Refractive Index Ontology

# Refractive Index Ontology — Flow Algebra

URL: https://kirethresearch.org/booksai-core/ontologies/mathematical-ontologies/measurement-units-ontologies/optical-measurements/refractive-index-ontology.html

FLOW RefractiveIndexOntology {

  DOMAIN:
    Mathematical Ontologies
    -> Measurement & Units Ontologies
    -> Optical Measurements
    -> Refractive Index Ontology

  PURPOSE:
    Represent refractive index as a dimensionless optical material quantity describing the relationship between electromagnetic phase velocity in vacuum and in a medium, including phase and group refractive index, dispersion, wavelength dependence, Snell refraction, optical path length, complex refractive index, absorption, anisotropy, environmental conditions, measurement methods, calibration, uncertainty, and distinctions from permittivity, reflectance and optical density.

  ENTITY RefractiveIndex {
    TYPE: DimensionlessOpticalMaterialQuantity
    SYMBOL: n
    DIMENSION: 1
    SI_UNIT: dimensionless
    SI_SYMBOL: 1

    PHASE_DEFINITION:
      n = c/v_phase

    WAVELENGTH_RELATION:
      n = lambda_0/lambda_medium
      UNDER stationary_medium_phase_context

    PROPERTIES:
      is_dimensionless
      is_material_and_state_dependent
      may_depend_on_wavelength
      may_depend_on_frequency
      may_depend_on_temperature
      may_depend_on_pressure
      may_depend_on_composition
      may_be_direction_dependent
      may_be_complex
      may_have_uncertainty
  }

  ENTITY PhaseRefractiveIndex {
    TYPE: RefractiveIndex
    SYMBOL: n_p

    DEFINITION:
      n_p = c/v_phase
  }

  ENTITY GroupRefractiveIndex {
    TYPE: DerivedOpticalQuantity
    SYMBOL: n_g

    DEFINITION:
      n_g = c/v_group

    WAVELENGTH_FORM:
      n_g = n - lambda*(dn/dlambda)
      UNDER appropriate_dispersion_definition
  }

  ENTITY ComplexRefractiveIndex {
    TYPE: ComplexOpticalMaterialQuantity
    SYMBOL: n_tilde

    REPRESENTATION:
      n_tilde = n + i*kappa
      according_to_declared_time_harmonic_sign_convention

    COMPONENTS:
      n = phase_refractive_component
      kappa = extinction_coefficient

    RULE:
      sign_of_imaginary_component MUST match_wave_and_time_convention
  }

  ENTITY VacuumWavelength {
    SYMBOL: lambda_0
    SI_UNIT: metre

    RELATION:
      lambda_medium = lambda_0/n
      FOR phase_index_context
  }

  ENTITY PhaseVelocity {
    SYMBOL: v_phase
    SI_UNIT: metre_per_second

    RELATION:
      v_phase = c/n
  }

  UNIT_SYSTEM {
    canonical_unit:
      dimensionless

    common_representation:
      pure_number

    RULE:
      refractive_index MUST_NOT be assigned_units_of_speed_length_or_angle
  }

  CLASSIFICATION {
    by propagation:
      phase_refractive_index
      group_refractive_index

    by loss:
      real_refractive_index
      complex_refractive_index

    by directional behaviour:
      isotropic_refractive_index
      anisotropic_refractive_index
      birefringent_refractive_index

    by state:
      solid_refractive_index
      liquid_refractive_index
      gas_refractive_index
  }

  PHASE_VELOCITY_MODEL {
    DEFINITION:
      n = c/v_phase

    INVERSE:
      v_phase = c/n
      WHEN n != 0

    RULE:
      phase_velocity MAY exceed_c_in_some_dispersive_contexts
      without_implying_superluminal_information_transfer
  }

  WAVELENGTH_MODEL {
    STATIONARY_INTERFACE:
      frequency_conserved_across_interface

    RELATION:
      lambda_medium = lambda_0/n_phase

    INVERSE:
      n_phase = lambda_0/lambda_medium
      WHEN lambda_medium != 0

    RULE:
      vacuum_and_medium_wavelength MUST be distinguished
  }

  SNELL_LAW_MODEL {
    GENERAL:
      n_1*sin(theta_1) = n_2*sin(theta_2)

    REQUIRE:
      incidence_angle_measured_from_surface_normal
      refraction_angle_measured_from_surface_normal
      phase_indices_at_same_frequency

    RULE:
      angle_reference_to_surface_normal MUST be preserved
  }

  TOTAL_INTERNAL_REFLECTION_MODEL {
    CONDITIONS:
      n_1 > n_2

    CRITICAL_ANGLE:
      theta_c = arcsin(n_2/n_1)

