# 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.refraction_angle(incident, medium1, medium2, normal=None, plane=None)[source]

This function calculates transmitted vector after refraction at planar surface. $$medium1$$ and $$medium2$$ can be $$Medium$$ or any sympifiable object. If $$incident$$ 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. 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, 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 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.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 and medium2 can be Medium 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 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

https://en.wikipedia.org/wiki/Fresnel_equations

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.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.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.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
>>> 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.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