Utilities¶
Contains
refraction_angle
fresnel_coefficients
deviation
brewster_angle
critical_angle
lens_makers_formula
mirror_formula
lens_formula
hyperfocal_distance
transverse_magnification
- sympy.physics.optics.utils.brewster_angle(medium1, medium2)[source]¶
This function calculates the Brewster’s angle of incidence to Medium 2 from Medium 1 in radians.
- Parameters
medium 1 : Medium or sympifiable
Refractive index of Medium 1
medium 2 : Medium or sympifiable
Refractive index of Medium 1
Examples
>>> from sympy.physics.optics import brewster_angle >>> brewster_angle(1, 1.33) 0.926093295503462
- sympy.physics.optics.utils.critical_angle(medium1, medium2)[source]¶
This function calculates the critical angle of incidence (marking the onset of total internal) to Medium 2 from Medium 1 in radians.
- Parameters
medium 1 : Medium or sympifiable
Refractive index of Medium 1.
medium 2 : Medium or sympifiable
Refractive index of Medium 1.
Examples
>>> from sympy.physics.optics import critical_angle >>> critical_angle(1.33, 1) 0.850908514477849
- sympy.physics.optics.utils.deviation(incident, medium1, medium2, normal=None, plane=None)[source]¶
This function calculates the angle of deviation of a ray due to refraction at planar surface.
- Parameters
incident : Matrix, Ray3D, sequence or float
Incident vector or angle of incidence
medium1 : sympy.physics.optics.medium.Medium or sympifiable
Medium 1 or its refractive index
medium2 : sympy.physics.optics.medium.Medium or sympifiable
Medium 2 or its refractive index
normal : Matrix, Ray3D, or sequence
Normal vector
plane : Plane
Plane of separation of the two media.
Returns angular deviation between incident and refracted rays
Examples
>>> from sympy.physics.optics import deviation >>> from sympy.geometry import Point3D, Ray3D, Plane >>> from sympy.matrices import Matrix >>> from sympy import symbols >>> n1, n2 = symbols('n1, n2') >>> n = Matrix([0, 0, 1]) >>> P = Plane(Point3D(0, 0, 0), normal_vector=[0, 0, 1]) >>> r1 = Ray3D(Point3D(-1, -1, 1), Point3D(0, 0, 0)) >>> deviation(r1, 1, 1, n) 0 >>> deviation(r1, n1, n2, plane=P) -acos(-sqrt(-2*n1**2/(3*n2**2) + 1)) + acos(-sqrt(3)/3) >>> round(deviation(0.1, 1.2, 1.5), 5) -0.02005
- sympy.physics.optics.utils.fresnel_coefficients(angle_of_incidence, medium1, medium2)[source]¶
This function uses Fresnel equations to calculate reflection and transmission coefficients. Those are obtained for both polarisations when the electric field vector is in the plane of incidence (labelled ‘p’) and when the electric field vector is perpendicular to the plane of incidence (labelled ‘s’). There are four real coefficients unless the incident ray reflects in total internal in which case there are two complex ones. Angle of incidence is the angle between the incident ray and the surface normal.
medium1
andmedium2
can beMedium
or any sympifiable object.- Parameters
angle_of_incidence : sympifiable
medium1 : Medium or sympifiable
Medium 1 or its refractive index
medium2 : Medium or sympifiable
Medium 2 or its refractive index
- Returns
Returns a list with four real Fresnel coefficients:
[reflection p (TM), reflection s (TE),
transmission p (TM), transmission s (TE)]
If the ray is undergoes total internal reflection then returns a
list of two complex Fresnel coefficients:
[reflection p (TM), reflection s (TE)]
Examples
>>> from sympy.physics.optics import fresnel_coefficients >>> fresnel_coefficients(0.3, 1, 2) [0.317843553417859, -0.348645229818821, 0.658921776708929, 0.651354770181179] >>> fresnel_coefficients(0.6, 2, 1) [-0.235625382192159 - 0.971843958291041*I, 0.816477005968898 - 0.577377951366403*I]
References
- sympy.physics.optics.utils.hyperfocal_distance(f, N, c)[source]¶
- Parameters
f: sympifiable
Focal length of a given lens.
N: sympifiable
F-number of a given lens.
c: sympifiable
Circle of Confusion (CoC) of a given image format.
Example
>>> from sympy.physics.optics import hyperfocal_distance >>> round(hyperfocal_distance(f = 0.5, N = 8, c = 0.0033), 2) 9.47
- sympy.physics.optics.utils.lens_formula(focal_length=None, u=None, v=None)[source]¶
This function provides one of the three parameters when two of them are supplied. This is valid only for paraxial rays.
