Utilities

Contains

• refraction_angle

• deviation

• brewster_angle

• lens_makers_formula

• mirror_formula

• lens_formula

• hyperfocal_distance

• transverse_magnification

sympy.physics.optics.utils.brewster_angle(medium1, medium2)

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.deviation(incident, medium1, medium2, normal=None, plane=None)

This function calculates the angle of deviation of a ray due to refraction at planar surface.

Parameters

incident : Matrix, Ray3D, or sequence

Incident vector

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.

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)

sympy.physics.optics.utils.hyperfocal_distance(f, N, c)

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
>>> from sympy.abc import f, N, c
>>> 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)

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)

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)

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)

This function calculates transmitted vector after refraction at planar surface. \(medium1\) and \(medium2\) can be \(Medium\) or any sympifiable object.

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. If \(incident\) is an instance of \(Ray3D\) and \(plane\) has not been provided and \(normal\) is not \(Ray3D\), output will be a \(Matrix\).

Parameters

incident : Matrix, Ray3D, or sequence

Incident vector

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.

Examples

>>> from sympy.physics.optics import refraction_angle
>>> from sympy.geometry import Point3D, Ray3D, Plane
>>> from sympy.matrices import Matrix
>>> from sympy import symbols
>>> 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)))

sympy.physics.optics.utils.transverse_magnification(si, so)

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