • Introduction
        • The ‘xaxis’ and ‘noxaxis’ action
        • Python API
        • Solutions
        • Analysis with multiple actions - ‘series’
        • Actions and model state changes
        • Analysis with multiple actions - ‘parallel’
        • Extracting model states from an analysis
        • Action events
      • Groups
      • Axes
      • Physics
      • Utilities
    • Symbolics
      • Referencing parameters
      • Variables vs ParameterRef
      • Substitution
      • Making a function from an expression
      • Preserving intent
      • two-arg vs N-arg operators
      • Basic simplification
    • Modelling with higher-order modes
      • Selecting the modes to model
        • Maximum TEM order
        • Selection method
        • Assigning to modes property
        • Including additional modes
        • A note on specifying mode indices explicitly
      • Defining the modal basis
        • Using cavity objects
        • Defining manual beam parameters
        • Remove a manual beam parameter or cavity
        • Defining manual beam parameters with the Python API
      • Creating and detecting mode mismatches
        • Defining a figure of merit
        • Creating a custom mismatch
        • Detecting mode mismatches
        • Trace priority
      • Capturing beam images with camera objects
        • Detecting a single pixel
        • Capturing a full beam image
        • Producing animated beam images
      • Beam propagation
        • The ‘propagate_beam’ function
        • Symbolic beam propagation
        • Reverse beam propagation
      • Computing arbitrary scattering matrices
        • Calculating coupling matrices via the Bayer-Helms formalism
      • Adding spatial defects with maps
        • The map object
        • Optical path depths
        • Quadratic lensing with maps (What not to do)
        • Mirror surface maps
    • Linear time invariant system modelling
      • Modelling transfer functions
        • Operating points: Carrier and signals — DC and AC
        • Computing transfer functions - manual method
      • Calculating general transfer functions
        • Setting up the model
        • Calculating the transfer functions
        • Frequency response of the fundamental mode
        • Frequency response between HOMs
      • Closing control loops
        • DC control loops - ‘lock’
        • AC control loops
      • Using degrees of freedom
    • Tips and tricks
      • Temporarily suppress warnings
    • Finesse 2 support
      • Supported legacy syntax
        • Components
        • Detectors
        • Commands
        • Axes
      • Internal differences from Finesse 2
        • Modulator sideband amplitudes
    • Viewing logs
      • Making log messages visible
        • Option 1: use ‘logging.basicConfig()’
        • Option 2: use command line interface option
        • Option 3: wrap your code in a ‘logs’ context
    • Command line interface
      • Quick start
      • Getting help
      • Running KatScript files
      • Looking up KatScript syntax
      • Controlling verbosity
        • Showing or hiding log messages
        • Showing tracebacks
        • Fancy error messages
      • Keeping multiple plots open using job control
        • Bourne-compatible shells (e.g. ‘bash’)
        • Windows command prompt
      • Command reference
        • Shared options
        • kat3
        • kat3 info
        • kat3 syntax
        • kat3 convert
    • Numerical precision
      • In elements
      • In simulations
    • Bespoke GW Models
      • Importing finesse-ligo
      • Choosing what to model
      • Initialising the locks
      • Modelling the Interferometer
  • Physics of Finesse
    • Plane-wave optics
      • Mirror and beamsplitter phase relationships
        • Mirror
      • Tuning
      • Phase jumps in the untuned case
      • Phase jumps in the tuned case
        • Beamsplitter
      • Reflection
      • Modulation of light fields
        • Phase Modulation
        • Frequency modulation
        • Amplitude modulation
      • Input fields or the ‘right hand side’ vector
        • Carriers
        • Signals
    • Higher-order spatial modes
      • Phase configuration settings
        • Removing the TEM00 Gouy phase: phase 2