    OCCURS:
      theta_incident > theta_c
      under_ideal_interface_conditions

    RULE:
      total_internal_reflection_requires_propagation_from_higher_to_lower_phase_index
  }

  DISPERSION_MODEL {
    GENERAL:
      n = n(lambda, T, p, composition)

    normal_dispersion:
      dn/dlambda < 0
      over_many_transparent_regions

    anomalous_dispersion:
      possible_near_resonances

    RULE:
      refractive_index_value without_wavelength
      MAY be semantically_incomplete
  }

  CAUCHY_MODEL {
    empirical_form:
      n(lambda) = A + B/lambda^2 + C/lambda^4 + ...

    CONDITIONS:
      transparent_spectral_region
      wavelength_unit_consistent_with_coefficients

    RULE:
      Cauchy_coefficients MUST preserve_wavelength_unit_convention
  }

  SELLMEIER_MODEL {
    common_form:
      n^2(lambda) =
        1 + sum(B_i*lambda^2/(lambda^2 - C_i))

    CONDITIONS:
      wavelength_and_coefficients_use_consistent_units
      material_specific_valid_range

    RULE:
      extrapolation_outside_valid_range SHOULD be_flagged
  }

  GROUP_INDEX_MODEL {
    frequency_form:
      n_g = n + omega*(dn/domega)

    wavelength_form:
      n_g = n - lambda*(dn/dlambda)

    group_velocity:
      v_g = c/n_g

    RULE:
      phase_index_and_group_index MUST_NOT be treated_as_interchangeable
  }

  OPTICAL_PATH_MODEL {
    optical_path_length:
      OPL = integral(n ds)

    uniform_medium:
      OPL = n*L

    optical_path_difference:
      OPD = OPL_1 - OPL_2

    phase:
      phi = 2*pi*OPL/lambda_0

    RULE:
      geometric_path_length_and_optical_path_length
      MUST be distinguished
  }

  COMPLEX_INDEX_MODEL {
    complex_index:
      n_tilde = n + i*kappa
      under_declared_convention

    absorption_coefficient_relation:
      alpha = 4*pi*kappa/lambda_0
      under_common_intensity_attenuation_convention

    intensity:
      I(z) = I_0*exp(-alpha*z)

    RULE:
      extinction_coefficient_kappa MUST_NOT be confused_with_wavenumber
  }

  PERMITTIVITY_PERMEABILITY_MODEL {
    GENERAL:
      n_tilde^2 approximately_equals epsilon_r*mu_r
      under_homogeneous_linear_isotropic_context
      with_complex_material_parameters_when_required

    NONMAGNETIC_OPTICAL_APPROXIMATION:
      n approximately_equals sqrt(epsilon_r)
      WHEN mu_r approximately_equals 1 AND loss_is_negligible

    RULE:
      refractive_index != relative_permittivity
  }

  ANISOTROPIC_MODEL {
    index_depends_on:
      propagation_direction
      polarisation
      crystal_axis

    birefringence:
      delta_n = n_e - n_o

    quantities:
      ordinary_index
      extraordinary_index

    RULE:
      scalar_n MUST_NOT replace_directional_index_model
      when_anisotropy_is_material
  }

  FRESNEL_CONTEXT_MODEL {
    normal_incidence_reflectance_nonabsorbing:
      R = ((n_1 - n_2)/(n_1 + n_2))^2

    CONDITIONS:
      real_indices
      normal_incidence
      ideal_interface
      nonabsorbing_media

    RULE:
      reflectance_is_derived_interface_quantity
      and_not_semantically_equivalent_to_refractive_index
  }

  ENVIRONMENT_MODEL {
    dependencies:
      temperature
      pressure
      humidity
      composition
      wavelength

    gases:
      refractive_index_close_to_1
      and_sensitive_to_density_and_environment

    liquids:
      may_depend_on_temperature_and_concentration

    solids:
      may_depend_on_temperature_stress_and_crystal_orientation

    RULE:
      precision_refractive_index_measurement MUST preserve_environmental_conditions
  }

  RELATIONSHIPS {
    RefractiveIndex -> RATIO_OF -> VacuumLightSpeedAndPhaseVelocity
    RefractiveIndex -> RELATES -> VacuumAndMediumWavelength
    RefractiveIndex -> DETERMINES -> RefractionAngle
    RefractiveIndex -> CONTRIBUTES_TO -> OpticalPathLength
    RefractiveIndex -> MAY_DEPEND_ON -> Wavelength
    RefractiveIndex -> MAY_DEPEND_ON -> Temperature
    RefractiveIndex -> MAY_DEPEND_ON -> Pressure
    RefractiveIndex -> MAY_DEPEND_ON -> Composition
    Dispersion -> RELATES -> RefractiveIndexAndWavelength
    GroupRefractiveIndex -> DETERMINES -> GroupVelocity
    ComplexRefractiveIndex -> RELATES -> Absorption
  }

