Gaussian Optics¶
Gaussian optics.
The module implements:
Ray transfer matrices for geometrical and gaussian optics.
See RayTransferMatrix, GeometricRay and BeamParameter
Conjugation relations for geometrical and gaussian optics.
See geometric_conj*, gauss_conj and conjugate_gauss_beams
The conventions for the distances are as follows:
- focal distance
- positive for convergent lenses
- object distance
- positive for real objects
- image distance
- positive for real images
- class sympy.physics.gaussopt.BeamParameter[source]¶
Representation for a gaussian ray in the Ray Transfer Matrix formalism.
Parameters : wavelen : the wavelength,
z : the distance to waist, and
w : the waist, or
z_r : the rayleigh range
See also
References
[R274] http://en.wikipedia.org/wiki/Complex_beam_parameter Examples
>>> from sympy.physics.gaussopt import BeamParameter >>> p = BeamParameter(530e-9, 1, w=1e-3) >>> p.q 1 + 1.88679245283019*I*pi
>>> p.q.n() 1.0 + 5.92753330865999*I >>> p.w_0.n() 0.00100000000000000 >>> p.z_r.n() 5.92753330865999
>>> from sympy.physics.gaussopt import FreeSpace >>> fs = FreeSpace(10) >>> p1 = fs*p >>> p.w.n() 0.00101413072159615 >>> p1.w.n() 0.00210803120913829
- divergence[source]¶
Half of the total angular spread.
Examples
>>> from sympy.physics.gaussopt import BeamParameter >>> p = BeamParameter(530e-9, 1, w=1e-3) >>> p.divergence 0.00053/pi
- gouy[source]¶
The Gouy phase.
Examples
>>> from sympy.physics.gaussopt import BeamParameter >>> p = BeamParameter(530e-9, 1, w=1e-3) >>> p.gouy atan(0.53/pi)
- q[source]¶
The complex parameter representing the beam.
Examples
>>> from sympy.physics.gaussopt import BeamParameter >>> p = BeamParameter(530e-9, 1, w=1e-3) >>> p.q 1 + 1.88679245283019*I*pi
- radius[source]¶
The radius of curvature of the phase front.
Examples
>>> from sympy.physics.gaussopt import BeamParameter >>> p = BeamParameter(530e-9, 1, w=1e-3) >>> p.radius 0.2809/pi**2 + 1
- w[source]¶
The beam radius at \(1/e^2\) intensity.
See also
- w_0
- the minimal radius of beam
Examples
>>> from sympy.physics.gaussopt import BeamParameter >>> p = BeamParameter(530e-9, 1, w=1e-3) >>> p.w 0.001*sqrt(0.2809/pi**2 + 1)
- w_0[source]¶
The beam waist (minimal radius).
See also
- w
- the beam radius at \(1/e^2\) intensity
Examples
>>> from sympy.physics.gaussopt import BeamParameter >>> p = BeamParameter(530e-9, 1, w=1e-3) >>> p.w_0 0.00100000000000000
- waist_approximation_limit[source]¶
The minimal waist for which the gauss beam approximation is valid.
The gauss beam is a solution to the paraxial equation. For curvatures that are too great it is not a valid approximation.
Examples
>>> from sympy.physics.gaussopt import BeamParameter >>> p = BeamParameter(530e-9, 1, w=1e-3) >>> p.waist_approximation_limit 1.06e-6/pi
- class sympy.physics.gaussopt.CurvedMirror[source]¶
Ray Transfer Matrix for reflection from curved surface.
Parameters : R : radius of curvature (positive for concave) See also
Examples
>>> from sympy.physics.gaussopt import CurvedMirror >>> from sympy import symbols >>> R = symbols('R') >>> CurvedMirror(R) Matrix([ [ 1, 0], [-2/R, 1]])
- class sympy.physics.gaussopt.CurvedRefraction[source]¶
Ray Transfer Matrix for refraction on curved interface.
Parameters : R : radius of curvature (positive for concave)
n1 : refractive index of one medium
n2 : refractive index of other medium
See also
Examples
>>> from sympy.physics.gaussopt import CurvedRefraction >>> from sympy import symbols >>> R, n1, n2 = symbols('R n1 n2') >>> CurvedRefraction(R, n1, n2) Matrix([ [ 1, 0], [(n1 - n2)/(R*n2), n1/n2]])
- class sympy.physics.gaussopt.FlatMirror[source]¶
Ray Transfer Matrix for reflection.
