U -eQ@sjdZddddddddd d d d d dddgZddlmZddlmZmZddlmZddl m Z m Z ddl m Z ddlmZddlmZmZddlmZddlmZGdddeZGdddeZGdddeZGdddeZGdddeZGd ddeZGd!ddeZGd"ddeZGd#d d eZd,d%d Z d&d Z!d'd Z"d(d Z#e#Z$d)dZ%d*dZ&d+S)-a 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 RayTransferMatrix FreeSpaceFlatRefractionCurvedRefraction FlatMirror CurvedMirrorThinLens GeometricRay BeamParameterwaist2rayleighrayleigh2waistgeometric_conj_abgeometric_conj_afgeometric_conj_bf gaussian_conjconjugate_gauss_beams)Expr)Ipi)sympify)imre)sqrt)atan2)MatrixMutableDenseMatrix)together) filldedentc@sPeZdZdZddZddZeddZedd Zed d Z ed d Z dS)ra Base class for a Ray Transfer Matrix. It should be used if there is not 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])) Examples ======== >>> from sympy.physics.optics 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 See Also ======== GeometricRay, BeamParameter, FreeSpace, FlatRefraction, CurvedRefraction, FlatMirror, CurvedMirror, ThinLens References ========== .. [1] https://en.wikipedia.org/wiki/Ray_transfer_matrix_analysis cGst|dkr.|d|df|d|dff}nFt|dkr`t|dtr`|djdkr`|d}nttdt|t||S)Nr)r r z` Expecting 2x2 Matrix or the 4 elements of the Matrix but got %slen isinstancershape ValueErrorrstr__new__clsargstempr-^/var/www/html/Darija-Ai-Train/env/lib/python3.8/site-packages/sympy/physics/optics/gaussopt.pyr(ss "    zRayTransferMatrix.__new__cCst|trtt||St|tr4tt||St|tr|t|jfdf}|d|djdd}t|jt t |t t |dSt||SdS)N)rrrT)complex)z_r) r$rr__mul__rr qexpandwavelenrrr)selfotherr,r2r-r-r.r1s     zRayTransferMatrix.__mul__cCs|dS)z The A parameter of the Matrix. Examples ======== >>> from sympy.physics.optics import RayTransferMatrix >>> mat = RayTransferMatrix(1, 2, 3, 4) >>> mat.A 1 )rrr-r5r-r-r.As zRayTransferMatrix.AcCs|dS)z The B parameter of the Matrix. Examples ======== >>> from sympy.physics.optics import RayTransferMatrix >>> mat = RayTransferMatrix(1, 2, 3, 4) >>> mat.B 2 )rrr-r7r-r-r.Bs zRayTransferMatrix.BcCs|dS)z The C parameter of the Matrix. Examples ======== >>> from sympy.physics.optics import RayTransferMatrix >>> mat = RayTransferMatrix(1, 2, 3, 4) >>> mat.C 3 )rrr-r7r-r-r.Cs zRayTransferMatrix.CcCs|dS)z The D parameter of the Matrix. Examples ======== >>> from sympy.physics.optics import RayTransferMatrix >>> mat = RayTransferMatrix(1, 2, 3, 4) >>> mat.D 4 )rrr-r7r-r-r.Ds zRayTransferMatrix.DN) __name__ __module__ __qualname____doc__r(r1propertyr8r9r:r;r-r-r-r.r;s7   c@seZdZdZddZdS)raQ Ray Transfer Matrix for free space. Parameters ========== distance See Also ======== RayTransferMatrix Examples ======== >>> from sympy.physics.optics import FreeSpace >>> from sympy import symbols >>> d = symbols('d') >>> FreeSpace(d) Matrix([ [1, d], [0, 1]]) cCst|d|ddSNrrrr()r*dr-r-r.r(szFreeSpace.__new__Nr<r=r>r?r(r-r-r-r.rsc@seZdZdZddZdS)ra Ray Transfer Matrix for refraction. Parameters ========== n1 : Refractive index of one medium. n2 : Refractive index of other medium. See Also ======== RayTransferMatrix Examples ======== >>> from sympy.physics.optics import FlatRefraction >>> from sympy import symbols >>> n1, n2 = symbols('n1 n2') >>> FlatRefraction(n1, n2) Matrix([ [1, 0], [0, n1/n2]]) cCs(tt||f\}}t|ddd||SrAmaprrr()r*n1n2r-r-r.r(szFlatRefraction.__new__NrDr-r-r-r.rsc@seZdZdZddZdS)raB 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 ======== RayTransferMatrix Examples ======== >>> from sympy.physics.optics 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]]) cCs8tt|||f\}}}t|dd||||||SrArE)r*RrGrHr-r-r.r((szCurvedRefraction.__new__NrDr-r-r-r.r sc@seZdZdZddZdS)rz Ray Transfer Matrix for reflection. See Also ======== RayTransferMatrix Examples ======== >>> from sympy.physics.optics import FlatMirror >>> FlatMirror() Matrix([ [1, 0], [0, 1]]) cCst|ddddSrArB)r*r-r-r.r(?szFlatMirror.