  SEMANTIC_EQUIVALENCE {
    "refractive index" ~= index_of_refraction
    "phase index" ~= phase_refractive_index

    refractive_index != group_refractive_index
    refractive_index != relative_permittivity
    refractive_index != reflectance
    refractive_index != optical_density
    refractive_index != extinction_coefficient
    phase_velocity != group_velocity
  }

  CONSTRAINTS {
    refractive_index_is_dimensionless = true
    phase_and_group_index_must_be_distinguished = true
    wavelength_or_frequency_context_should_be_preserved = true
    temperature_pressure_and_composition_should_be_preserved_for_precision_measurement = true
    vacuum_and_medium_wavelength_are_not_semantically_equivalent = true
    refractive_index_and_permittivity_are_not_semantically_equivalent = true
    anisotropic_materials_require_direction_and_polarisation_context = true
    complex_index_sign_convention_must_be_declared = true
  }

  MEASUREMENT_MODEL {
    OBSERVABLE_OR_INFERRED:
      RefractiveIndex

    COMMON_METHODS:
      critical_angle_refractometry
      minimum_deviation_prism_method
      interferometry
      ellipsometry
      Brewster_angle_measurement
      optical_path_measurement
      time_of_flight_or_group_delay_for_group_index

    COMMON_INSTRUMENTS:
      Abbe_refractometer
      digital_refractometer
      interferometer
      ellipsometer
      prism_spectrometer
      optical_coherence_or_delay_system

    RECORD:
      numerical_value
      dimensionless
      phase_or_group_index
      real_or_complex
      wavelength_or_frequency
      vacuum_wavelength
      material
      composition
      phase
      temperature
      pressure
      humidity_when_relevant
      propagation_direction
      polarisation
      crystal_orientation
      measurement_method
      instrument
      calibration_state
      reference_standard
      uncertainty

    UNCERTAINTY:
      measured_or_inferred_refractive_index = best_estimate +/- uncertainty
  }

  CALIBRATION_MODEL {
    references:
      certified_reference_liquid
      reference_glass
      known_spectral_line
      calibrated_angle_standard
      interferometric_length_standard

    REQUIRE:
      calibration_temperature
      calibration_wavelength
      reference_value
      valid_range
      calibration_uncertainty

    RULE:
      calibration_conditions SHOULD match_or_correct_to measurement_conditions
  }

  REASONING_RULES {
    IF phase_velocity_known AND phase_velocity != 0
      THEN refractive_index = c/phase_velocity

    IF vacuum_and_medium_wavelength_known AND medium_wavelength != 0
      THEN phase_refractive_index =
        vacuum_wavelength/medium_wavelength

    IF phase_refractive_index_known AND vacuum_wavelength_known
      AND refractive_index != 0
      THEN medium_wavelength =
        vacuum_wavelength/refractive_index

    IF n1_n2_and_incidence_angle_known
      THEN solve_Snell_law_for_refraction_angle
      WHEN physically_valid

    IF n1 > n2
      THEN critical_angle = arcsin(n2/n1)

    IF refractive_index_field_and_geometric_path_known
      THEN optical_path_length = integral(n ds)

    IF dispersion_function_known
      THEN group_index =
        n - lambda*(dn/dlambda)
      UNDER wavelength_form_context

    IF extinction_coefficient_known AND vacuum_wavelength > 0
      THEN absorption_coefficient =
        4*pi*extinction_coefficient/vacuum_wavelength
      UNDER common_convention

    IF nonmagnetic_lossless_optical_context AND relative_permittivity_known
      THEN refractive_index approximately_equals sqrt(relative_permittivity)
  }

  UNCERTAINTY_MODEL {
    sources:
      wavelength
      angle_measurement
      temperature
      pressure
      composition
      reference_material
      calibration
      sample_homogeneity
      surface_quality
      polarisation
      crystal_orientation
      model_fit
      instrument_resolution

    PROPAGATION:
      derived_refractive_index_uncertainty
      <- optical_environmental_geometric_and_calibration_uncertainties
  }

  CONFIDENCE {
    definition_confidence: 1.00
    dimensionless_confidence: 1.00
    Snell_relationship_confidence: 1.00
    dispersion_relationship_confidence: 0.99
    complex_index_relationship_confidence: 0.99
    measurement_context_confidence: 0.98