See also
Examples
>>> from sympy.physics.gaussopt import FlatMirror >>> FlatMirror() Matrix([ [1, 0], [0, 1]])
- class sympy.physics.gaussopt.FlatRefraction[source]¶
Ray Transfer Matrix for refraction.
Parameters : n1 : refractive index of one medium
n2 : refractive index of other medium
See also
Examples
>>> from sympy.physics.gaussopt import FlatRefraction >>> from sympy import symbols >>> n1, n2 = symbols('n1 n2') >>> FlatRefraction(n1, n2) Matrix([ [1, 0], [0, n1/n2]])
- class sympy.physics.gaussopt.FreeSpace[source]¶
Ray Transfer Matrix for free space.
Parameters : distance : See also
Examples
>>> from sympy.physics.gaussopt import FreeSpace >>> from sympy import symbols >>> d = symbols('d') >>> FreeSpace(d) Matrix([ [1, d], [0, 1]])
- class sympy.physics.gaussopt.GeometricRay[source]¶
Representation for a geometric ray in the Ray Transfer Matrix formalism.
Parameters : h : height, and
angle : angle, or
matrix : a 2x1 matrix (Matrix(2, 1, [height, angle]))
Examples :
======= :
>>> from sympy.physics.gaussopt import GeometricRay, FreeSpace :
>>> from sympy import symbols, Matrix :
>>> d, h, angle = symbols(‘d, h, angle’) :
>>> GeometricRay(h, angle) :
Matrix([ :
[ h], :
[angle]]) :
>>> FreeSpace(d)*GeometricRay(h, angle) :
Matrix([ :
[angle*d + h], :
[ angle]]) :
>>> GeometricRay( Matrix( ((h,), (angle,)) ) ) :
Matrix([ :
[ h], :
[angle]]) :
See also
- class sympy.physics.gaussopt.RayTransferMatrix[source]¶
Base class for a Ray Transfer Matrix.
It should be used if there isn’t already a more specific subclass mentioned in See Also.
Parameters : parameters : A, B, C and D or 2x2 matrix (Matrix(2, 2, [A, B, C, D])) See also
GeometricRay, BeamParameter, FreeSpace, FlatRefraction, CurvedRefraction, FlatMirror, CurvedMirror, ThinLens
References
[R275] http://en.wikipedia.org/wiki/Ray_transfer_matrix_analysis Examples
>>> from sympy.physics.gaussopt import RayTransferMatrix, ThinLens >>> from sympy import Symbol, Matrix
>>> mat = RayTransferMatrix(1, 2, 3, 4) >>> mat Matrix([ [1, 2], [3, 4]])
>>> RayTransferMatrix(Matrix([[1, 2], [3, 4]])) Matrix([ [1, 2], [3, 4]])
>>> mat.A 1
>>> f = Symbol('f') >>> lens = ThinLens(f) >>> lens Matrix([ [ 1, 0], [-1/f, 1]])
>>> lens.C -1/f
- A[source]¶
The A parameter of the Matrix.
Examples
>>> from sympy.physics.gaussopt import RayTransferMatrix >>> mat = RayTransferMatrix(1, 2, 3, 4) >>> mat.A 1
- B[source]¶
The B parameter of the Matrix.
Examples
>>> from sympy.physics.gaussopt import RayTransferMatrix >>> mat = RayTransferMatrix(1, 2, 3, 4) >>> mat.B 2
- class sympy.physics.gaussopt.ThinLens[source]¶
Ray Transfer Matrix for a thin lens.
Parameters : f : the focal distance See also
Examples
>>> from sympy.physics.gaussopt import ThinLens >>> from sympy import symbols >>> f = symbols('f') >>> ThinLens(f) Matrix([ [ 1, 0], [-1/f, 1]])
- sympy.physics.gaussopt.conjugate_gauss_beams(wavelen, waist_in, waist_out, **kwargs)[source]¶
Find the optical setup conjugating the object/image waists.