__new__NrDr-r-r-r.r-sc@seZdZdZddZdS)ra Ray Transfer Matrix for reflection from curved surface. Parameters ========== R : radius of curvature (positive for concave) See Also ======== RayTransferMatrix Examples ======== >>> from sympy.physics.optics import CurvedMirror >>> from sympy import symbols >>> R = symbols('R') >>> CurvedMirror(R) Matrix([ [ 1, 0], [-2/R, 1]]) cCst|}t|ddd|dS)Nrrrrr()r*rIr-r-r.r(\szCurvedMirror.__new__NrDr-r-r-r.rCsc@seZdZdZddZdS)ram Ray Transfer Matrix for a thin lens. Parameters ========== f : The focal distance. See Also ======== RayTransferMatrix Examples ======== >>> from sympy.physics.optics import ThinLens >>> from sympy import symbols >>> f = symbols('f') >>> ThinLens(f) Matrix([ [ 1, 0], [-1/f, 1]]) cCst|}t|ddd|dS)NrrrK)r*fr-r-r.r({szThinLens.__new__NrDr-r-r-r.rasc@s0eZdZdZddZeddZeddZdS) ra 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.optics 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 ======== RayTransferMatrix cGstt|dkr2t|dtr2|djdkr2|d}n6t|dkrT|df|dff}nttdt|t||S)Nrr)r rr z` Expecting 2x1 Matrix or the 2 elements of the Matrix but got %sr"r)r-r-r.r(s   zGeometricRay.__new__cCs|dS)a0 The distance from the optical axis. Examples ======== >>> from sympy.physics.optics import GeometricRay >>> from sympy import symbols >>> h, angle = symbols('h, angle') >>> gRay = GeometricRay(h, angle) >>> gRay.height h rr-r7r-r-r.heightszGeometricRay.heightcCs|dS)a0 The angle with the optical axis. Examples ======== >>> from sympy.physics.optics import GeometricRay >>> from sympy import symbols >>> h, angle = symbols('h, angle') >>> gRay = GeometricRay(h, angle) >>> gRay.angle angle rr-r7r-r-r.angleszGeometricRay.angleN)r<r=r>r?r(r@rNrOr-r-r-r.rs '  c@seZdZdZdddZeddZedd Zed d Zed d Z eddZ eddZ eddZ eddZ eddZeddZeddZdS)r a 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. n : the refractive index of medium. Examples ======== >>> from sympy.physics.optics 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.optics import FreeSpace >>> fs = FreeSpace(10) >>> p1 = fs*p >>> p.w.n() 0.00101413072159615 >>> p1.w.n() 0.00210803120913829 See Also ======== RayTransferMatrix References ========== .. [1] https://en.wikipedia.org/wiki/Complex_beam_parameter .. [2] https://en.wikipedia.org/wiki/Gaussian_beam NrcCs~t|}t|}t|}|dk r2|dkr2t|}n:|dk rT|dkrTtt|||}n|dkrl|dkrltdt|||||S)NzMust specify one of w and z_r.)rr r&rr()r*r4zr0wnr-r-r.r(s zBeamParameter.__new__cCs |jdS)Nrr+r7r-r-r.r4 szBeamParameter.wavelencCs |jdS)NrrSr7r-r-r.rP$szBeamParameter.zcCs |jdS)Nr rSr7r-r-r.r0(szBeamParameter.z_rcCs |jdS)Nr!rSr7r-r-r.rR,szBeamParameter.ncCs|jt|jS)a The complex parameter representing the beam. Examples ======== >>> from sympy.physics.optics import BeamParameter >>> p = BeamParameter(530e-9, 1, w=1e-3) >>> p.q 1 + 1.88679245283019*I*pi )rPrr0r7r-r-r.r20s zBeamParameter.qcCs|jd|j|jdS)a The radius of curvature of the phase front. Examples ======== >>> from sympy.physics.optics import BeamParameter >>> p = BeamParameter(530e-9, 1, w=1e-3) >>> p.radius 1 + 3.55998576005696*pi**2 rr )rPr0r7r-r-r.radius?s zBeamParameter.radiuscCs|jtd|j|jdS)a The radius of the beam w(z), at any position z along the beam. The beam radius at `1/e^2` intensity (axial value). See Also ======== w_0 : The minimal radius of beam. Examples ======== >>> from sympy.physics.optics import BeamParameter >>> p = BeamParameter(530e-9, 1, w=1e-3) >>> p.w 0.001*sqrt(0.2809/pi**2 + 1) rr )w_0rrPr0r7r-r-r.rQNszBeamParameter.wcCst|jt|j|jS)ar The minimal radius of beam at `1/e^2` intensity (peak value). See Also ======== w : the beam radius