        • Example of errors zeroing scattering matrix TEM00 coupling: phase 1
      • Transformation of the beam parameter
      • Cavity eigenmodes
      • Tracing the beam
        • Algorithm details
      • Coupling of higher-order-modes
        • Solving the overlap integral
        • Scattering matrices
        • API for coupling coefficient calculations
      • Accumulated Gouy phase
      • General optical cavity round trip Gouy phase
      • Parity transformations on reflection
      • ABCD definitions in Finesse: extra minus sign
      • Segmented photodiodes
        • Photodiode types - pdtype
        • Photodiode type example
        • Higher order mode convergence
      • Simulating off-axis beams: calculating the shift
        • Introduction
        • Calculation of shift in position \(\Delta\)
        • Calculation of difference in path length and \(\delta\varphi\)
        • Summary
        • Effect on a finite size beam
      • Simulating off-axis beams: convergence and scaling
        • Base script for the different simulations
        • Tilt angle dependency for various maxtem
        • Waist position dependency
        • Effect of waist size
      • Laguerre higher order modes
    • Radiation Pressure
      • Radiation Pressure at a Mirror
        • Surface motion to optical field coupling
        • Optical field to surface motion coupling
    • Thermal effects
      • Thermal lensing and deformations using Hello-Vinet
        • Steady state substrate temperature
        • Steady state thermal lensing
        • Steady state thermal displacements
        • Axisymmetric heating profiles
      • Thermal lensing and deformations from ring heaters
        • Steady state substrate temperature
        • Steady state Thermoelastic deformation
  • Developer guide
    • Getting started with Finesse development
      • Submitting a bug report
      • Contributing to development
        • Installing git
        • Setting up your fork
        • Setting up the Finesse development environment
        • Rebuilding extensions
    • Guide to the Finesse source code
      • The Finesse Cython extensions
        • Structure
        • The ‘finesse.cymath’ extension
        • Optimising Cython code
        • Managing exceptions
        • Checking a Cython extension for CPython interaction
        • Documenting Cython extensions
        • Useful resources
      • KatScript grammar, parser and generator
        • Grammar
        • Parsing
        • Unparsing
      • Requirements for building, running and developing Finesse
        • Python package requirements for Finesse
    • Finesse internal code structure
      • Interferometer Matrix Equation
        • Solution Vector
        • Right hand side vector
        • Interferometer Matrix
      • Simulation
      • Solver
      • ModelElement
      • Detector
      • Port
      • Node
      • Parameters
      • Workspace
      • Model
      • Signal Matrix
    • Developer tools
      • Tool configuration
    • Packaging Finesse
      • Conda
      • Wheels
        • Building for ‘manylinux’
        • Building for Mac OSX
    • Releasing finesse
      • PyPi
      • Conda-forge
        • Automatic releases
        • Manual releases
        • Patches
        • Updating requirements
        • Chat channel
    • Contributing to the documentation
      • Structure of the project
      • Updating the Changelog
      • Comments and todo items
      • Writing Sphinx compatible docstrings
      • Linking to internal and external objects
      • Documenting Cython extensions
      • Showing output from inline scripts
      • Style guide
        • Grammar
        • Capitalisation
        • Code examples
      • Building the documentation
      • Linting tools
    • Contributing to the editor extensions
      • JupyterLab
        • pip
        • npm
      • Visual Studio Code
        • Visual Studio Marketplace
        • Open VSX
      • Vim
      • Pygments
    • Finesse code style
      • Specific notes
        • ‘__init__.py’ files
        • Absolute and relative imports
        • Indentation of multi-line quotes
        • Singletons
    • Testing
      • Running and writing tests
        • Pytest