Parameters : wavelen : the wavelength of the beam
waist_in and waist_out : the waists to be conjugated
f : the focal distance of the element used in the conjugation
Returns : a tuple containing (s_in, s_out, f) :
s_in : the distance before the optical element
s_out : the distance after the optical element
f : the focal distance of the optical element
Examples
>>> from sympy.physics.gaussopt import conjugate_gauss_beams >>> from sympy import symbols, factor >>> l, w_i, w_o, f = symbols('l w_i w_o f')
>>> conjugate_gauss_beams(l, w_i, w_o, f=f)[0] f*(-sqrt(w_i**2/w_o**2 - pi**2*w_i**4/(f**2*l**2)) + 1)
>>> factor(conjugate_gauss_beams(l, w_i, w_o, f=f)[1]) f*w_o**2*(w_i**2/w_o**2 - sqrt(w_i**2/w_o**2 - pi**2*w_i**4/(f**2*l**2)))/w_i**2
>>> conjugate_gauss_beams(l, w_i, w_o, f=f)[2] f
- sympy.physics.gaussopt.gaussian_conj(s_in, z_r_in, f)[source]¶
Conjugation relation for gaussian beams.
Parameters : s_in : the distance to optical element from the waist
z_r_in : the rayleigh range of the incident beam
f : the focal length of the optical element
Returns : a tuple containing (s_out, z_r_out, m) :
s_out : the distance between the new waist and the optical element
z_r_out : the rayleigh range of the emergent beam
m : the ration between the new and the old waists
Examples
>>> from sympy.physics.gaussopt import gaussian_conj >>> from sympy import symbols >>> s_in, z_r_in, f = symbols('s_in z_r_in f')
>>> gaussian_conj(s_in, z_r_in, f)[0] 1/(-1/(s_in + z_r_in**2/(-f + s_in)) + 1/f)
>>> gaussian_conj(s_in, z_r_in, f)[1] z_r_in/(1 - s_in**2/f**2 + z_r_in**2/f**2)
>>> gaussian_conj(s_in, z_r_in, f)[2] 1/sqrt(1 - s_in**2/f**2 + z_r_in**2/f**2)
- sympy.physics.gaussopt.geometric_conj_ab(a, b)[source]¶
Conjugation relation for geometrical beams under paraxial conditions.
Takes the distances to the optical element and returns the needed focal distance.
See also
Examples
>>> from sympy.physics.gaussopt import geometric_conj_ab >>> from sympy import symbols >>> a, b = symbols('a b') >>> geometric_conj_ab(a, b) a*b/(a + b)
- sympy.physics.gaussopt.geometric_conj_af(a, f)[source]¶
Conjugation relation for geometrical beams under paraxial conditions.
Takes the object distance (for geometric_conj_af) or the image distance (for geometric_conj_bf) to the optical element and the focal distance. Then it returns the other distance needed for conjugation.
See also
Examples
>>> from sympy.physics.gaussopt import geometric_conj_af, geometric_conj_bf >>> from sympy import symbols >>> a, b, f = symbols('a b f') >>> geometric_conj_af(a, f) a*f/(a - f) >>> geometric_conj_bf(b, f) b*f/(b - f)
- sympy.physics.gaussopt.geometric_conj_bf(a, f)¶
Conjugation relation for geometrical beams under paraxial conditions.
Takes the object distance (for geometric_conj_af) or the image distance (for geometric_conj_bf) to the optical element and the focal distance. Then it returns the other distance needed for conjugation.
See also
Examples
>>> from sympy.physics.gaussopt import geometric_conj_af, geometric_conj_bf >>> from sympy import symbols >>> a, b, f = symbols('a b f') >>> geometric_conj_af(a, f) a*f/(a - f) >>> geometric_conj_bf(b, f) b*f/(b - f)
- sympy.physics.gaussopt.rayleigh2waist(z_r, wavelen)[source]¶
Calculate the waist from the rayleigh range of a gaussian beam.
See also
Examples
>>> from sympy.physics.gaussopt import rayleigh2waist >>> from sympy import symbols >>> z_r, wavelen = symbols('z_r wavelen') >>> rayleigh2waist(z_r, wavelen) sqrt(wavelen*z_r)/sqrt(pi)
- sympy.physics.gaussopt.waist2rayleigh(w, wavelen)[source]¶
Calculate the rayleigh range from the waist of a gaussian beam.
See also
Examples
>>> from sympy.physics.gaussopt import waist2rayleigh >>> from sympy import symbols >>> w, wavelen = symbols('w wavelen') >>> waist2rayleigh(w, wavelen) pi*w**2/wavelen