at `1/e^2` intensity (axial value). Examples ======== >>> from sympy.physics.optics import BeamParameter >>> p = BeamParameter(530e-9, 1, w=1e-3) >>> p.w_0 0.00100000000000000 )rr0rrRr4r7r-r-r.rUdszBeamParameter.w_0cCs|jt|jS)z Half of the total angular spread. Examples ======== >>> from sympy.physics.optics import BeamParameter >>> p = BeamParameter(530e-9, 1, w=1e-3) >>> p.divergence 0.00053/pi )r4rrUr7r-r-r. divergencexs zBeamParameter.divergencecCst|j|jS)z The Gouy phase. Examples ======== >>> from sympy.physics.optics import BeamParameter >>> p = BeamParameter(530e-9, 1, w=1e-3) >>> p.gouy atan(0.53/pi) )rrPr0r7r-r-r.gouys zBeamParameter.gouycCsd|jtS)a The minimal waist for which the gauss beam approximation is valid. Explanation =========== 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.optics import BeamParameter >>> p = BeamParameter(530e-9, 1, w=1e-3) >>> p.waist_approximation_limit 1.06e-6/pi r )r4rr7r-r-r.waist_approximation_limitsz'BeamParameter.waist_approximation_limit)NNr)r<r=r>r?r(r@r4rPr0rRr2rTrQrUrVrWrXr-r-r-r.r s02           rcCs&tt||f\}}|d|t|S)a^ Calculate the rayleigh range from the waist of a gaussian beam. See Also ======== rayleigh2waist, BeamParameter Examples ======== >>> from sympy.physics.optics import waist2rayleigh >>> from sympy import symbols >>> w, wavelen = symbols('w wavelen') >>> waist2rayleigh(w, wavelen) pi*w**2/wavelen r )rFrr)rQr4rRr-r-r.r scCs"tt||f\}}t|t|S)ajCalculate the waist from the rayleigh range of a gaussian beam. See Also ======== waist2rayleigh, BeamParameter Examples ======== >>> from sympy.physics.optics import rayleigh2waist >>> from sympy import symbols >>> z_r, wavelen = symbols('z_r wavelen') >>> rayleigh2waist(z_r, wavelen) sqrt(wavelen*z_r)/sqrt(pi) )rFrrr)r0r4r-r-r.r scCs@tt||f\}}|js|jr,|jr(|S|S||||SdS)a Conjugation relation for geometrical beams under paraxial conditions. Explanation =========== Takes the distances to the optical element and returns the needed focal distance. See Also ======== geometric_conj_af, geometric_conj_bf Examples ======== >>> from sympy.physics.optics import geometric_conj_ab >>> from sympy import symbols >>> a, b = symbols('a b') >>> geometric_conj_ab(a, b) a*b/(a + b) N)rFr is_infinite)abr-r-r.r s cCs tt||f\}}t||  S)a Conjugation relation for geometrical beams under paraxial conditions. Explanation =========== 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 ======== geometric_conj_ab Examples ======== >>> from sympy.physics.optics.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) )rFrr )rZrMr-r-r.r scCstt|||f\}}}dd||d||d|}dtd||d||d}|d||d||d}|||fS)a 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.optics 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) rrLr )rFrr)s_inZz_r_inrMs_outmZz_r_outr-r-r.rs )$$ c Kstt|||f\}}}||}t||}t|dkr>tdnd|krTttdnpd|krt|d}|dtd|d|d|d}t|||d}n"d|krttdn ttd |||fS) a 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.optics 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*(1 - sqrt(w_i**2/w_o**2 - pi**2*w_i**4/(f**2*l**2))) >>> 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 rz,The function expects only one named argumentdistzD Currently only focal length is supported as a parameterrMr rr\zG The functions expects the focal length as a named argument) rFrr r#r&NotImplementedErrorrrr) r4Zwaist_inZ waist_outkwargsr^rPrMr\r]r-r-r.rKs+    ( N)r)'r?__all__Zsympy.core.exprrZsympy.core.numbersrrZsympy.core.sympifyrZ$sympy.functions.elementary.complexesrrZ(sympy.functions.elementary.miscellaneousrZ(sympy.functions.elementary.trigonometricrZsympy.matrices.denserrZsympy.polys.rationaltoolsrZsympy.utilities.miscrrrrrrrrrr r r r r rrrr-r-r-r.sX      !##[R 0