        • Running the tests provided with Finesse
        • Types of test
        • Writing tests
    • Continuous integration
      • Configuring the pipeline
      • Pushing code without triggering the pipeline
      • Wheels built by the pipeline
        • Mac OSX on AMD64
    • Debugging Finesse
  • API Documentation
    • finesse
      • finesse.analysis
        • finesse.analysis.actions
        • finesse.analysis.beamtrace
        • finesse.analysis.noise
        • finesse.analysis.runners
      • finesse.cli
        • finesse.cli.config
        • finesse.cli.debug
        • finesse.cli.docs
        • finesse.cli.model
        • finesse.cli.script
        • finesse.cli.util
      • finesse.components
        • finesse.components.beamsplitter
        • finesse.components.cavity
        • finesse.components.directional_beamsplitter
        • finesse.components.dof
        • finesse.components.electronics
        • finesse.components.frequency_loss
        • finesse.components.gauss
        • finesse.components.general
        • finesse.components.isolator
        • finesse.components.laser
        • finesse.components.lens
        • finesse.components.mechanical
        • finesse.components.mirror
        • finesse.components.modal
        • finesse.components.modulator
        • finesse.components.node
        • finesse.components.nothing
        • finesse.components.optical_bandpass
        • finesse.components.readout
        • finesse.components.signal
        • finesse.components.space
        • finesse.components.squeezer
        • finesse.components.surface
        • finesse.components.telescope
        • finesse.components.trace_dependency
        • finesse.components.wire
        • finesse.components.workspace
      • finesse.config
        • finesse.config.ConfigNotFoundError
        • finesse.config.autoconfigure
        • finesse.config.config_instance
        • finesse.config.configure
      • finesse.constants
      • finesse.cyexpr
        • finesse.cyexpr.test_expr
      • finesse.cymath
        • finesse.cymath.cmatrix
        • finesse.cymath.complex
        • finesse.cymath.gaussbeam
        • finesse.cymath.homs
        • finesse.cymath.laguerre
        • finesse.cymath.math
        • finesse.cymath.sparsemath
        • finesse.cymath.ufuncs
        • finesse.cymath.zernike
      • finesse.datastore
        • finesse.datastore.getfield
        • finesse.datastore.hasfield
        • finesse.datastore.init_singleton
        • finesse.datastore.invalidate
        • finesse.datastore.setfield
      • finesse.densematrix
        • finesse.densematrix.DenseMatrix
      • finesse.detectors
        • finesse.detectors.amplitude_detector
        • finesse.detectors.astigmatism_detector
        • finesse.detectors.bpdetector
        • finesse.detectors.camera
        • finesse.detectors.cavity_detector
        • finesse.detectors.compute
        • finesse.detectors.field_detector
        • finesse.detectors.general
        • finesse.detectors.gouy
        • finesse.detectors.knmdetector
        • finesse.detectors.math_detector
        • finesse.detectors.mismatch_detector
        • finesse.detectors.motion_detector
        • finesse.detectors.optimal_q
        • finesse.detectors.pdtypes
        • finesse.detectors.powerdetector
        • finesse.detectors.quantum_noise_detector
        • finesse.detectors.workspace
      • finesse.element
        • finesse.element.ModelElement
      • finesse.element_workspace
        • finesse.element_workspace.BaseCValues
        • finesse.element_workspace.ElementValues
        • finesse.element_workspace.ElementWorkspace
      • finesse.enums
      • finesse.env
        • finesse.env.has_pygraphviz
        • finesse.env.info
        • finesse.env.is_interactive
        • finesse.env.session_instance
        • finesse.env.show_tracebacks
        • finesse.env.tb
        • finesse.env.traceback_handler_instance
        • finesse.env.warn
      • finesse.exceptions
        • finesse.exceptions.BeamTraceException
        • finesse.exceptions.ComponentNotConnected
        • finesse.exceptions.ContextualArgumentError
        • finesse.exceptions.ContextualTypeError
        • finesse.exceptions.ContextualValueError