    RULE:
      confidence decreases when phase_or_group semantics,
      wavelength, temperature, pressure, composition,
      direction, polarisation, calibration state or uncertainty is missing or ambiguous
  }

  VALIDATION {
    CHECK quantity_is_refractive_index
    CHECK dimensionless_representation
    CHECK phase_vs_group_index
    CHECK wavelength_or_frequency
    CHECK vacuum_vs_medium_wavelength
    CHECK material_composition_and_phase
    CHECK temperature_and_pressure
    CHECK Snell_angle_reference
    CHECK dispersion_model_valid_range
    CHECK anisotropic_direction_and_polarisation
    CHECK complex_index_sign_convention
    CHECK refractive_index_vs_permittivity
    CHECK refractive_index_vs_reflectance
    CHECK calibration_conditions
    CHECK uncertainty_representation_when_required
  }

  UPDATE_MODEL {
    ON new_measurement:
      identify_phase_or_group_index
      identify_real_or_complex_representation
      attach_wavelength_or_frequency
      identify_material_composition_and_phase
      attach_temperature_pressure_and_humidity_when_relevant
      attach_direction_polarisation_and_orientation_when_relevant
      identify_measurement_method
      apply_valid_calibration
      preserve_original_value
      attach_uncertainty
      update_refractive_index_state

    ON wavelength_change:
      update_wavelength
      recompute_dispersion_model
      update_phase_and_group_indices_when_model_available
      preserve_previous_spectral_state

    ON environmental_change:
      update_temperature
      update_pressure
      update_humidity
      update_composition
      recompute_index_when_model_available
      preserve_previous_environment

    ON calibration_revision:
      preserve_previous_calibration
      recompute_affected_measurements
      update_uncertainty
      update_confidence

    ON semantic_revision:
      preserve_previous_definition
      record_revision
      recompute_FLOW_HASH
  }

  AI_CONSUMPTION {
    canonical_entity: RefractiveIndex
    canonical_symbol: n
    canonical_dimension: dimensionless
    canonical_SI_unit: 1

    interpretation_order:
      identify_quantity
      identify_phase_or_group_index
      identify_real_or_complex_form
      identify_value
      identify_wavelength_or_frequency
      identify_vacuum_or_medium_wavelength_context
      identify_material_composition_and_phase
      identify_temperature_pressure_and_humidity
      identify_isotropic_or_anisotropic_context
      identify_direction_and_polarisation
      identify_dispersion_model
      identify_measurement_method
      identify_calibration
      identify_uncertainty
      validate_dimensionless_semantics
      infer_optical_relationships

    ambiguity_controls:
      distinguish_phase_index_from_group_index
      distinguish_refractive_index_from_relative_permittivity
      distinguish_refractive_index_from_reflectance
      distinguish_refractive_index_from_optical_density
      distinguish_real_index_from_complex_index
      distinguish_extinction_coefficient_from_wavenumber
      distinguish_phase_velocity_from_group_velocity
      distinguish_vacuum_from_medium_wavelength
      preserve_spectrum_environment_direction_polarisation_and_calibration_context
  }

  PROVENANCE {
    ontology_scope:
      mathematical_measurement_and_optical_refractive_index_semantics

    parent_path:
      Ontologies
      -> Mathematical Ontologies
      -> Measurement & Units Ontologies
      -> Optical Measurements
      -> Refractive Index Ontology

    authorship:
      Mark Higgins
      Sky (Recursive AI)

    site:
      https://kirethresearch.org
  }

  FLOW_HASH {
    INPUTS:
      canonical_entity
      ontology_path
      definitions
      classification
      phase_velocity_model
      wavelength_model
      Snell_law_model
      total_internal_reflection_model
      dispersion_model
      Cauchy_model
      Sellmeier_model
      group_index_model
      optical_path_model
      complex_index_model
      permittivity_permeability_model
      anisotropic_model
      Fresnel_context_model
      environment_model
      relationships
      constraints
      measurement_model
      calibration_model
      uncertainty_model
      reasoning_rules
      confidence_model
      update_model

    PURPOSE:
      deterministic_semantic_integrity_identifier

    RECOMPUTE_WHEN:
      ontology_definition_changes
      phase_or_group_index_model_changes
      dispersion_model_changes
      complex_index_model_changes
      anisotropic_model_changes
      environment_model_changes
      measurement_model_changes
      calibration_model_changes
      constraint_model_changes
      reasoning_rule_changes
  }

}

END FLOW