- Parameters
focal_length : sympifiable
Focal length of the mirror.
u : sympifiable
Distance of object from the optical center on the principal axis.
v : sympifiable
Distance of the image from the optical center on the principal axis.
Examples
>>> from sympy.physics.optics import lens_formula >>> from sympy.abc import f, u, v >>> lens_formula(focal_length=f, u=u) f*u/(f + u) >>> lens_formula(focal_length=f, v=v) f*v/(f - v) >>> lens_formula(u=u, v=v) u*v/(u - v)
- sympy.physics.optics.utils.lens_makers_formula(n_lens, n_surr, r1, r2)[source]¶
This function calculates focal length of a thin lens. It follows cartesian sign convention.
- Parameters
n_lens : Medium or sympifiable
Index of refraction of lens.
n_surr : Medium or sympifiable
Index of reflection of surrounding.
r1 : sympifiable
Radius of curvature of first surface.
r2 : sympifiable
Radius of curvature of second surface.
Examples
>>> from sympy.physics.optics import lens_makers_formula >>> lens_makers_formula(1.33, 1, 10, -10) 15.1515151515151
- sympy.physics.optics.utils.mirror_formula(focal_length=None, u=None, v=None)[source]¶
This function provides one of the three parameters when two of them are supplied. This is valid only for paraxial rays.
- Parameters
focal_length : sympifiable
Focal length of the mirror.
u : sympifiable
Distance of object from the pole on the principal axis.
v : sympifiable
Distance of the image from the pole on the principal axis.
Examples
>>> from sympy.physics.optics import mirror_formula >>> from sympy.abc import f, u, v >>> mirror_formula(focal_length=f, u=u) f*u/(-f + u) >>> mirror_formula(focal_length=f, v=v) f*v/(-f + v) >>> mirror_formula(u=u, v=v) u*v/(u + v)
- sympy.physics.optics.utils.refraction_angle(incident, medium1, medium2, normal=None, plane=None)[source]¶
This function calculates transmitted vector after refraction at planar surface.
medium1
andmedium2
can beMedium
or any sympifiable object. Ifincident
is a number then treated as angle of incidence (in radians) in which case refraction angle is returned.If
incident
is an object of \(Ray3D\), \(normal\) also has to be an instance of \(Ray3D\) in order to get the output as a \(Ray3D\). Please note that if plane of separation is not provided and normal is an instance of \(Ray3D\),normal
will be assumed to be intersecting incident ray at the plane of separation. This will not be the case when \(normal\) is a \(Matrix\) or any other sequence. Ifincident
is an instance of \(Ray3D\) and \(plane\) has not been provided andnormal
is not \(Ray3D\), output will be a \(Matrix\).- Parameters
incident : Matrix, Ray3D, sequence or a number
Incident vector or angle of incidence
medium1 : sympy.physics.optics.medium.Medium or sympifiable
Medium 1 or its refractive index
medium2 : sympy.physics.optics.medium.Medium or sympifiable
Medium 2 or its refractive index
normal : Matrix, Ray3D, or sequence
Normal vector
plane : Plane
Plane of separation of the two media.
- Returns
Returns an angle of refraction or a refracted ray depending on inputs.
Examples
>>> from sympy.physics.optics import refraction_angle >>> from sympy.geometry import Point3D, Ray3D, Plane >>> from sympy.matrices import Matrix >>> from sympy import symbols, pi >>> n = Matrix([0, 0, 1]) >>> P = Plane(Point3D(0, 0, 0), normal_vector=[0, 0, 1]) >>> r1 = Ray3D(Point3D(-1, -1, 1), Point3D(0, 0, 0)) >>> refraction_angle(r1, 1, 1, n) Matrix([ [ 1], [ 1], [-1]]) >>> refraction_angle(r1, 1, 1, plane=P) Ray3D(Point3D(0, 0, 0), Point3D(1, 1, -1))
With different index of refraction of the two media
>>> n1, n2 = symbols('n1, n2') >>> refraction_angle(r1, n1, n2, n) Matrix([ [ n1/n2], [ n1/n2], [-sqrt(3)*sqrt(-2*n1**2/(3*n2**2) + 1)]]) >>> refraction_angle(r1, n1, n2, plane=P) Ray3D(Point3D(0, 0, 0), Point3D(n1/n2, n1/n2, -sqrt(3)*sqrt(-2*n1**2/(3*n2**2) + 1))) >>> round(refraction_angle(pi/6, 1.2, 1.5), 5) 0.41152
- sympy.physics.optics.utils.transverse_magnification(si, so)[source]¶
Calculates the transverse magnification, which is the ratio of the image size to the object size.
- Parameters
so: sympifiable
Lens-object distance.
si: sympifiable
Lens-image distance.
Example
>>> from sympy.physics.optics import transverse_magnification >>> transverse_magnification(30, 15) -2