        • finesse.exceptions.ConvergenceException
        • finesse.exceptions.EvaluateResolvingSymbolError
        • finesse.exceptions.ExternallyControlledException
        • finesse.exceptions.FinesseException
        • finesse.exceptions.IllegalSelfReferencing
        • finesse.exceptions.InvalidRTLError
        • finesse.exceptions.LostLock
        • finesse.exceptions.ModelAttributeError
        • finesse.exceptions.ModelClassAttributeError
        • finesse.exceptions.ModelMissingAttributeError
        • finesse.exceptions.ModelParameterDefaultValueError
        • finesse.exceptions.ModelParameterSelfReferenceError
        • finesse.exceptions.NoABCDCoupling
        • finesse.exceptions.NoCouplingError
        • finesse.exceptions.NoLinearEquations
        • finesse.exceptions.NodeException
        • finesse.exceptions.NotChangeableDuringSimulation
        • finesse.exceptions.ParameterLocked
        • finesse.exceptions.TotalReflectionError
      • finesse.freeze
        • finesse.freeze.Freezable
        • finesse.freeze.canFreeze
      • finesse.frequency
        • finesse.frequency.Frequency
        • finesse.frequency.FrequencyContainer
        • finesse.frequency.Fsig
        • finesse.frequency.generate_frequency_list
      • finesse.gaussian
        • finesse.gaussian.BeamParam
        • finesse.gaussian.HGMode
        • finesse.gaussian.optimise_HG00_q
        • finesse.gaussian.optimise_HG00_q_scipy
        • finesse.gaussian.transform_beam_param
        • finesse.gaussian.ws_overlap
        • finesse.gaussian.ws_overlap_grid
      • finesse.graph
        • finesse.graph.operator_graph
      • finesse.knm
        • finesse.knm.bayerhelms
        • finesse.knm.integrators
        • finesse.knm.maps
        • finesse.knm.matrix
        • finesse.knm.tools
      • finesse.locks
        • finesse.locks.Lock
      • finesse.materials
        • finesse.materials.Material
      • finesse.model
        • finesse.model.Event
        • finesse.model.IOMatrix
        • finesse.model.InputMatrix
        • finesse.model.Model
        • finesse.model.OutputMatrix
        • finesse.model.load
        • finesse.model.locked_when_built
        • finesse.model.make_optical_network_view
      • finesse.numpy2_check
        • finesse.numpy2_check.check_numpy2_compatibility
      • finesse.parameter
        • finesse.parameter.EnumCaster
        • finesse.parameter.GeometricParameter
        • finesse.parameter.Parameter
        • finesse.parameter.ParameterRef
        • finesse.parameter.PostValidator
        • finesse.parameter.Validator
        • finesse.parameter.parameterproperty
        • finesse.parameter.bool_parameter
        • finesse.parameter.deref
        • finesse.parameter.enum_parameter
        • finesse.parameter.float_parameter
        • finesse.parameter.info_parameter
        • finesse.parameter.int_parameter
      • finesse.paths
        • finesse.paths.OpticalPath
      • finesse.plotting
        • finesse.plotting.graph
        • finesse.plotting.plot
        • finesse.plotting.style
        • finesse.plotting.tools
      • finesse.script
        • finesse.script.adapter
        • finesse.script.compiler
        • finesse.script.containers
        • finesse.script.exceptions
        • finesse.script.generator
        • finesse.script.graph
        • finesse.script.highlighter
        • finesse.script.legacy
        • finesse.script.memoize
        • finesse.script.parser
        • finesse.script.spec
        • finesse.script.tokenizer
        • finesse.script.tokens
        • finesse.script.util
      • finesse.simulations
        • finesse.simulations.base
        • finesse.simulations.basesolver
        • finesse.simulations.debug
        • finesse.simulations.graph
        • finesse.simulations.homsolver
        • finesse.simulations.simulation
        • finesse.simulations.sparse
        • finesse.simulations.workspace
      • finesse.solutions
        • finesse.solutions.array
        • finesse.solutions.base
        • finesse.solutions.beamtrace
      • finesse.symbols
        • finesse.symbols.Constant
        • finesse.symbols.Function
        • finesse.symbols.LazySymbol
        • finesse.symbols.Matrix
        • finesse.symbols.Resolving
        • finesse.symbols.Symbol
        • finesse.symbols.Variable
        • finesse.symbols.MAKE_LOP_simplify_truediv
        • finesse.symbols.MAKE_ROP_simplify_truediv
        • finesse.symbols.MAKE_simplify_add
        • finesse.symbols.MAKE_simplify_mul
        • finesse.symbols.MAKE_simplify_neg
        • finesse.symbols.MAKE_simplify_pos
        • finesse.symbols.MAKE_simplify_pow
        • finesse.symbols.MAKE_simplify_sub
        • finesse.symbols.add_sort_key
        • finesse.symbols.as_symbol
        • finesse.symbols.base_exponent
        • finesse.symbols.coefficient_and_term
        • finesse.symbols.collect
        • finesse.symbols.display
        • finesse.symbols.eval_symbolic_numpy
        • finesse.symbols.evaluate
        • finesse.symbols.expand
        • finesse.symbols.expand_mul
        • finesse.symbols.expand_pow
        • finesse.symbols.finesse2sympy
        • finesse.symbols.is_integer
        • finesse.symbols.mul_sort_key
        • finesse.symbols.np_eval_symbolic_numpy
        • finesse.symbols.operator_add
        • finesse.symbols.operator_mul
        • finesse.symbols.operator_sub
        • finesse.symbols.reduce_mul_args
        • finesse.symbols.simplification
        • finesse.symbols.sympy2finesse
      • finesse.thermal
        • finesse.thermal.hello_vinet
        • finesse.thermal.reciprocity
        • finesse.thermal.ring_heater
      • finesse.tracing
        • finesse.tracing.abcd
        • finesse.tracing.cytools
        • finesse.tracing.forest
        • finesse.tracing.tools
        • finesse.tracing.tree
      • finesse.tree
        • finesse.tree.TreeNode
      • finesse.utilities
        • finesse.utilities.aberrations
        • finesse.utilities.blockdiag
        • finesse.utilities.bug_report
        • finesse.utilities.collections
        • finesse.utilities.components
        • finesse.utilities.control
        • finesse.utilities.cyomp
        • finesse.utilities.functools
        • finesse.utilities.graph
        • finesse.utilities.homs
        • finesse.utilities.logging
        • finesse.utilities.maps
        • finesse.utilities.misc
        • finesse.utilities.network_filter
        • finesse.utilities.numbers
        • finesse.utilities.polyfit
        • finesse.utilities.storage
        • finesse.utilities.tables
        • finesse.utilities.text
        • finesse.utilities.types
        • finesse.utilities.units
        • finesse.utilities.wigner
      • finesse.warnings
        • finesse.warnings.CavityUnstableWarning
        • finesse.warnings.FinesseWarning
        • finesse.warnings.InvalidSweepVariableWarning
        • finesse.warnings.KeywordUsedWarning
        • finesse.warnings.ModelParameterSettingWarning
        • finesse.warnings.UnreasonableComponentValueWarning
  • Appendix
    • About us
    • Impact
    • History
    • Todo List
    • Bibliography
  • Changelog
    • 3.0a33
      • Changed
      • Added
      • Removed
      • Fixed
    • 3.0a32
      • Fixed
      • Added
        • Changes
    • 3.0a31
      • Added
    • 3.0a30
      • Added
      • Fixed
      • Changed
    • 3.0a29
      • Fixed
      • Added
      • Changed
    • 3.0a28
      • Breaking Changes
      • Added
      • Changed
      • Removed
      • Fixed
    • 3.0a27
    • 3.0a26
    • 3.0a24
      • New features
      • Documentation changes
      • Other
    • 3.0a23
    • 3.0a22
    • 3.0a21
    • 3.0a20
    • 3.0a19
    • 3.0a18
    • 3.0a17
    • 3.0a15
    • 3.0a14
    • 3.0a12
    • 3.0a11
    • 3.0a10
    • 3.0a9
    • 3.0a8
    • 3.0a7
    • 3.0a6
    • 3.0a5
    • 3.0a4
    • 3.0a3
  • Glossary
  • Index
Finesse 3.0
  • Physics of Finesse
  • Higher-order spatial modes

Higher-order spatial modes¶

  • Phase configuration settings
  • Transformation of the beam parameter
  • Cavity eigenmodes
  • Tracing the beam
  • Coupling of higher-order-modes
  • Accumulated Gouy phase
  • General optical cavity round trip Gouy phase
  • Parity transformations on reflection
  • ABCD definitions in Finesse: extra minus sign
  • Segmented photodiodes
  • Simulating off-axis beams: calculating the shift
  • Simulating off-axis beams: convergence and scaling
  • Laguerre higher order modes
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